Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D9 |
| commutator, cosocle | C2×C4×C9 | C23 |
| Frattini | C2×C3×C4 | C22×D3 |
| Fitting | C2×C9×D8 | C2 |
| socle | C22×C3 | D3×D4 |
| hypercenter | C2×D8 | D9 |
| hypocenter | C9 | #496 |
| LCS | G ▷ C2×C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C9×D8 | C2 | semidirect |
| 2. #63502 | C2 | semidirect |
| 3. #63498 | C2 | semidirect |
| 4. C2×#3673 | C2 | semidirect |
| 5. C2×#3672 | C2 | semidirect |
| 6. C2×#3664 | C2 | semidirect |
| 7. #63375 | C2 | semidirect |
| 44. C8×C9 | D4 | semidirect |
| 45. C9⋊C8 | D4 | semidirect |
| 74. C2×D8 | D9 | semidirect |
| 124. C9 | #496 | semidirect |
| 141. C8 | #611 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×D8 | 2 |
| #63502 | 2 |
| #63498 | 2 |
| C2×#3673 | 2 |
| C2×#3672 | 2 |
| C2×#3664 | 2 |
| #63375 | 2 |
| #5154 | 3 |