Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1268 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 1268 conjugacy classes of subgroups — too many to draw. Only its 240 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | D32 |
| commutator, cosocle | C2×C32 | C23×C32 |
| Frattini | C2 | C2×C32×D32 |
| Fitting, socle | C22×C34 | C22 |
| hypocenter | C32 | C2×C32×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C32×C3⋊D3 | C2 | semidirect |
| 2. C22×C33×D3 | C2 | semidirect |
| 3. C2×C33×Dic3 | C2 | semidirect |
| 4. C32×#626 | C2 | direct (N × H) |
| 7. C2×C3×#626 | C3 | direct (N × H) |
| 10. C2×C32×C3⋊D3 | C22 | semidirect |
| 12. C2×C33×D3 | C22 | semidirect |
| 13. C33×Dic3 | C22 | semidirect |
| 15. C22×C3×C3⋊D3 | C2×C3 | semidirect |
| 20. C22×C32×D3 | D3 | semidirect |
| 21. C22×C32×D3 | C2×C3 | semidirect |
| 25. C2×C32×Dic3 | D3 | semidirect |
| 26. C2×C32×Dic3 | C2×C3 | semidirect |
| 28. C3×#626 | C2×C3 | direct (N × H) |
| 29. C2×C34 | D4 | semidirect |
| 36. C2×#626 | C32 | direct (N × H) |
| 47. C2×C3×C3⋊D3 | C22×C3 | semidirect |
| 58. C2×C32×D3 | C2×D3 | semidirect |
| 59. C2×C32×D3 | C22×C3 | semidirect |
| 62. C32×Dic3 | C2×D3 | semidirect |
| 63. C32×Dic3 | C22×C3 | semidirect |
| 64. C34 | C2×D4 | semidirect |
| 69. C22×C3⋊D3 | C2×C32 | semidirect |
| 74. C22×C3×D3 | C3×D3 | semidirect |
| 75. C22×C3×D3 | C2×C32 | semidirect |
| 82. C2×C3×Dic3 | C3×D3 | semidirect |
| 83. C2×C3×Dic3 | C2×C32 | semidirect |
| 85. #626 | C2×C32 | direct (N × H) |
| 95. C2×C33 | D12 | semidirect |
| 96. C2×C33 | #67 | semidirect |
| 97. C2×C33 | C3×D4 | semidirect |
| 132. C2×C3⋊D3 | C22×C32 | semidirect |
| 143. C2×C3×D3 | C2×C3×D3 | semidirect |
| 144. C2×C3×D3 | C22×C32 | semidirect |
| 149. C3×Dic3 | C2×C3×D3 | semidirect |
| 150. C3×Dic3 | C22×C32 | semidirect |
| 155. C33 | C2×D12 | semidirect |
| 156. C33 | C2×#67 | semidirect |
| 157. C33 | C2×C3×D4 | semidirect |
| 165. C22×D3 | C32×D3 | semidirect |
| 171. C2×Dic3 | C32×D3 | semidirect |
| 193. C2×C32 | #626 | direct (N × H) |
| 194. C2×C32 | C3×D12 | semidirect |
| 195. C2×C32 | C3×#67 | semidirect |
| 196. C2×C32 | C32×D4 | semidirect |
| 231. C2×D3 | C2×C32×D3 | semidirect |
| 235. Dic3 | C2×C32×D3 | semidirect |
| 244. C32 | C2×#626 | direct (N × H) |
| 245. C32 | C2×C3×D12 | semidirect |
| 246. C32 | C2×C3×#67 | semidirect |
| 247. C32 | C2×C32×D4 | semidirect |
| 264. C2×C3 | C3×#626 | direct (N × H) |
| 265. C2×C3 | C32×D12 | semidirect |
| 266. C2×C3 | C32×#67 | semidirect |
| 284. C3 | C2×C3×#626 | direct (N × H) |
| 285. C3 | C2×C32×D12 | semidirect |
| 286. C3 | C2×C32×#67 | semidirect |
| 289. C2 | C32×#626 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×C3⋊D3 | 2 |
| C22×C33×D3 | 2 |
| C2×C33×Dic3 | 2 |
| C32×#626 | 2 |
| C2×C32×#67 | 3 |
| C2×C32×D12 | 3 |
| C2×C3×#626 | 3 |