Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
622 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 622 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×#3828 | 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 |
|---|---|---|
| 2. C22×C3×Dic6 | C2 | semidirect |
| 3. C3×#926 | C2 | semidirect |
| 4. #63966 | C2 | semidirect |
| 5. #63871 | C2 | semidirect |
| 6. #63866 | C2 | semidirect |
| 7. #63864 | C2 | semidirect |
| 29. C22×Dic6 | D3 | semidirect |
| 35. #926 | D3 | semidirect |
| 196. C32 | #536 | semidirect |
| 235. C3 | #5526 | semidirect |
| 236. C3 | #5816 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3828 | 2 |
| C22×C3×Dic6 | 2 |
| C3×#926 | 2 |
| #63966 | 2 |
| #63871 | 2 |
| #63866 | 2 |
| #63864 | 2 |
| #5816 | 3 |
| #5526 | 3 |