Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
126 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×D9 |
| commutator, cosocle | C8×C9 | C23 |
| Frattini | C3×C8 | C22×D3 |
| Fitting | C9×SD32 | C2 |
| socle | C2×C3 | D3×D8 |
| hypercenter | SD32 | D9 |
| hypocenter | C9 | #508 |
| LCS | G ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #63481 | C2 | semidirect |
| 2. #63474 | C2 | semidirect |
| 3. C9×SD32 | C2 | semidirect |
| 4. #63388 | C2 | semidirect |
| 5. #63385 | C2 | semidirect |
| 6. #63359 | C2 | semidirect |
| 7. C16×D9 | C2 | semidirect |
| 61. SD32 | D9 | semidirect |
| 97. C9 | #508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63481 | 2 |
| #63474 | 2 |
| C9×SD32 | 2 |
| #63388 | 2 |
| #63385 | 2 |
| #63359 | 2 |
| C16×D9 | 2 |
| #4916 | 3 |