Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
426 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 120 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 |
|---|---|---|
| 14. C2×#627 | C4 | semidirect |
| 28. C2×C32×C4 | Q8 | semidirect |
| 34. C2×C3⋊Dic3 | Q8 | semidirect |
| 89. C2×C3×C4 | Dic6 | semidirect |
| 150. C32 | #391 | semidirect |
| 167. C2×C4 | #627 | semidirect |
| 195. C3 | #4646 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#627 | 2 |
| C2×C4×C3⋊Dic3 | 2 |
| C2×#3724 | 2 |
| C3×C3⋊#95 | 2 |
| #63579 | 2 |
| #4646 | 3 |