Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
615 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 164 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 | C2×C32×Q8 | C22 |
| hypercenter | C2×Q8 | D32 |
| hypocenter | C32 | #548 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#239 | C2 | semidirect |
| 2. C2×C3×D3×Q8 | C2 | semidirect |
| 3. Q8×C3⋊Dic3 | C2 | semidirect |
| 4. #63908 | C2 | semidirect |
| 5. C4×#625 | C2 | semidirect |
| 6. #63863 | C2 | semidirect |
| 7. #63856 | C2 | semidirect |
| 26. C2×#239 | D3 | semidirect |
| 27. C2×D3×Q8 | D3 | semidirect |
| 41. C32×Q8 | D4 | semidirect |
| 90. C2×C3×D3 | #41 | semidirect |
| 109. C3×Q8 | #67 | semidirect |
| 110. C3×Q8 | #67 | semidirect |
| 166. C2×D3 | #239 | semidirect |
| 180. C32 | #548 | semidirect |
| 184. Q8 | #625 | semidirect |
| 220. C3 | #5817 | semidirect |
| 221. C3 | #5831 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#239 | 2 |
| C2×C3×D3×Q8 | 2 |
| Q8×C3⋊Dic3 | 2 |
| #63908 | 2 |
| C4×#625 | 2 |
| #63863 | 2 |
| #63856 | 2 |
| #5831 | 3 |
| #5817 | 3 |