Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
674 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 674 conjugacy classes of subgroups — too many to draw. Only its 140 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 |
| hypercenter | C42 | D32 |
| hypocenter | C32 | #532 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3796 | C2 | semidirect |
| 2. C42×C3⋊D3 | C2 | semidirect |
| 3. C3×#875 | C2 | semidirect |
| 4. #63914 | C2 | semidirect |
| 5. #63912 | C2 | semidirect |
| 6. #63902 | C2 | semidirect |
| 7. #63873 | C2 | semidirect |
| 29. #875 | D3 | semidirect |
| 39. C4×C3⋊D3 | D4 | semidirect |
| 79. C3⋊Dic3 | #41 | semidirect |
| 152. C32 | #532 | semidirect |
| 186. C3 | #5668 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3796 | 2 |
| C42×C3⋊D3 | 2 |
| C3×#875 | 2 |
| #63914 | 2 |
| #63912 | 2 |
| #63902 | 2 |
| #63873 | 2 |
| #5668 | 3 |