Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
338 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×S4 |
| commutator, cosocle | C2×A4 | C23×C3 |
| Frattini | C2 | C22×C3×S4 |
| Fitting | C22×C3×Q8 | D3 |
| socle | C23×C3 | C22×D3 |
| hypercenter | C3×Q8 | S4 |
| hypocenter | A4 | C3×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×Q8×A4 | C2 | semidirect |
| 2. C3×C4⋊S4 | C2 | semidirect |
| 3. C3×C4×S4 | C2 | semidirect |
| 4. #5919 | C3 | direct (N × H) |
| 11. C22×C3×Q8 | D3 | semidirect |
| 13. Q8×A4 | C2×C3 | semidirect |
| 14. C4⋊S4 | C2×C3 | semidirect |
| 15. C4×S4 | C2×C3 | semidirect |
| 47. C3×A4 | #41 | semidirect |
| 51. C22×Q8 | C3×D3 | semidirect |
| 66. C3×Q8 | S4 | semidirect |
| 101. C22×C3 | #239 | semidirect |
| 104. A4 | C3×#41 | semidirect |
| 118. Q8 | C3×S4 | semidirect |
| 142. C22 | C3×#239 | semidirect |
| 150. C3 | #5919 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q8×A4 | 2 |
| C3×C4⋊S4 | 2 |
| C3×C4×S4 | 2 |
| #5919 | 3 |
| C3×#548 | 3 |
| C3×#239 | 4 |