Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1015 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | A4×D9 |
| commutator, hypocenter | C22×C9 | C22×C3×C4 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C25×C9 | C2×C3 |
| socle | C25×C3 | C3×D3 |
| prime radical | C23×C3×A4 | D3 |
| cosocle | C23×C9 | C23×C3 |
| LCS | G ▷ C22×C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×A4×Dic9 | C2 | direct (N × H) |
| 4. C24×Dic9 | C3 | semidirect |
| 5. C22×C9×A4 | C4 | semidirect |
| 8. A4×Dic9 | C22 | direct (N × H) |
| 16. C23×Dic9 | C2×C3 | semidirect |
| 18. C2×C9×A4 | C2×C4 | semidirect |
| 30. C24×C9 | C3×C4 | semidirect |
| 41. C22×Dic9 | A4 | direct (N × H) |
| 42. C22×Dic9 | C22×C3 | semidirect |
| 45. C9×A4 | C22×C4 | semidirect |
| 69. C23×C9 | C2×C3×C4 | semidirect |
| 79. C2×Dic9 | C2×A4 | direct (N × H) |
| 90. C22×A4 | Dic9 | direct (N × H) |
| 118. C22×C9 | C4×A4 | semidirect |
| 119. C22×C9 | C22×C3×C4 | semidirect |
| 123. Dic9 | C22×A4 | direct (N × H) |
| 146. C2×A4 | C2×Dic9 | direct (N × H) |
| 162. C2×C9 | C2×C4×A4 | semidirect |
| 167. C24 | C3×Dic9 | semidirect |
| 195. A4 | C22×Dic9 | direct (N × H) |
| 201. C9 | C22×C4×A4 | semidirect |
| 211. C23 | C2×C3×Dic9 | semidirect |
| 238. C22 | A4×Dic9 | direct (N × H) |
| 239. C22 | C22×C3×Dic9 | semidirect |
| 249. C2 | C2×A4×Dic9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C9×A4 | 2 |
| C2×A4×Dic9 | 2 |
| C22×Dic3×A4 | 3 |
| C24×Dic9 | 3 |
| C22×C3×Dic9 | 4 |