Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1192 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#1004 | C2 | semidirect |
| 2. C32×#940 | C2 | semidirect |
| 3. C32×#908 | C2 | semidirect |
| 4. C33×#100 | C2 | semidirect |
| 5. C32×#825 | C2 | semidirect |
| 7. C3×#4747 | C3 | direct (N × H) |
| 14. C2×C32×D12 | C22 | semidirect |
| 15. C33×M16 | C22 | semidirect |
| 16. C32×C4×Dic3 | C22 | semidirect |
| 20. C3×#1004 | C2×C3 | semidirect |
| 23. C3×#940 | C2×C3 | semidirect |
| 24. C3×#908 | C2×C3 | semidirect |
| 27. C32×#100 | D3 | semidirect |
| 28. C32×#100 | C2×C3 | semidirect |
| 29. C3×#825 | C2×C3 | semidirect |
| 41. C32×D12 | D4 | semidirect |
| 43. C32×Dic6 | D4 | semidirect |
| 46. #4747 | C32 | direct (N × H) |
| 61. C2×C3×D12 | C22×C3 | semidirect |
| 65. C32×M16 | C2×D3 | semidirect |
| 66. C32×M16 | C22×C3 | semidirect |
| 68. C3×C4×Dic3 | C22×C3 | semidirect |
| 82. #1004 | C2×C32 | semidirect |
| 85. #940 | C2×C32 | semidirect |
| 86. #908 | C2×C32 | semidirect |
| 89. C3×#100 | C3×D3 | semidirect |
| 90. C3×#100 | C2×C32 | semidirect |
| 91. #825 | C2×C32 | semidirect |
| 120. C3×D12 | C3×D4 | semidirect |
| 122. C3×Dic6 | C3×D4 | semidirect |
| 146. C2×D12 | C22×C32 | semidirect |
| 150. C3×M16 | C2×C3×D3 | semidirect |
| 151. C3×M16 | C22×C32 | semidirect |
| 153. C4×Dic3 | C22×C32 | semidirect |
| 184. #100 | C32×D3 | semidirect |
| 185. C33 | #453 | semidirect |
| 214. D12 | C32×D4 | semidirect |
| 216. Dic6 | C32×D4 | semidirect |
| 242. M16 | C2×C32×D3 | semidirect |
| 269. C32 | #4747 | direct (N × H) |
| 270. C32 | C3×#453 | semidirect |
| 310. C3 | C3×#4747 | direct (N × H) |
| 311. C3 | C32×#453 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#1004 | 2 |
| C32×#940 | 2 |
| C32×#908 | 2 |
| C33×#100 | 2 |
| C32×#825 | 2 |
| C32×#453 | 3 |
| C3×#4747 | 3 |