Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
248 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D27⋊C3 |
| commutator, cosocle | C2×C27 | C23×C3 |
| Frattini | C2×C9 | C22×C3×D3 |
| Fitting | C23×C27⋊C3 | C2 |
| socle | C22×C3 | C2×C3×D9 |
| hypercenter | C23 | D27⋊C3 |
| hypocenter | C27 | C2×C3×D4 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C27⋊C3 | C2 | semidirect |
| 2. C22×D27⋊C3 | C2 | semidirect |
| 3. #6171⋊C3 | C2 | direct (N × H) |
| 4. C2×(C27⋊C3)⋊C4 | C2 | semidirect |
| 6. C2×#6171 | C3 | semidirect |
| 9. C22×C27⋊C3 | C22 | semidirect |
| 11. C2×D27⋊C3 | C22 | semidirect |
| 12. (C27⋊C3)⋊C4 | C22 | semidirect |
| 18. C23×C27 | C2×C3 | semidirect |
| 19. C22×D27 | C2×C3 | semidirect |
| 20. #6171 | C2×C3 | semidirect |
| 21. C2×Dic27 | C2×C3 | semidirect |
| 23. C2×C27⋊C3 | D4 | semidirect |
| 39. C22×C27 | C22×C3 | semidirect |
| 41. C2×D27 | C22×C3 | semidirect |
| 42. Dic27 | C22×C3 | semidirect |
| 43. C27⋊C3 | C2×D4 | semidirect |
| 61. C2×C27 | C3×D4 | semidirect |
| 83. C27 | C2×C3×D4 | semidirect |
| 120. C23 | D27⋊C3 | semidirect |
| 135. C22 | C2×D27⋊C3 | semidirect |
| 141. C2 | #6171⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C27⋊C3 | 2 |
| C22×D27⋊C3 | 2 |
| #6171⋊C3 | 2 |
| C2×(C27⋊C3)⋊C4 | 2 |
| C2×C3×#611 | 3 |
| C2×#6171 | 3 |