Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
464 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#66321 | C2 | direct (N × H) |
| 4. C22×#1084 | C3 | semidirect |
| 5. C22×AΣL(1,𝔽27) | C4 | semidirect |
| 7. #66321 | C22 | direct (N × H) |
| 12. C2×#1084 | C2×C3 | semidirect |
| 13. C2×AΣL(1,𝔽27) | C2×C4 | semidirect |
| 19. C22×C33 | Dic3 | semidirect |
| 24. C22×H(3,𝔽3) | C3×C4 | semidirect |
| 26. #1084 | C22×C3 | semidirect |
| 27. AΣL(1,𝔽27) | C22×C4 | semidirect |
| 37. C2×C33 | C2×Dic3 | semidirect |
| 41. C2×H(3,𝔽3) | C2×C3×C4 | semidirect |
| 61. C33 | C22×Dic3 | semidirect |
| 63. H(3,𝔽3) | C22×C3×C4 | semidirect |
| 110. C22 | #66321 | direct (N × H) |
| 117. C2 | C2×#66321 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×AΣL(1,𝔽27) | 2 |
| C2×#66321 | 2 |
| C22×C32×Dic3 | 3 |
| C22×#1084 | 3 |