Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
372 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D36 |
| commutator, cosocle | C4×C9 | C24 |
| Frattini | C3×C4 | C23×D3 |
| Fitting | C9×#131 | C2 |
| socle | C2×C3 | C22×D12 |
| hypercenter | #131 | D9 |
| hypocenter | C9 | #578 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #63716 | C2 | semidirect |
| 2. C9×#131 | C2 | semidirect |
| 3. #63471 | C2 | semidirect |
| 4. #63467 | C2 | semidirect |
| 5. C2×Dic36 | C2 | semidirect |
| 55. #131 | D9 | semidirect |
| 86. C9 | #578 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63716 | 2 |
| C9×#131 | 2 |
| #63471 | 2 |
| #63467 | 2 |
| C2×Dic36 | 2 |
| #5753 | 3 |