Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
756 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 180 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×C32 | D32 |
| commutator | C2×C32 | C2×C32×C4 |
| Frattini | C22 | C32×D32 |
| Fitting, socle, prime radical | C22×C34 | C22 |
| cosocle | C22×C32 | C22×C32 |
| hypocenter | C32 | C32×C4⋊C4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C3×Dic3⋊Dic3 | C3 | direct (N × H) |
| 10. C33×Dic3 | C4 | semidirect |
| 29. Dic3⋊Dic3 | C32 | direct (N × H) |
| 47. C32×Dic3 | Dic3 | semidirect |
| 48. C32×Dic3 | C3×C4 | semidirect |
| 49. C34 | C4⋊C4 | semidirect |
| 118. C3×Dic3 | C3×Dic3 | semidirect |
| 119. C3×Dic3 | C32×C4 | semidirect |
| 124. C33 | Dic3⋊C4 | semidirect |
| 125. C33 | C4⋊Dic3 | semidirect |
| 126. C33 | C3×C4⋊C4 | semidirect |
| 194. Dic3 | C32×Dic3 | semidirect |
| 203. C32 | Dic3⋊Dic3 | direct (N × H) |
| 204. C32 | C3×Dic3⋊C4 | semidirect |
| 205. C32 | C3×C4⋊Dic3 | semidirect |
| 206. C32 | C32×C4⋊C4 | semidirect |
| 239. C3 | C3×Dic3⋊Dic3 | direct (N × H) |
| 240. C3 | C32×Dic3⋊C4 | semidirect |
| 241. C3 | C32×C4⋊Dic3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×C3⋊Dic3 | 2 |
| C2×C33×Dic3 | 2 |
| C32×C4⋊Dic3 | 3 |
| C32×Dic3⋊C4 | 3 |
| C3×Dic3⋊Dic3 | 3 |