Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
342 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 131 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×C4⋊Q8 | C2 |
| prime radical | C2×D36 | C22 |
| hypercenter | C4⋊Q8 | D9 |
| hypocenter | C9 | #554 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×C4⋊Q8 | C2 | semidirect |
| 2. #63508 | C2 | semidirect |
| 3. #63459 | C2 | semidirect |
| 5. C4⋊C4×D9 | C2 | semidirect |
| 6. #63452 | C2 | semidirect |
| 8. C4×Dic18 | C2 | semidirect |
| 43. D36 | Q8 | semidirect |
| 75. C4⋊Q8 | D9 | semidirect |
| 127. C9 | #554 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C4⋊Q8 | 2 |
| #63508 | 2 |
| #63459 | 2 |
| #63455 | 2 |
| C4⋊C4×D9 | 2 |
| #63452 | 2 |
| C4×D36 | 2 |
| C4×Dic18 | 2 |
| #5730 | 3 |