# Rational groups

A group G is rational if all of its characters are rational valued. (Equivalently every g and gk are conjugate for k coprime to |G|.) It is an unsolved problem to determine what cyclic groups of order p can be composition factors of rational groups.

Rational groups are sometimes also known as Q-groups, though some authors reserve that term for a stronger property that every representation of G can be realized over ℚ. (According to this terminology, Q8 is rational but not a Q-group.)

### Groups of order 1

dρLabelID
C1Trivial group11+C11,1

### Groups of order 2

dρLabelID
C2Cyclic group21+C22,1

### Groups of order 4

dρLabelID
C22Klein 4-group V4 = elementary abelian group of type [2,2]; = rectangle symmetries4C2^24,2

### Groups of order 6

dρLabelID
S3Symmetric group on 3 letters; = D3 = GL2(𝔽2) = triangle symmetries = 1st non-abelian group32+S36,1

### Groups of order 8

dρLabelID
D4Dihedral group; = He2 = AΣL1(𝔽4) = 2+ 1+2 = square symmetries42+D48,3
Q8Quaternion group; = C4.C2 = Dic2 = 2- 1+282-Q88,4
C23Elementary abelian group of type [2,2,2]8C2^38,5

### Groups of order 12

dρLabelID
D6Dihedral group; = C2×S3 = hexagon symmetries62+D612,4

### Groups of order 16

dρLabelID
C24Elementary abelian group of type [2,2,2,2]16C2^416,14
C2×D4Direct product of C2 and D48C2xD416,11
C2×Q8Direct product of C2 and Q816C2xQ816,12

### Groups of order 18

dρLabelID
C3⋊S3The semidirect product of C3 and S3 acting via S3/C3=C29C3:S318,4

### Groups of order 24

dρLabelID
S4Symmetric group on 4 letters; = PGL2(𝔽3) = Aut(Q8) = Hol(C22) = tetrahedron symmetries = cube/octahedron rotations43+S424,12
C22×S3Direct product of C22 and S312C2^2xS324,14

### Groups of order 32

dρLabelID
2+ 1+4Extraspecial group; = D4D484+ES+(2,2)32,49
2- 1+4Gamma matrices = Extraspecial group; = D4Q8164-ES-(2,2)32,50
C22≀C2Wreath product of C22 by C28C2^2wrC232,27
C8⋊C22The semidirect product of C8 and C22 acting faithfully; = Aut(D8) = Hol(C8)84+C8:C2^232,43
C41D4The semidirect product of C4 and D4 acting via D4/C4=C216C4:1D432,34
C4⋊Q8The semidirect product of C4 and Q8 acting via Q8/C4=C232C4:Q832,35
C8.C22The non-split extension by C8 of C22 acting faithfully164-C8.C2^232,44
C25Elementary abelian group of type [2,2,2,2,2]32C2^532,51
C22×D4Direct product of C22 and D416C2^2xD432,46
C22×Q8Direct product of C22 and Q832C2^2xQ832,47

### Groups of order 36

dρLabelID
S32Direct product of S3 and S3; = Spin+4(𝔽2) = Hol(S3)64+S3^236,10
C2×C3⋊S3Direct product of C2 and C3⋊S318C2xC3:S336,13

### Groups of order 48

dρLabelID
C2×S4Direct product of C2 and S4; = O3(𝔽3) = cube/octahedron symmetries63+C2xS448,48
S3×D4Direct product of S3 and D4; = Aut(D12) = Hol(C12)124+S3xD448,38
S3×Q8Direct product of S3 and Q8244-S3xQ848,40
S3×C23Direct product of C23 and S324S3xC2^348,51

### Groups of order 54

dρLabelID
C33⋊C23rd semidirect product of C33 and C2 acting faithfully27C3^3:C254,14

