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×#4175 | C2 | semidirect |
| 3. #10619642 | C2 | direct (N × H) |
| 4. C2×#66322 | C2 | semidirect |
| 5. C2×#6312 | C3 | semidirect |
| 9. C22×AΣL(1,𝔽27) | C22 | semidirect |
| 11. C2×#4175 | C22 | semidirect |
| 12. #66322 | C22 | semidirect |
| 13. C23×C33 | C2×C3 | semidirect |
| 14. C22×C3⋊(C3⋊D3) | C2×C3 | semidirect |
| 15. #6312 | C2×C3 | semidirect |
| 16. C2×C3⋊(C3⋊Dic3) | C2×C3 | semidirect |
| 18. C23×H(3,𝔽3) | D3 | semidirect |
| 23. C2×AΣL(1,𝔽27) | D4 | semidirect |
| 32. C22×C33 | C22×C3 | semidirect |
| 34. C2×C3⋊(C3⋊D3) | C22×C3 | semidirect |
| 35. C3⋊(C3⋊Dic3) | C22×C3 | semidirect |
| 41. C22×H(3,𝔽3) | C2×D3 | semidirect |
| 45. AΣL(1,𝔽27) | C2×D4 | semidirect |
| 64. C2×C33 | C3×D4 | semidirect |
| 71. C2×H(3,𝔽3) | #67 | semidirect |
| 105. C33 | C2×C3×D4 | semidirect |
| 107. H(3,𝔽3) | C2×#67 | semidirect |
| 174. C23 | #4175 | semidirect |
| 197. C22 | C2×#4175 | semidirect |
| 205. C2 | #10619642 | 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×#4175 | 2 |
| #10619642 | 2 |
| C2×#66322 | 2 |
| C2×#6312 | 3 |
| C2×#638⋊C3 | 3 |