Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
514 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 120 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 |
| 2. #11724362 | C2 | semidirect |
| 3. C3×C4⋊PSU(3,𝔽2) | C2 | semidirect |
| 8. #11724148 | C2 | semidirect |
| 9. Dic3×PSU(3,𝔽2) | C2 | semidirect |
| 33. C4⋊PSU(3,𝔽2) | D3 | semidirect |
| 48. C4×C3⋊(C3⋊D3) | Q8 | semidirect |
| 50. C32×C4×D3 | Q8 | semidirect |
| 51. #6289 | Q8 | semidirect |
| 53. D3×C3⋊Dic3 | Q8 | semidirect |
| 182. C33 | #533 | semidirect |
| 206. C4×D3 | PSU(3,𝔽2) | semidirect |
| 266. C32 | #5728 | semidirect |
| 313. C3 | #10596542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724365 | 2 |
| #11724362 | 2 |
| C3×C4⋊PSU(3,𝔽2) | 2 |
| C4×D3×#157 | 2 |
| #11724247 | 2 |
| #11724150 | 2 |
| #11724149 | 2 |
| #11724148 | 2 |
| Dic3×PSU(3,𝔽2) | 2 |
| #10596542 | 3 |
| #5728 | 9 |