### Groups of order 64

dρLabelID
C2≀C22Wreath product of C2 by C22; = Hol(C2×C4)84+C2wrC2^264,138
D44D43rd semidirect product of D4 and D4 acting via D4/C22=C2; = Hol(D4)84+D4:4D464,134
C233D42nd semidirect product of C23 and D4 acting via D4/C2=C2216C2^3:3D464,215
C232Q82nd semidirect product of C23 and Q8 acting via Q8/C2=C2216C2^3:2Q864,224
C24⋊C224th semidirect product of C24 and C22 acting faithfully16C2^4:C2^264,242
C83D43rd semidirect product of C8 and D4 acting via D4/C2=C2232C8:3D464,177
C8⋊Q8The semidirect product of C8 and Q8 acting via Q8/C2=C2264C8:Q864,182
D4.10D45th non-split extension by D4 of D4 acting via D4/C22=C2164-D4.10D464,137
C22.29C2415th central stem extension by C22 of C2416C2^2.29C2^464,216
C22.54C2440th central stem extension by C22 of C2416C2^2.54C2^464,241
C8.2D42nd non-split extension by C8 of D4 acting via D4/C2=C2232C8.2D464,178
C23.38C2311st non-split extension by C23 of C23 acting via C23/C22=C232C2^3.38C2^364,217
C22.31C2417th central stem extension by C22 of C2432C2^2.31C2^464,218
C23.41C2314th non-split extension by C23 of C23 acting via C23/C22=C232C2^3.41C2^364,225
C22.56C2442nd central stem extension by C22 of C2432C2^2.56C2^464,243
C22.57C2443rd central stem extension by C22 of C2432C2^2.57C2^464,244
C22.58C2444th central stem extension by C22 of C2464C2^2.58C2^464,245
C26Elementary abelian group of type [2,2,2,2,2,2]64C2^664,267
D42Direct product of D4 and D416D4^264,226
C2×2+ 1+4Direct product of C2 and 2+ 1+416C2xES+(2,2)64,264
D4×Q8Direct product of D4 and Q832D4xQ864,230
D4×C23Direct product of C23 and D432D4xC2^364,261
C2×2- 1+4Direct product of C2 and 2- 1+432C2xES-(2,2)64,265
Q82Direct product of Q8 and Q864Q8^264,239
Q8×C23Direct product of C23 and Q864Q8xC2^364,262
C2×C8⋊C22Direct product of C2 and C8⋊C2216C2xC8:C2^264,254
C2×C22≀C2Direct product of C2 and C22≀C216C2xC2^2wrC264,202
C2×C41D4Direct product of C2 and C41D432C2xC4:1D464,211
C2×C8.C22Direct product of C2 and C8.C2232C2xC8.C2^264,255
C2×C4⋊Q8Direct product of C2 and C4⋊Q864C2xC4:Q864,212

### Groups of order 72

dρLabelID
PSU3(𝔽2)Projective special unitary group on 𝔽23; = C32Q8 = M998+PSU(3,2)72,41
S3≀C2Wreath product of S3 by C2; = SO+4(𝔽2)64+S3wrC272,40
C3⋊S4The semidirect product of C3 and S4 acting via S4/A4=C2126+C3:S472,43
C2×S32Direct product of C2, S3 and S3124+C2xS3^272,46
C22×C3⋊S3Direct product of C22 and C3⋊S336C2^2xC3:S372,49

### Groups of order 96

dρLabelID
C22⋊S4The semidirect product of C22 and S4 acting via S4/C22=S386+C2^2:S496,227
C22×S4Direct product of C22 and S412C2^2xS496,226
S3×C24Direct product of C24 and S348S3xC2^496,230
C2×S3×D4Direct product of C2, S3 and D424C2xS3xD496,209
C2×S3×Q8Direct product of C2, S3 and Q848C2xS3xQ896,212

### Groups of order 108

dρLabelID
C32⋊D6The semidirect product of C32 and D6 acting faithfully96+C3^2:D6108,17
S3×C3⋊S3Direct product of S3 and C3⋊S318S3xC3:S3108,39
C2×C33⋊C2Direct product of C2 and C33⋊C254C2xC3^3:C2108,44

### Groups of order 120

dρLabelID
S5Symmetric group on 5 letters; = PGL2(𝔽5) = Aut(A5) = 5-cell symmetries; almost simple54+S5120,34

