Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
286 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 52 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. #11724365 | C2 | semidirect |
| 3. #11722808 | C2 | semidirect |
| 4. C3×#63744 | C2 | semidirect |
| 20. #63744 | D3 | semidirect |
| 28. C4⋊(C3⋊(C3⋊D3)) | Q8 | semidirect |
| 30. C32×D12 | Q8 | semidirect |
| 119. C33 | #474 | semidirect |
| 133. D12 | PSU(3,𝔽2) | semidirect |
| 181. C32 | #4846 | semidirect |
| 219. C3 | #10593558 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724365 | 2 |
| D12×#157 | 2 |
| #11722808 | 2 |
| C3×#63744 | 2 |
| #11722707 | 2 |
| #10593558 | 3 |
| #4846 | 9 |