Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
540 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 75 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. C22×#4183 | C2 | semidirect |
| 2. C2×C4×AΣL(1,𝔽27) | C2 | semidirect |
| 3. C4⋊#4183 | C2 | direct (N × H) |
| 9. C2×#4183 | C22 | semidirect |
| 10. C4×AΣL(1,𝔽27) | C22 | semidirect |
| 12. C2×C33×C4 | D3 | semidirect |
| 16. C2×C4×3−1+2 | D3 | semidirect |
| 17. C2×C4×H(3,𝔽3) | D3 | semidirect |
| 20. C2×AΣL(1,𝔽27) | D4 | semidirect |
| 31. C33×C4 | C2×D3 | semidirect |
| 38. C4×3−1+2 | C2×D3 | semidirect |
| 39. C4×H(3,𝔽3) | C2×D3 | semidirect |
| 40. AΣL(1,𝔽27) | C2×D4 | semidirect |
| 56. C2×C33 | D12 | semidirect |
| 59. C2×3−1+2 | D12 | semidirect |
| 61. C2×H(3,𝔽3) | D12 | semidirect |
| 89. C33 | C2×D12 | semidirect |
| 90. 3−1+2 | C2×D12 | semidirect |
| 91. H(3,𝔽3) | C2×D12 | semidirect |
| 143. C2×C4 | #4183 | semidirect |
| 159. C4 | C2×#4183 | semidirect |
| 165. C2 | C4⋊#4183 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#4183 | 2 |
| C2×C4×AΣL(1,𝔽27) | 2 |
| C4⋊#4183 | 2 |
| C2×C3×C4⋊(C3⋊D3) | 3 |
| C2×C4⋊AGL(1,ℤ/9) | 3 |
| C2×C4⋊((C3⋊D3)⋊C3) | 3 |