Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
438 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 100 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, socle, cosocle | C23×C32 | C23 |
| Fitting | C22×C32×C4 | C22 |
| prime radical | C23×C3⋊D3 | C22 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #396 |
| LCS | G ▷ C23×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×Dic3⋊C4 | C2 | semidirect |
| 3. C2×#3724 | C2 | semidirect |
| 4. C2×Dic3⋊Dic3 | C2 | semidirect |
| 7. C3×C3⋊#95 | C2 | semidirect |
| 35. C2×Dic3⋊C4 | D3 | semidirect |
| 37. C3⋊#95 | D3 | semidirect |
| 265. C32 | #396 | semidirect |
| 337. C3 | #4676 | semidirect |
| 338. C3 | #4994 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3752 | 2 |
| C2×C3×Dic3⋊C4 | 2 |
| C2×#3724 | 2 |
| C2×Dic3⋊Dic3 | 2 |
| C2×#3722 | 2 |
| C3×C3⋊#95 | 2 |
| #4994 | 3 |
| #4676 | 3 |