Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
243 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | D9 |
| commutator, hypocenter | C9 | C2×C4×C8 |
| Frattini | C2×C3×C4 | C22×D3 |
| Fitting | C4×C8×C9 | C2 |
| socle | C22×C3 | C2×C4×D3 |
| prime radical | C22×D9 | C2×C4 |
| cosocle | C2×C4×C9 | C23 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C4×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×C8×C9 | C2 | semidirect |
| 4. C8×Dic9 | C2 | semidirect |
| 5. C4×C9⋊C8 | C2 | semidirect |
| 14. C8×D9 | C4 | direct (N × H) |
| 25. C4×D9 | C8 | direct (N × H) |
| 27. C8×C9 | C2×C4 | semidirect |
| 28. C9⋊C8 | C2×C4 | semidirect |
| 41. C4×C9 | C2×C8 | semidirect |
| 44. Dic9 | C2×C8 | semidirect |
| 49. C4×C8 | D9 | direct (N × H) |
| 61. D9 | C4×C8 | direct (N × H) |
| 77. C9 | C2×C4×C8 | semidirect |
| 86. C8 | C4×D9 | direct (N × H) |
| 95. C4 | C8×D9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D9 | 2 |
| C42×D9 | 2 |
| C4×C8×C9 | 2 |
| C8×Dic9 | 2 |
| C4×C9⋊C8 | 2 |
| C4×D3×C8 | 3 |