### Groups of order 128

dρLabelID
2+ 1+6Extraspecial group; = D42+ 1+4168+ES+(2,3)128,2326
2- 1+6Extraspecial group; = D42- 1+4328-ES-(2,3)128,2327
D4≀C2Wreath product of D4 by C284+D4wrC2128,928
Q8≀C2Wreath product of Q8 by C2164-Q8wrC2128,937
C23≀C2Wreath product of C23 by C216C2^3wrC2128,1578
D811D45th semidirect product of D8 and D4 acting via D4/C22=C2168+D8:11D4128,2020
C425D45th semidirect product of C42 and D4 acting faithfully168+C4^2:5D4128,931
C426D46th semidirect product of C42 and D4 acting faithfully168+C4^2:6D4128,932
D8⋊C236th semidirect product of D8 and C23 acting via C23/C22=C2168+D8:C2^3128,2317
C24⋊C232nd semidirect product of C24 and C23 acting faithfully168+C2^4:C2^3128,1758
C42⋊C234th semidirect product of C42 and C23 acting faithfully16C4^2:C2^3128,2264
M4(2)⋊C233rd semidirect product of M4(2) and C23 acting via C23/C2=C22168+M4(2):C2^3128,1751
C247D42nd semidirect product of C24 and D4 acting via D4/C2=C2232C2^4:7D4128,1135
C249D44th semidirect product of C24 and D4 acting via D4/C2=C2232C2^4:9D4128,1345
C246Q85th semidirect product of C24 and Q8 acting via Q8/C2=C2232C2^4:6Q8128,1572
C2411D46th semidirect product of C24 and D4 acting via D4/C2=C2232C2^4:11D4128,1544
M4(2)⋊7D41st semidirect product of M4(2) and D4 acting via D4/C4=C232M4(2):7D4128,1883
C25⋊C222nd semidirect product of C25 and C22 acting faithfully32C2^5:C2^2128,1411
C4315C215th semidirect product of C43 and C2 acting faithfully64C4^3:15C2128,1599
C4231D425th semidirect product of C42 and D4 acting via D4/C2=C2264C4^2:31D4128,1389
C4233D427th semidirect product of C42 and D4 acting via D4/C2=C2264C4^2:33D4128,1550
C4234D428th semidirect product of C42 and D4 acting via D4/C2=C2264C4^2:34D4128,1551
C4235D429th semidirect product of C42 and D4 acting via D4/C2=C2264C4^2:35D4128,1555
M4(2)⋊8D42nd semidirect product of M4(2) and D4 acting via D4/C4=C264M4(2):8D4128,1884
M4(2)⋊5Q83rd semidirect product of M4(2) and Q8 acting via Q8/C4=C264M4(2):5Q8128,1897
C427Q87th semidirect product of C42 and Q8 acting via Q8/C2=C22128C4^2:7Q8128,1283
C4210Q810th semidirect product of C42 and Q8 acting via Q8/C2=C22128C4^2:10Q8128,1392
C4212Q812nd semidirect product of C42 and Q8 acting via Q8/C2=C22128C4^2:12Q8128,1575
C4213Q813rd semidirect product of C42 and Q8 acting via Q8/C2=C22128C4^2:13Q8128,1576
C4219Q86th semidirect product of C42 and Q8 acting via Q8/C4=C2128C4^2:19Q8128,1600
C42.15D415th non-split extension by C42 of D4 acting faithfully168+C4^2.15D4128,934
C24.177D432nd non-split extension by C24 of D4 acting via D4/C22=C216C2^4.177D4128,1735
C23.9C249th non-split extension by C23 of C24 acting via C24/C22=C22168+C2^3.9C2^4128,1759
C42.12C2312nd non-split extension by C42 of C23 acting faithfully168+C4^2.12C2^3128,1753
C22.73C2554th central stem extension by C22 of C2516C2^2.73C2^5128,2216
D8.13D45th non-split extension by D8 of D4 acting via D4/C22=C2328-D8.13D4128,2021
C4.C2513rd non-split extension by C4 of C25 acting via C25/C24=C2328-C4.C2^5128,2318
C42.14D414th non-split extension by C42 of D4 acting faithfully328-C4^2.14D4128,933
C42.16D416th non-split extension by C42 of D4 acting faithfully328-C4^2.16D4128,935
