Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1440 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1440 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. C33×#127 | C2 | semidirect |
| 3. C32×#932 | C2 | semidirect |
| 4. C32×#833 | C2 | semidirect |
| 5. C32×#805 | C2 | semidirect |
| 7. C3×#5077 | C3 | direct (N × H) |
| 10. C2×C33×D4 | C22 | semidirect |
| 15. C33×C42 | C22 | semidirect |
| 18. C32×#208 | C22 | semidirect |
| 20. C3×#1004 | C2×C3 | semidirect |
| 23. C32×#127 | D3 | semidirect |
| 24. C32×#127 | C2×C3 | semidirect |
| 25. C3×#932 | C2×C3 | semidirect |
| 28. C3×#833 | C2×C3 | semidirect |
| 29. C3×#805 | C2×C3 | semidirect |
| 41. C32×D12 | D4 | semidirect |
| 43. C32×Dic6 | D4 | semidirect |
| 46. #5077 | C32 | direct (N × H) |
| 57. C2×C32×D4 | C2×D3 | semidirect |
| 59. C2×C32×D4 | C22×C3 | semidirect |
| 68. C32×C42 | C2×D3 | semidirect |
| 69. C32×C42 | C22×C3 | semidirect |
| 72. C3×#208 | C22×C3 | semidirect |
| 84. #1004 | C2×C32 | semidirect |
| 87. C3×#127 | C3×D3 | semidirect |
| 88. C3×#127 | C2×C32 | semidirect |
| 89. #932 | C2×C32 | semidirect |
| 92. #833 | C2×C32 | semidirect |
| 93. #805 | C2×C32 | semidirect |
| 126. C3×D12 | C3×D4 | semidirect |
| 128. C3×Dic6 | C3×D4 | semidirect |
| 147. C2×C3×D4 | C2×C3×D3 | semidirect |
| 149. C2×C3×D4 | C22×C32 | semidirect |
| 158. C3×C42 | C2×C3×D3 | semidirect |
| 159. C3×C42 | C22×C32 | semidirect |
| 162. #208 | C22×C32 | semidirect |
| 192. #127 | C32×D3 | semidirect |
| 195. C33 | #453 | semidirect |
| 228. D12 | C32×D4 | semidirect |
| 230. Dic6 | C32×D4 | semidirect |
| 253. C2×D4 | C2×C32×D3 | semidirect |
| 257. C42 | C2×C32×D3 | semidirect |
| 286. C32 | #5077 | direct (N × H) |
| 287. C32 | C3×#453 | semidirect |
| 328. C3 | C3×#5077 | direct (N × H) |
| 329. 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 |
| C33×#127 | 2 |
| C32×#932 | 2 |
| C32×#833 | 2 |
| C32×#805 | 2 |
| C32×#453 | 3 |
| C3×#5077 | 3 |