Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
459 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 132 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 | C2×C32×Q8 | C22 |
| prime radical | C2×C3×D12 | C22 |
| hypercenter | C2×Q8 | D32 |
| hypocenter | C32 | #556 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×C3⋊Dic3 | C2 | semidirect |
| 3. C3×#944 | C2 | semidirect |
| 4. C4×#627 | C2 | semidirect |
| 5. #63893 | C2 | semidirect |
| 7. #63878 | C2 | semidirect |
| 9. #63862 | C2 | semidirect |
| 10. #63861 | C2 | semidirect |
| 11. Dic3×Dic6 | C2 | semidirect |
| 45. #944 | D3 | semidirect |
| 252. C32 | #556 | semidirect |
| 295. C3 | #5671 | semidirect |
| 296. C3 | #5732 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C3⋊Dic3 | 2 |
| C3×#947 | 2 |
| C3×#944 | 2 |
| C4×#627 | 2 |
| #63893 | 2 |
| Dic3×D12 | 2 |
| #63878 | 2 |
| #63863 | 2 |
| #63862 | 2 |
| #63861 | 2 |
| Dic3×Dic6 | 2 |
| #5732 | 3 |
| #5671 | 3 |