C24.15Q814th non-split extension by C24 of Q8 acting via Q8/C2=C2232C2^4.15Q8128,1574
C24.178D433rd non-split extension by C24 of D4 acting via D4/C22=C232C2^4.178D4128,1736
C24.125D480th non-split extension by C24 of D4 acting via D4/C2=C2232C2^4.125D4128,1924
C24.126D481st non-split extension by C24 of D4 acting via D4/C2=C2232C2^4.126D4128,1925
C24.128D483rd non-split extension by C24 of D4 acting via D4/C2=C2232C2^4.128D4128,1927
C42.275D4257th non-split extension by C42 of D4 acting via D4/C2=C2232C4^2.275D4128,1949
C42.C231st non-split extension by C42 of C23 acting faithfully32C4^2.C2^3128,387
C42.13C2313rd non-split extension by C42 of C23 acting faithfully328-C4^2.13C2^3128,1754
C23.10C2410th non-split extension by C23 of C24 acting via C24/C22=C22328-C2^3.10C2^4128,1760
C22.75C2556th central stem extension by C22 of C2532C2^2.75C2^5128,2218
C22.87C2568th central stem extension by C22 of C2532C2^2.87C2^5128,2230
C23.333C2450th central stem extension by C23 of C2432C2^3.333C2^4128,1165
C23.569C24286th central stem extension by C23 of C2432C2^3.569C2^4128,1401
C22.125C25106th central stem extension by C22 of C2532C2^2.125C2^5128,2268
C22.126C25107th central stem extension by C22 of C2532C2^2.126C2^5128,2269
C22.127C25108th central stem extension by C22 of C2532C2^2.127C2^5128,2270
C22.132C25113rd central stem extension by C22 of C2532C2^2.132C2^5128,2275
C22.138C25119th central stem extension by C22 of C2532C2^2.138C2^5128,2281
M4(2).C234th non-split extension by M4(2) of C23 acting via C23/C2=C22328-M4(2).C2^3128,1752
C42.196D4178th non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.196D4128,1390
C42.199D4181st non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.199D4128,1552
C42.200D4182nd non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.200D4128,1553
C42.276D4258th non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.276D4128,1950
C42.290D4272nd non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.290D4128,1970
C42.291D4273rd non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.291D4128,1971
C42.301D4283rd non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.301D4128,1985
C42.302D4284th non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.302D4128,1986
C42.303D4285th non-split extension by C42 of D4 acting via D4/C2=C2264C4^2.303D4128,1987
C42.6C236th non-split extension by C42 of C23 acting faithfully64C4^2.6C2^3128,392
C22.88C2569th central stem extension by C22 of C2564C2^2.88C2^5128,2231
C22.92C2573rd central stem extension by C22 of C2564C2^2.92C2^5128,2235
C23.334C2451st central stem extension by C23 of C2464C2^3.334C2^4128,1166
C23.514C24231st central stem extension by C23 of C2464C2^3.514C2^4128,1346
C24.360C23200th non-split extension by C24 of C23 acting via C23/C2=C2264C2^4.360C2^3128,1347
C24.361C23201st non-split extension by C24 of C23 acting via C23/C2=C2264C2^4.361C2^3128,1348
C23.556C24273rd central stem extension by C23 of C2464C2^3.556C2^4128,1388
C23.559C24276th central stem extension by C23 of C2464C2^3.559C2^4128,1391
C24.384C23224th non-split extension by C24 of C23 acting via C23/C2=C2264C2^4.384C2^3128,1407
C24.385C23225th non-split extension by C24 of C23 acting via C23/C2=C2264C2^4.385C2^3128,1409
