Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
496 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #66431 |
| commutator, cosocle | C2×C32×C9 | C23 |
| Frattini | C2×C3 | C2×#1113 |
| Fitting | C22×C32×C9 | C22 |
| socle | C22×C33 | C2×D3 |
| hypocenter | C32×C9 | C2×D4 |
| LCS | G ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C3⋊D9 | C2 | semidirect |
| 2. C2×C9×C3⋊Dic3 | C2 | semidirect |
| 3. #10619897 | C2 | direct (N × H) |
| 8. C2×C3×C3⋊D9 | C22 | semidirect |
| 9. C9×C3⋊Dic3 | C22 | semidirect |
| 13. C22×C3⋊D9 | D3 | semidirect |
| 19. C2×C32×C9 | D4 | semidirect |
| 34. C2×C3⋊D9 | C2×D3 | semidirect |
| 41. C32×C9 | C2×D4 | semidirect |
| 46. C2×C3⋊Dic3 | D9 | semidirect |
| 64. C2×C3×C9 | #67 | semidirect |
| 84. C3⋊Dic3 | C2×D9 | semidirect |
| 99. C3×C9 | C2×#67 | semidirect |
| 115. C2×C32 | D36 | semidirect |
| 127. C2×C9 | #625 | semidirect |
| 151. C32 | C2×D36 | semidirect |
| 155. C9 | C2×#625 | semidirect |
| 166. C2×C3 | #6192 | semidirect |
| 180. C3 | C2×#6192 | semidirect |
| 183. C2 | #10619897 | 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⋊D9 | 2 |
| C2×C9×C3⋊Dic3 | 2 |
| #10619897 | 2 |
| C2×#6295 | 3 |
| C2×#6192 | 3 |