Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
534 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×C42 | C22 |
| hypercenter | C42 | D32 |
| hypocenter | C32 | #540 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #64083 | C2 | semidirect |
| 2. C3×#872 | C2 | semidirect |
| 3. C3×#870 | C2 | semidirect |
| 4. #63910 | C2 | semidirect |
| 5. #63909 | C2 | semidirect |
| 6. #63897 | C2 | semidirect |
| 7. #63894 | C2 | semidirect |
| 36. #872 | D3 | semidirect |
| 37. #870 | D3 | semidirect |
| 172. C32 | #540 | semidirect |
| 208. C3 | #5535 | semidirect |
| 209. C3 | #5697 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64083 | 2 |
| C3×#872 | 2 |
| C3×#870 | 2 |
| #63910 | 2 |
| #63909 | 2 |
| #63897 | 2 |
| #63894 | 2 |
| #5697 | 3 |
| #5535 | 3 |