Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
526 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 114 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. #11724364 | C2 | semidirect |
| 2. C4×C3⋊PSU(3,𝔽2) | C2 | semidirect |
| 3. #11724362 | C2 | semidirect |
| 4. C3×#64243 | C2 | semidirect |
| 8. #11724151 | C2 | semidirect |
| 9. #11724151 | C2 | semidirect |
| 10. #11724150 | C2 | semidirect |
| 11. D3×#3777 | C2 | semidirect |
| 46. #64243 | D3 | semidirect |
| 62. C4⋊(C3⋊(C3⋊D3)) | Q8 | semidirect |
| 64. C32×D12 | Q8 | semidirect |
| 65. #6291 | Q8 | semidirect |
| 209. C33 | #554 | semidirect |
| 228. D12 | PSU(3,𝔽2) | semidirect |
| 297. C32 | #5690 | semidirect |
| 347. C3 | #10596545 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724364 | 2 |
| C4×C3⋊PSU(3,𝔽2) | 2 |
| #11724362 | 2 |
| C3×#64243 | 2 |
| #11724259 | 2 |
| D12×#157 | 2 |
| #11724151 | 2 |
| #11724150 | 2 |
| D3×#3777 | 2 |
| #10596545 | 3 |
| #5690 | 9 |