C23.583C24300th central stem extension by C23 of C2464C2^3.583C2^4128,1415
C24.459C23299th non-split extension by C24 of C23 acting via C23/C2=C2264C2^4.459C2^3128,1545
C23.714C24431st central stem extension by C23 of C2464C2^3.714C2^4128,1546
C23.715C24432nd central stem extension by C23 of C2464C2^3.715C2^4128,1547
C23.716C24433rd central stem extension by C23 of C2464C2^3.716C2^4128,1548
C24.462C23302nd non-split extension by C24 of C23 acting via C23/C2=C2264C2^4.462C2^3128,1549
C23.741C24458th central stem extension by C23 of C2464C2^3.741C2^4128,1573
C22.133C25114th central stem extension by C22 of C2564C2^2.133C2^5128,2276
C22.137C25118th central stem extension by C22 of C2564C2^2.137C2^5128,2280
C22.139C25120th central stem extension by C22 of C2564C2^2.139C2^5128,2282
C22.141C25122nd central stem extension by C22 of C2564C2^2.141C2^5128,2284
C22.145C25126th central stem extension by C22 of C2564C2^2.145C2^5128,2288
C42.40Q840th non-split extension by C42 of Q8 acting via Q8/C2=C22128C4^2.40Q8128,1577
C42.201D4183rd non-split extension by C42 of D4 acting via D4/C2=C22128C4^2.201D4128,1554
C23.634C24351st central stem extension by C23 of C24128C2^3.634C2^4128,1466
C23.711C24428th central stem extension by C23 of C24128C2^3.711C2^4128,1543
C27Elementary abelian group of type [2,2,2,2,2,2,2]128C2^7128,2328
C22×2+ 1+4Direct product of C22 and 2+ 1+432C2^2xES+(2,2)128,2323
D4×C24Direct product of C24 and D464D4xC2^4128,2320
C22×2- 1+4Direct product of C22 and 2- 1+464C2^2xES-(2,2)128,2324
Q8×C24Direct product of C24 and Q8128Q8xC2^4128,2321
C2×C2≀C22Direct product of C2 and C2≀C2216C2xC2wrC2^2128,1755
C2×D44D4Direct product of C2 and D44D416C2xD4:4D4128,1746
C2×D42Direct product of C2, D4 and D432C2xD4^2128,2194
C2×C233D4Direct product of C2 and C233D432C2xC2^3:3D4128,2177
C22×C8⋊C22Direct product of C22 and C8⋊C2232C2^2xC8:C2^2128,2310
C2×C232Q8Direct product of C2 and C232Q832C2xC2^3:2Q8128,2188
C2×D4.10D4Direct product of C2 and D4.10D432C2xD4.10D4128,1749
C22×C22≀C2Direct product of C22 and C22≀C232C2^2xC2^2wrC2128,2163
C2×C24⋊C22Direct product of C2 and C24⋊C2232C2xC2^4:C2^2128,2258
C2×C22.29C24Direct product of C2 and C22.29C2432C2xC2^2.29C2^4128,2178
C2×C22.54C24Direct product of C2 and C22.54C2432C2xC2^2.54C2^4128,2257
C2×D4×Q8Direct product of C2, D4 and Q864C2xD4xQ8128,2198
C2×C83D4Direct product of C2 and C83D464C2xC8:3D4128,1880
C2×C8.2D4Direct product of C2 and C8.2D464C2xC8.2D4128,1881
C22×C41D4Direct product of C22 and C41D464C2^2xC4:1D4128,2172
C22×C8.C22Direct product of C22 and C8.C2264C2^2xC8.C2^2128,2311
C2×C23.38C23Direct product of C2 and C23.38C2364C2xC2^3.38C2^3128,2179
C2×C22.31C24Direct product of C2 and C22.31C2464C2xC2^2.31C2^4128,2180
C2×C23.41C23Direct product of C2 and C23.41C2364C2xC2^3.41C2^3128,2189
C2×C22.56C24Direct product of C2 and C22.56C2464C2xC2^2.56C2^4128,2259
C2×C22.57C24Direct product of C2 and C22.57C2464C2xC2^2.57C2^4128,2260
C2×Q82Direct product of C2, Q8 and Q8128C2xQ8^2128,2209
C2×C8⋊Q8Direct product of C2 and C8⋊Q8128C2xC8:Q8128,1893
C22×C4⋊Q8Direct product of C22 and C4⋊Q8128C2^2xC4:Q8128,2173
C2×C22.58C24Direct product of C2 and C22.58C24128C2xC2^2.58C2^4128,2262

