Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
450 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×D32 |
| commutator | C22×C32 | C22×C4 |
| Fitting | C22×C32×C4 | C22 |
| socle, prime radical, cosocle | C23×C32 | C23 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #391 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. C2×#634 | C4 | semidirect |
| 37. C2×C3×Dic3 | Q8 | semidirect |
| 105. C2×Dic3 | Dic6 | semidirect |
| 148. C32 | #391 | semidirect |
| 187. C3 | #4929 | semidirect |
| 188. C3 | #4973 | 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×C4⋊Dic3 | 2 |
| C2×C3×C4×Dic3 | 2 |
| #63579 | 2 |
| #4973 | 3 |
| #4929 | 3 |