Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1140 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 288 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, hypocenter | C32 | C22×C32×C4 |
| Frattini | C2 | C2×C32×D32 |
| Fitting, socle | C22×C34 | C22 |
| prime radical | C22×C32×C3⋊D3 | C2 |
| cosocle | C2×C32 | C23×C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C33×Dic3 | C2 | semidirect |
| 3. C32×#624 | C2 | direct (N × H) |
| 5. C2×C3×#624 | C3 | direct (N × H) |
| 7. C2×C32×C3⋊D3 | C4 | semidirect |
| 9. C33×Dic3 | C22 | semidirect |
| 15. C2×C32×Dic3 | D3 | semidirect |
| 16. C2×C32×Dic3 | C2×C3 | semidirect |
| 18. C3×#624 | C2×C3 | direct (N × H) |
| 19. C2×C34 | C2×C4 | semidirect |
| 21. C32×C3⋊D3 | C2×C4 | semidirect |
| 24. C2×#624 | C32 | direct (N × H) |
| 30. C2×C3×C3⋊D3 | C3×C4 | semidirect |
| 38. C32×Dic3 | C2×D3 | semidirect |
| 39. C32×Dic3 | C22×C3 | semidirect |
| 40. C34 | C22×C4 | semidirect |
| 50. C2×C3×Dic3 | C3×D3 | semidirect |
| 51. C2×C3×Dic3 | C2×C32 | semidirect |
| 53. #624 | C2×C32 | direct (N × H) |
| 60. C2×C33 | C4×D3 | semidirect |
| 61. C2×C33 | C2×C3×C4 | semidirect |
| 64. C3×C3⋊D3 | C2×C3×C4 | semidirect |
| 80. C2×C3⋊D3 | C32×C4 | semidirect |
| 90. C3×Dic3 | C2×C3×D3 | semidirect |
| 91. C3×Dic3 | C22×C32 | semidirect |
| 95. C33 | C2×C4×D3 | semidirect |
| 96. C33 | C22×C3×C4 | semidirect |
| 104. C2×Dic3 | C32×D3 | semidirect |
| 119. C2×C32 | #624 | direct (N × H) |
| 120. C2×C32 | C3×C4×D3 | semidirect |
| 121. C2×C32 | C2×C32×C4 | semidirect |
| 125. C3⋊D3 | C2×C32×C4 | semidirect |
| 143. Dic3 | C2×C32×D3 | semidirect |
| 150. C32 | C2×#624 | direct (N × H) |
| 151. C32 | C2×C3×C4×D3 | semidirect |
| 152. C32 | C22×C32×C4 | semidirect |
| 163. C2×C3 | C3×#624 | direct (N × H) |
| 164. C2×C3 | C32×C4×D3 | semidirect |
| 176. C3 | C2×C3×#624 | direct (N × H) |
| 177. C3 | C2×C32×C4×D3 | semidirect |
| 179. C2 | C32×#624 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×C3⋊D3 | 2 |
| C2×C33×Dic3 | 2 |
| C32×#624 | 2 |
| C2×C32×C4×D3 | 3 |
| C2×C3×#624 | 3 |