Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
514 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 116 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. #11724361 | C2 | semidirect |
| 3. C3×#64243 | C2 | semidirect |
| 7. #11724148 | C2 | semidirect |
| 28. #64243 | D3 | semidirect |
| 36. C4×C3⋊(C3⋊D3) | Q8 | semidirect |
| 38. C32×C4×D3 | Q8 | semidirect |
| 39. #6289 | Q8 | semidirect |
| 41. D3×C3⋊Dic3 | Q8 | semidirect |
| 146. C33 | #533 | semidirect |
| 166. C4×D3 | PSU(3,𝔽2) | semidirect |
| 216. C32 | #5688 | semidirect |
| 254. C3 | #10596542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724364 | 2 |
| #11724361 | 2 |
| C3×#64243 | 2 |
| C4×D3×#157 | 2 |
| #11724247 | 2 |
| #11724151 | 2 |
| #11724148 | 2 |
| #10596542 | 3 |
| #5688 | 9 |