Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2786 conjugacy classes of subgroups; only the 913 normal subgroups are shown.
This group has 2786 conjugacy classes of subgroups — too many to draw. Only its 913 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×D9 |
| commutator, cosocle | C2×C3×C9 | C25 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×C3×C9 | C22 |
| socle, prime radical | C24×C32 | C2×D3 |
| hypocenter | C3×C9 | C23×Q8 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C22×#6189 | C2 | direct (N × H) |
| 11. C2×#6189 | C22 | direct (N × H) |
| 21. C23×C3×C9 | Q8 | semidirect |
| 26. #6189 | C23 | direct (N × H) |
| 41. C22×C3×C9 | C2×Q8 | semidirect |
| 63. C23×C9 | Dic6 | semidirect |
| 71. C2×C3×C9 | C22×Q8 | semidirect |
| 92. C22×C9 | C2×Dic6 | semidirect |
| 102. C3×C9 | C23×Q8 | semidirect |
| 105. C23×C3 | Dic18 | semidirect |
| 122. C2×C9 | C22×Dic6 | semidirect |
| 131. C22×C3 | C2×Dic18 | semidirect |
| 144. C9 | C23×Dic6 | semidirect |
| 145. C23 | #6189 | direct (N × H) |
| 153. C2×C3 | C22×Dic18 | semidirect |
| 157. C22 | C2×#6189 | direct (N × H) |
| 163. C3 | C23×Dic18 | semidirect |
| 165. C2 | C22×#6189 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3⋊Dic9 | 2 |
| C23×C9×Dic3 | 2 |
| C23×C3×Dic9 | 2 |
| C22×#6189 | 2 |
| C23×#627 | 3 |
| C23×Dic18 | 3 |