Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
153 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×D12 |
| commutator, cosocle | C3×#41 | C22×C3 |
| Frattini | C4 | C2×D3×A4 |
| Fitting | C3×#131 | C2×C3 |
| socle | C2×C3 | D4×A4 |
| hypercenter | C8 | D3×A4 |
| hypocenter | C3×Q8 | C3×D4 |
| LCS | G ▷ C3×#41 ▷ C3×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊#994 | C2 | semidirect |
| 2. C3×#131⋊C3 | C2 | semidirect |
| 3. #5752 | C3 | semidirect |
| 15. #131⋊C3 | D3 | semidirect |
| 104. C3 | #576⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊#994 | 2 |
| C3×#131⋊C3 | 2 |
| #5752 | 3 |
| #576⋊C3 | 3 |
| C3×Dic12 | 4 |