Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
900 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 900 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C23×C3 |
| Frattini | C2 | C2×C3×D3×C3⋊D3 |
| Fitting, socle, prime radical | C22×C34 | C22 |
| hypocenter | C33 | C2×C3×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×#6293 | C2 | direct (N × H) |
| 6. C2×#6293 | C3 | direct (N × H) |
| 25. #6293 | C2×C3 | direct (N × H) |
| 27. C2×C34 | Q8 | semidirect |
| 52. C34 | C2×Q8 | semidirect |
| 82. C2×C33 | Dic6 | semidirect |
| 83. C2×C33 | Dic6 | semidirect |
| 84. C2×C33 | C3×Q8 | semidirect |
| 124. C33 | C2×Dic6 | semidirect |
| 125. C33 | C2×Dic6 | semidirect |
| 126. C33 | C2×C3×Q8 | semidirect |
| 158. C2×C32 | #627 | semidirect |
| 159. C2×C32 | C3×Dic6 | semidirect |
| 160. C2×C32 | C3×Dic6 | semidirect |
| 161. C2×C32 | #634 | semidirect |
| 194. C32 | C2×#627 | semidirect |
| 195. C32 | C2×C3×Dic6 | semidirect |
| 196. C32 | C2×C3×Dic6 | semidirect |
| 197. C32 | C2×#634 | semidirect |
| 210. C2×C3 | C3×#627 | semidirect |
| 211. C2×C3 | #6293 | direct (N × H) |
| 212. C2×C3 | C3×#634 | semidirect |
| 224. C3 | C2×C3×#627 | semidirect |
| 225. C3 | C2×#6293 | direct (N × H) |
| 226. C3 | C2×C3×#634 | semidirect |
| 227. C2 | C3×#6293 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C3⋊(C3⋊Dic3) | 2 |
| C2×C32×C3⋊Dic3 | 2 |
| C2×C33×Dic3 | 2 |
| C3×#6293 | 2 |
| C2×C3×#634 | 3 |
| C2×#6293 | 3 |
| C2×C3×#627 | 3 |