### Groups of order 144

dρLabelID
Dic3⋊D62nd semidirect product of Dic3 and D6 acting via D6/S3=C2; = Hol(Dic3)124+Dic3:D6144,154
S3×S4Direct product of S3 and S4; = Hol(C2×C6)126+S3xS4144,183
C2×PSU3(𝔽2)Direct product of C2 and PSU3(𝔽2)188+C2xPSU(3,2)144,187
C2×S3≀C2Direct product of C2 and S3≀C2124+C2xS3wrC2144,186
C2×C3⋊S4Direct product of C2 and C3⋊S4186+C2xC3:S4144,189
C22×S32Direct product of C22, S3 and S324C2^2xS3^2144,192
D4×C3⋊S3Direct product of D4 and C3⋊S336D4xC3:S3144,172
Q8×C3⋊S3Direct product of Q8 and C3⋊S372Q8xC3:S3144,174
C23×C3⋊S3Direct product of C23 and C3⋊S372C2^3xC3:S3144,196

### Groups of order 162

dρLabelID
C34⋊C24th semidirect product of C34 and C2 acting faithfully81C3^4:C2162,54

### Groups of order 192

dρLabelID
Q82S42nd semidirect product of Q8 and S4 acting via S4/C22=S3; = Hol(Q8)84+Q8:2S4192,1494
C23⋊S42nd semidirect product of C23 and S4 acting faithfully; = Aut(C22×C4)84+C2^3:S4192,1493
C24⋊D61st semidirect product of C24 and D6 acting faithfully; = Aut(C2×Q8)86+C2^4:D6192,955
C42⋊D6The semidirect product of C42 and D6 acting faithfully126+C4^2:D6192,956
D4×S4Direct product of D4 and S4126+D4xS4192,1472
Q8×S4Direct product of Q8 and S4246-Q8xS4192,1477
C23×S4Direct product of C23 and S424C2^3xS4192,1537
S3×2+ 1+4Direct product of S3 and 2+ 1+4248+S3xES+(2,2)192,1524
S3×2- 1+4Direct product of S3 and 2- 1+4488-S3xES-(2,2)192,1526
S3×C25Direct product of C25 and S396S3xC2^5192,1542
C2×C22⋊S4Direct product of C2 and C22⋊S4126+C2xC2^2:S4192,1538
S3×C8⋊C22Direct product of S3 and C8⋊C22; = Aut(D24) = Hol(C24)248+S3xC8:C2^2192,1331
S3×C22≀C2Direct product of S3 and C22≀C224S3xC2^2wrC2192,1147
C22×S3×D4Direct product of C22, S3 and D448C2^2xS3xD4192,1514
S3×C41D4Direct product of S3 and C41D448S3xC4:1D4192,1273
S3×C8.C22Direct product of S3 and C8.C22488-S3xC8.C2^2192,1335
S3×C4⋊Q8Direct product of S3 and C4⋊Q896S3xC4:Q8192,1282
C22×S3×Q8Direct product of C22, S3 and Q896C2^2xS3xQ8192,1517

### Groups of order 200

dρLabelID
C52⋊Q8The semidirect product of C52 and Q8 acting faithfully108+C5^2:Q8200,44

### Groups of order 216

dρLabelID
C324S42nd semidirect product of C32 and S4 acting via S4/A4=C236C3^2:4S4216,165
S33Direct product of S3, S3 and S3; = Hol(C3×S3)128+S3^3216,162
C2×C32⋊D6Direct product of C2 and C32⋊D6186+C2xC3^2:D6216,102
C22×C33⋊C2Direct product of C22 and C33⋊C2108C2^2xC3^3:C2216,176
C2×S3×C3⋊S3Direct product of C2, S3 and C3⋊S336C2xS3xC3:S3216,171

