Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
616 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 186 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×#912 | C2 | semidirect |
| 2. C33×D16 | C2 | semidirect |
| 3. C32×#827 | C2 | semidirect |
| 4. C32×Dic24 | C2 | semidirect |
| 5. C32×D3×C16 | C2 | semidirect |
| 7. C3×#4912 | C3 | direct (N × H) |
| 16. C3×#912 | C2×C3 | semidirect |
| 20. C32×D16 | D3 | semidirect |
| 21. C32×D16 | C2×C3 | semidirect |
| 23. C3×#827 | C2×C3 | semidirect |
| 24. C3×Dic24 | C2×C3 | semidirect |
| 25. C3×D3×C16 | C2×C3 | semidirect |
| 34. #4912 | C32 | direct (N × H) |
| 57. #912 | C2×C32 | semidirect |
| 61. C3×D16 | C3×D3 | semidirect |
| 62. C3×D16 | C2×C32 | semidirect |
| 64. #827 | C2×C32 | semidirect |
| 65. Dic24 | C2×C32 | semidirect |
| 66. D3×C16 | C2×C32 | semidirect |
| 118. D16 | C32×D3 | semidirect |
| 120. C33 | #508 | semidirect |
| 165. C32 | #4912 | direct (N × H) |
| 166. C32 | C3×#508 | semidirect |
| 188. C3 | C3×#4912 | direct (N × H) |
| 189. C3 | C32×#508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#912 | 2 |
| C33×D16 | 2 |
| C32×#827 | 2 |
| C32×Dic24 | 2 |
| C32×D3×C16 | 2 |
| C32×#508 | 3 |
| C3×#4912 | 3 |