Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
652 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 652 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. C2×C32×Dic12 | C2 | semidirect |
| 3. C2×C33×C16 | C2 | semidirect |
| 4. C32×#800 | C2 | direct (N × H) |
| 6. C2×C3×#800 | C3 | direct (N × H) |
| 11. C32×Dic12 | C22 | semidirect |
| 14. C33×C16 | C22 | semidirect |
| 17. C2×C3×Dic12 | C2×C3 | semidirect |
| 21. C2×C32×C16 | D3 | semidirect |
| 22. C2×C32×C16 | C2×C3 | semidirect |
| 23. C3×#800 | C2×C3 | direct (N × H) |
| 34. C2×#800 | C32 | direct (N × H) |
| 47. C3×Dic12 | C22×C3 | semidirect |
| 51. C32×C16 | C2×D3 | semidirect |
| 52. C32×C16 | C22×C3 | semidirect |
| 61. C2×Dic12 | C2×C32 | semidirect |
| 65. C2×C3×C16 | C3×D3 | semidirect |
| 66. C2×C3×C16 | C2×C32 | semidirect |
| 67. #800 | C2×C32 | direct (N × H) |
| 90. C2×C33 | SD32 | semidirect |
| 105. Dic12 | C22×C32 | semidirect |
| 109. C3×C16 | C2×C3×D3 | semidirect |
| 110. C3×C16 | C22×C32 | semidirect |
| 129. C2×C16 | C32×D3 | semidirect |
| 130. C33 | C2×SD32 | semidirect |
| 155. C2×C32 | #800 | direct (N × H) |
| 156. C2×C32 | C3×SD32 | semidirect |
| 167. C16 | C2×C32×D3 | semidirect |
| 184. C32 | C2×#800 | direct (N × H) |
| 185. C32 | C2×C3×SD32 | semidirect |
| 198. C2×C3 | C3×#800 | direct (N × H) |
| 199. C2×C3 | C32×SD32 | semidirect |
| 210. C3 | C2×C3×#800 | direct (N × H) |
| 211. C3 | C2×C32×SD32 | semidirect |
| 213. C2 | C32×#800 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×Dic12 | 2 |
| C2×C32×D24 | 2 |
| C2×C33×C16 | 2 |
| C32×#800 | 2 |
| C2×C32×SD32 | 3 |
| C2×C3×#800 | 3 |