Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
464 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 70 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 |
|---|---|---|
| 1. C23×AΣL(1,𝔽27) | C2 | semidirect |
| 2. C22×#4174 | C2 | semidirect |
| 3. #10619641 | C2 | direct (N × H) |
| 4. C2×#66321 | C2 | semidirect |
| 6. C2×#6235 | C3 | semidirect |
| 9. C22×AΣL(1,𝔽27) | C22 | semidirect |
| 11. C2×#4174 | C22 | semidirect |
| 12. #66321 | C22 | semidirect |
| 13. C23×C33 | D3 | semidirect |
| 18. C23×H(3,𝔽3) | C2×C3 | semidirect |
| 19. C22×#272 | C2×C3 | semidirect |
| 20. #6235 | C2×C3 | semidirect |
| 21. C2×#1084 | C2×C3 | semidirect |
| 23. C2×AΣL(1,𝔽27) | D4 | semidirect |
| 32. C22×C33 | C2×D3 | semidirect |
| 41. C22×H(3,𝔽3) | C22×C3 | semidirect |
| 43. C2×#272 | C22×C3 | semidirect |
| 44. #1084 | C22×C3 | semidirect |
| 45. AΣL(1,𝔽27) | C2×D4 | semidirect |
| 64. C2×C33 | #67 | semidirect |
| 71. C2×H(3,𝔽3) | C3×D4 | semidirect |
| 105. C33 | C2×#67 | semidirect |
| 107. H(3,𝔽3) | C2×C3×D4 | semidirect |
| 174. C23 | #4174 | semidirect |
| 197. C22 | C2×#4174 | semidirect |
| 205. C2 | #10619641 | 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 |
| C22×#4174 | 2 |
| #10619641 | 2 |
| C2×#66321 | 2 |
| C2×C32×#67 | 3 |
| C2×#6235 | 3 |