Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1104 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C3 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C24×C3 |
| Frattini | C32 | C23×C3×D3 |
| Fitting | C23×AΣL(1,𝔽27) | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23×C3 | |
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 | direct (N × H) |
| 4. C23×#272 | C3 | semidirect |
| 5. C22×AΣL(1,𝔽27) | C22 | semidirect |
| 6. C2×#4174 | C22 | direct (N × H) |
| 7. C23×C33 | D3 | semidirect |
| 10. C23×H(3,𝔽3) | C2×C3 | semidirect |
| 11. C22×#272 | C2×C3 | semidirect |
| 12. C2×AΣL(1,𝔽27) | C23 | semidirect |
| 13. #4174 | C23 | direct (N × H) |
| 18. C22×C33 | C2×D3 | semidirect |
| 21. C22×H(3,𝔽3) | C22×C3 | semidirect |
| 22. C2×#272 | C22×C3 | semidirect |
| 23. AΣL(1,𝔽27) | C24 | semidirect |
| 33. C2×C33 | C22×D3 | semidirect |
| 36. C2×H(3,𝔽3) | C23×C3 | semidirect |
| 37. #272 | C23×C3 | semidirect |
| 51. C33 | C23×D3 | semidirect |
| 53. H(3,𝔽3) | C24×C3 | semidirect |
| 86. C23 | #4174 | direct (N × H) |
| 96. C22 | C2×#4174 | direct (N × H) |
| 102. C2 | C22×#4174 | 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 |
| C23×C32×D3 | 3 |
| C23×#272 | 3 |