Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
498 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 152 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 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×C42 | C22 |
| prime radical | C22×C3⋊D3 | C23 |
| hypercenter | C42 | D32 |
| hypocenter | C32 | #533 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#869 | C2 | semidirect |
| 3. #63872 | C2 | semidirect |
| 6. #63843 | C2 | semidirect |
| 28. #869 | D3 | semidirect |
| 35. C4×C3⋊D3 | Q8 | semidirect |
| 129. C32 | #533 | semidirect |
| 155. C3 | #5728 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C42×C3⋊D3 | 2 |
| C3×#869 | 2 |
| #63872 | 2 |
| #63871 | 2 |
| #63849 | 2 |
| #63843 | 2 |
| #5728 | 3 |