Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
6044 conjugacy classes of subgroups; only the 1468 normal subgroups are shown.
This group has 6044 conjugacy classes of subgroups — too many to draw. Only its 1468 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×C3⋊D3 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×D3×C3⋊D3 |
| Fitting, socle, prime radical | C24×C33 | C22 |
| hypocenter | C33 | C23×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C22×#6293 | C2 | direct (N × H) |
| 11. C2×#6293 | C22 | direct (N × H) |
| 21. C23×C33 | Q8 | semidirect |
| 26. #6293 | C23 | direct (N × H) |
| 41. C22×C33 | C2×Q8 | semidirect |
| 57. C23×C32 | Dic6 | semidirect |
| 58. C23×C32 | Dic6 | semidirect |
| 71. C2×C33 | C22×Q8 | semidirect |
| 87. C22×C32 | C2×Dic6 | semidirect |
| 88. C22×C32 | C2×Dic6 | semidirect |
| 102. C33 | C23×Q8 | semidirect |
| 105. C23×C3 | #627 | semidirect |
| 106. C23×C3 | #634 | semidirect |
| 120. C2×C32 | C22×Dic6 | semidirect |
| 121. C2×C32 | C22×Dic6 | semidirect |
| 131. C22×C3 | C2×#627 | semidirect |
| 132. C22×C3 | C2×#634 | semidirect |
| 143. C32 | C23×Dic6 | semidirect |
| 144. C32 | C23×Dic6 | semidirect |
| 145. C23 | #6293 | direct (N × H) |
| 153. C2×C3 | C22×#627 | semidirect |
| 154. C2×C3 | C22×#634 | semidirect |
| 157. C22 | C2×#6293 | direct (N × H) |
| 163. C3 | C23×#627 | semidirect |
| 164. C3 | C23×#634 | semidirect |
| 165. C2 | C22×#6293 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3⋊(C3⋊Dic3) | 2 |
| C23×C3×C3⋊Dic3 | 2 |
| C23×C32×Dic3 | 2 |
| C22×#6293 | 2 |
| C23×#634 | 3 |
| C23×#627 | 3 |