Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
324 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini, socle | C22×C3 | C23×D3 |
| Fitting | C9×#117 | C2 |
| prime radical | #63492 | C2 |
| hypercenter | #117 | D9 |
| hypocenter | C9 | #542 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#117 | C2 | semidirect |
| 3. #63486 | C2 | semidirect |
| 4. #63460 | C2 | semidirect |
| 5. #63458 | C2 | semidirect |
| 6. #63447 | C2 | semidirect |
| 7. #63445 | C2 | semidirect |
| 8. #63437 | C2 | semidirect |
| 9. C4×D36 | C2 | semidirect |
| 10. #63431 | C2 | semidirect |
| 11. C4×Dic18 | C2 | semidirect |
| 114. #117 | D9 | semidirect |
| 176. C9 | #542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#117 | 2 |
| #63492 | 2 |
| #63486 | 2 |
| #63460 | 2 |
| #63458 | 2 |
| #63447 | 2 |
| #63445 | 2 |
| #63437 | 2 |
| C4×D36 | 2 |
| #63431 | 2 |
| C4×Dic18 | 2 |
| #5534 | 3 |