Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
184 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D4×A4 |
| commutator, cosocle | #41 | C22×C32 |
| Frattini | C4 | C22×C3×A4 |
| Fitting | C3×#578 | C3 |
| hypercenter | C3×D8 | A4 |
| hypocenter | Q8 | C32×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×2−1+4⋊C3 | C2 | semidirect |
| 2. C3×#131⋊C3 | C2 | semidirect |
| 3. C3×#578 | C3 | semidirect |
| 4. #578⋊C3 | C3 | direct (N × H) |
| 8. C3×2−1+4 | C2×C3 | semidirect |
| 9. 2−1+4⋊C3 | C2×C3 | semidirect |
| 13. C3×#131 | C2×C3 | semidirect |
| 14. #131⋊C3 | C2×C3 | semidirect |
| 18. #578 | C32 | semidirect |
| 35. 2−1+4 | C2×C32 | semidirect |
| 39. #131 | C2×C32 | semidirect |
| 97. C3 | #578⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×2−1+4⋊C3 | 2 |
| C3×#131⋊C3 | 2 |
| C3×#578 | 3 |
| #578⋊C3 | 3 |
| C32×D8 | 4 |