Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1348 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 155 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⋊#1113 |
| commutator, cosocle | C2×C34 | C23 |
| Frattini | C2 | C2×C3⋊#1113 |
| Fitting, socle | C22×C34 | C22 |
| hypocenter | C34 | C2×D4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C3⋊(C3⋊D3) | C2 | semidirect |
| 2. C2×C32×C3⋊Dic3 | C2 | semidirect |
| 3. #10620156 | C2 | direct (N × H) |
| 8. C2×C3×C3⋊(C3⋊D3) | C22 | semidirect |
| 9. C32×C3⋊Dic3 | C22 | semidirect |
| 10. C22×C3⋊(C3⋊D3) | D3 | semidirect |
| 14. C2×C3×C3⋊Dic3 | D3 | semidirect |
| 19. C2×C34 | D4 | semidirect |
| 31. C2×C3⋊(C3⋊D3) | C2×D3 | semidirect |
| 38. C3×C3⋊Dic3 | C2×D3 | semidirect |
| 41. C34 | C2×D4 | semidirect |
| 49. C2×C3⋊Dic3 | C3⋊D3 | semidirect |
| 61. C2×C33 | D12 | semidirect |
| 62. C2×C33 | #67 | semidirect |
| 93. C3⋊Dic3 | C2×C3⋊D3 | semidirect |
| 99. C33 | C2×D12 | semidirect |
| 100. C33 | C2×#67 | semidirect |
| 122. C2×C32 | #625 | semidirect |
| 123. C2×C32 | #626 | semidirect |
| 124. C2×C32 | C4⋊(C3⋊D3) | semidirect |
| 157. C32 | C2×#625 | semidirect |
| 158. C32 | C2×#626 | semidirect |
| 159. C32 | C2×C4⋊(C3⋊D3) | semidirect |
| 170. C2×C3 | #6292 | semidirect |
| 171. C2×C3 | #6295 | semidirect |
| 184. C3 | C2×#6292 | semidirect |
| 185. C3 | C2×#6295 | semidirect |
| 187. C2 | #10620156 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C3⋊(C3⋊D3) | 2 |
| C2×C32×C3⋊Dic3 | 2 |
| #10620156 | 2 |
| C2×#6295 | 3 |
| C2×#6292 | 3 |