Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
588 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#125 | C2 |
| prime radical | C2×C4⋊(C3⋊D3) | C22 |
| hypercenter | #125 | C3⋊D3 |
| hypocenter | C32 | #563 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#125 | C2 | semidirect |
| 2. #64107 | C2 | semidirect |
| 3. #64107 | C2 | semidirect |
| 4. #64106 | C2 | semidirect |
| 5. #64105 | C2 | semidirect |
| 7. #64097 | C2 | semidirect |
| 9. #64079 | C2 | semidirect |
| 32. C3×#125 | D3 | semidirect |
| 92. #125 | C3⋊D3 | semidirect |
| 147. C32 | #563 | semidirect |
| 177. C3 | #5699 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#125 | 2 |
| #64107 | 2 |
| #64106 | 2 |
| #64105 | 2 |
| #64103 | 2 |
| #64097 | 2 |
| #64084 | 2 |
| #64079 | 2 |
| #5699 | 3 |