Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
540 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 61 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. D4×AΣL(1,𝔽27) | C2 | semidirect |
| 3. #10619679 | C2 | semidirect |
| 4. C4⋊#4183 | C2 | semidirect |
| 5. C4×#4183 | C2 | semidirect |
| 9. C22×AΣL(1,𝔽27) | C22 | semidirect |
| 12. C2×#4183 | C22 | semidirect |
| 13. C4×AΣL(1,𝔽27) | C22 | semidirect |
| 14. #66330 | C22 | semidirect |
| 16. C33×D4 | D3 | semidirect |
| 22. D4×3−1+2 | D3 | semidirect |
| 23. D4×H(3,𝔽3) | D3 | semidirect |
| 33. #4183 | D4 | direct (N × H) |
| 39. C22×C33 | C2×D3 | semidirect |
| 43. C33×C4 | C2×D3 | semidirect |
| 45. C22×3−1+2 | C2×D3 | semidirect |
| 46. C22×H(3,𝔽3) | C2×D3 | semidirect |
| 53. C4×3−1+2 | C2×D3 | semidirect |
| 54. C4×H(3,𝔽3) | C2×D3 | semidirect |
| 57. AΣL(1,𝔽27) | C2×D4 | semidirect |
| 129. C33 | D3×D4 | semidirect |
| 130. 3−1+2 | D3×D4 | semidirect |
| 131. H(3,𝔽3) | D3×D4 | semidirect |
| 205. D4 | #4183 | direct (N × H) |
| 226. C22 | C2×#4183 | semidirect |
| 228. C4 | C2×#4183 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#4183 | 2 |
| D4×AΣL(1,𝔽27) | 2 |
| #10619679 | 2 |
| C4⋊#4183 | 2 |
| C4×#4183 | 2 |
| C3×D4×C3⋊D3 | 3 |
| D4×AGL(1,ℤ/9) | 3 |
| D4×(C3⋊D3)⋊C3 | 3 |