Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
828 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D3×D9 |
| commutator, cosocle | C22×C3×C9 | C24 |
| Frattini | C22×C3 | C22×D32 |
| Fitting | C3×C42×C9 | C22 |
| socle | C22×C32 | C23×D3 |
| hypercenter | C42 | D3×D9 |
| hypocenter | C3×C9 | #540 |
| LCS | G ▷ C22×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11721711 | C2 | semidirect |
| 2. C9×#872 | C2 | semidirect |
| 3. C3×#63432 | C2 | semidirect |
| 4. #11721039 | C2 | semidirect |
| 5. #11721038 | C2 | semidirect |
| 6. #11720997 | C2 | semidirect |
| 7. #11720993 | C2 | semidirect |
| 37. #63432 | D3 | semidirect |
| 105. #872 | D9 | semidirect |
| 205. C3×C9 | #540 | semidirect |
| 289. C9 | #5697 | semidirect |
| 325. C3 | #10589637 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11721711 | 2 |
| C9×#872 | 2 |
| C3×#63432 | 2 |
| #11721039 | 2 |
| #11721038 | 2 |
| #11720997 | 2 |
| #11720993 | 2 |
| #10593989 | 3 |
| #10589637 | 3 |