Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1340 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #4183 |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C24 |
| Frattini | C32 | C23×C3⋊D3 |
| Fitting | C23×AΣL(1,𝔽27) | C2 |
| socle | C23×C3 | #272 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×AΣL(1,𝔽27) | C2 | semidirect |
| 2. C22×#4183 | C2 | direct (N × H) |
| 6. C22×AΣL(1,𝔽27) | C22 | semidirect |
| 7. C2×#4183 | C22 | direct (N × H) |
| 8. C23×C33 | D3 | semidirect |
| 10. C23×3−1+2 | D3 | semidirect |
| 11. C23×H(3,𝔽3) | D3 | semidirect |
| 14. C2×AΣL(1,𝔽27) | C23 | semidirect |
| 15. #4183 | C23 | direct (N × H) |
| 21. C22×C33 | C2×D3 | semidirect |
| 23. C22×3−1+2 | C2×D3 | semidirect |
| 24. C22×H(3,𝔽3) | C2×D3 | semidirect |
| 27. AΣL(1,𝔽27) | C24 | semidirect |
| 38. C2×C33 | C22×D3 | semidirect |
| 40. C2×3−1+2 | C22×D3 | semidirect |
| 41. C2×H(3,𝔽3) | C22×D3 | semidirect |
| 58. C33 | C23×D3 | semidirect |
| 59. 3−1+2 | C23×D3 | semidirect |
| 60. H(3,𝔽3) | C23×D3 | semidirect |
| 94. C23 | #4183 | direct (N × H) |
| 104. C22 | C2×#4183 | direct (N × H) |
| 110. C2 | C22×#4183 | 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×#4183 | 2 |
| C23×C3×C3⋊D3 | 3 |
| C23×AGL(1,ℤ/9) | 3 |
| C23×(C3⋊D3)⋊C3 | 3 |