### Groups of order 240

dρLabelID
C2×S5Direct product of C2 and S5; = O3(𝔽5)104+C2xS5240,189

### Groups of order 288

dρLabelID
PSO4+ (𝔽3)Projective special orthogonal group of + type on 𝔽34; = A4S4 = Hol(A4)129+PSO+(4,3)288,1026
D6≀C2Wreath product of D6 by C2124+D6wrC2288,889
C4⋊S3≀C2The semidirect product of C4 and S3≀C2 acting via S3≀C2/C32⋊C4=C2248+C4:S3wrC2288,879
C32⋊C4⋊Q81st semidirect product of C32⋊C4 and Q8 acting via Q8/C4=C2488-C3^2:C4:Q8288,870
C22×PSU3(𝔽2)Direct product of C22 and PSU3(𝔽2)36C2^2xPSU(3,2)288,1032
C2×S3×S4Direct product of C2, S3 and S4; = Aut(S3×SL2(𝔽3))186+C2xS3xS4288,1028
S32×D4Direct product of S3, S3 and D4248+S3^2xD4288,958
C22×S3≀C2Direct product of C22 and S3≀C224C2^2xS3wrC2288,1031
C2×Dic3⋊D6Direct product of C2 and Dic3⋊D624C2xDic3:D6288,977
C22×C3⋊S4Direct product of C22 and C3⋊S436C2^2xC3:S4288,1034
S32×Q8Direct product of S3, S3 and Q8488-S3^2xQ8288,965
S32×C23Direct product of C23, S3 and S348S3^2xC2^3288,1040
C24×C3⋊S3Direct product of C24 and C3⋊S3144C2^4xC3:S3288,1044
C2×D4×C3⋊S3Direct product of C2, D4 and C3⋊S372C2xD4xC3:S3288,1007
C2×Q8×C3⋊S3Direct product of C2, Q8 and C3⋊S3144C2xQ8xC3:S3288,1010

### Groups of order 324

dρLabelID
He3⋊D6The semidirect product of He3 and D6 acting faithfully96+He3:D6324,39
He35D61st semidirect product of He3 and D6 acting via D6/C3=C221812+He3:5D6324,121
S3×C33⋊C2Direct product of S3 and C33⋊C254S3xC3^3:C2324,168
C2×C34⋊C2Direct product of C2 and C34⋊C2162C2xC3^4:C2324,175
C3⋊S32Direct product of C3⋊S3 and C3⋊S318C3:S3^2324,169

### Groups of order 400

dρLabelID
C2×C52⋊Q8Direct product of C2 and C52⋊Q8208+C2xC5^2:Q8400,212

### Groups of order 432

dρLabelID
C625D65th semidirect product of C62 and D6 acting faithfully186+C6^2:5D6432,523
C6223D64th semidirect product of C62 and D6 acting via D6/C3=C2236C6^2:23D6432,686
S3×PSU3(𝔽2)Direct product of S3 and PSU3(𝔽2)2416+S3xPSU(3,2)432,742
S3×S3≀C2Direct product of S3 and S3≀C2128+S3xS3wrC2432,741
S3×C3⋊S4Direct product of S3 and C3⋊S42412+S3xC3:S4432,747
C3⋊S3×S4Direct product of C3⋊S3 and S436C3:S3xS4432,746
C22×C32⋊D6Direct product of C22 and C32⋊D636C2^2xC3^2:D6432,545
C2×C324S4Direct product of C2 and C324S454C2xC3^2:4S4432,762
D4×C33⋊C2Direct product of D4 and C33⋊C2108D4xC3^3:C2432,724
Q8×C33⋊C2Direct product of Q8 and C33⋊C2216Q8xC3^3:C2432,726
C23×C33⋊C2Direct product of C23 and C33⋊C2216C2^3xC3^3:C2432,774
C2×S33Direct product of C2, S3, S3 and S3248+C2xS3^3432,759
C22×S3×C3⋊S3Direct product of C22, S3 and C3⋊S372C2^2xS3xC3:S3432,768

### Groups of order 480

dρLabelID
C22×S5Direct product of C22 and S520C2^2xS5480,1186

### Groups of order 486

dρLabelID
C35⋊C25th semidirect product of C35 and C2 acting faithfully243C3^5:C2486,260
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