Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
810 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 231 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 | C22×C32×Q8 | C2 |
| prime radical | C2×#3828 | C2 |
| hypercenter | C22×Q8 | C3⋊D3 |
| hypocenter | C32 | #536 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C32×Q8 | C2 | semidirect |
| 3. #64156 | C2 | semidirect |
| 4. #64155 | C2 | semidirect |
| 5. #64153 | C2 | semidirect |
| 6. #64138 | C2 | semidirect |
| 7. #64135 | C2 | semidirect |
| 24. C22×C3×Q8 | D3 | semidirect |
| 73. C22×Q8 | C3⋊D3 | semidirect |
| 123. C32 | #536 | semidirect |
| 153. C3 | #5816 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×Q8 | 2 |
| C2×#3828 | 2 |
| #64156 | 2 |
| #64155 | 2 |
| #64153 | 2 |
| #64138 | 2 |
| #64135 | 2 |
| #5816 | 3 |