Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
578 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 578 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 |
|---|---|---|
| 2. C22×C3×Dic6 | C2 | semidirect |
| 3. #64135 | C2 | semidirect |
| 4. #63961 | C2 | semidirect |
| 5. #63873 | C2 | semidirect |
| 6. #63863 | C2 | semidirect |
| 7. #63856 | C2 | semidirect |
| 28. C22×Dic6 | D3 | semidirect |
| 187. C32 | #536 | semidirect |
| 225. C3 | #5816 | semidirect |
| 226. C3 | #5825 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#235 | 2 |
| C22×C3×Dic6 | 2 |
| #64135 | 2 |
| #63961 | 2 |
| #63873 | 2 |
| #63863 | 2 |
| #63856 | 2 |
| #5825 | 3 |
| #5816 | 3 |