Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
333 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 129 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×#125 | C2 |
| prime radical | C22×D9 | C23 |
| hypercenter | #125 | D9 |
| hypocenter | C9 | #533 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#125 | C2 | semidirect |
| 4. #63450 | C2 | semidirect |
| 5. #63449 | C2 | semidirect |
| 7. #63432 | C2 | semidirect |
| 37. C4×D9 | Q8 | semidirect |
| 64. #125 | D9 | semidirect |
| 104. C9 | #533 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#125 | 2 |
| #63458 | 2 |
| #63454 | 2 |
| #63450 | 2 |
| #63449 | 2 |
| C42×D9 | 2 |
| #63432 | 2 |
| #5688 | 3 |