Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
409 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 60 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. C2×#66363 | C2 | direct (N × H) |
| 2. C22×C32×A4 | C3 | semidirect |
| 3. C24×H(3,𝔽3) | C3 | semidirect |
| 4. C22×#1094 | C3 | semidirect |
| 6. #66363 | C22 | direct (N × H) |
| 7. C2×C32×A4 | C2×C3 | semidirect |
| 10. C23×H(3,𝔽3) | C2×C3 | semidirect |
| 11. C2×#1094 | C2×C3 | semidirect |
| 19. C32×A4 | C22×C3 | semidirect |
| 25. C22×H(3,𝔽3) | A4 | semidirect |
| 26. C22×H(3,𝔽3) | C22×C3 | semidirect |
| 27. #1094 | C22×C3 | semidirect |
| 42. C2×H(3,𝔽3) | C2×A4 | semidirect |
| 67. H(3,𝔽3) | C22×A4 | semidirect |
| 87. C24 | AΣL(1,𝔽27) | semidirect |
| 115. C23 | C2×AