Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
618 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 618 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C22×C32×C4 | C22 |
| prime radical | C2×C3×#235 | C2 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #536 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#634 | C2 | semidirect |
| 3. C3×#926 | C2 | semidirect |
| 4. #63962 | C2 | semidirect |
| 5. #63866 | C2 | semidirect |
| 6. #63860 | C2 | semidirect |
| 7. #63857 | C2 | semidirect |
| 34. #926 | D3 | semidirect |
| 191. C32 | #536 | semidirect |
| 229. C3 | #5526 | semidirect |
| 230. C3 | #5825 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#634 | 2 |
| C2×C3×#235 | 2 |
| C3×#926 | 2 |
| #63962 | 2 |
| #63866 | 2 |
| #63860 | 2 |
| #63857 | 2 |
| #5825 | 3 |
| #5526 | 3 |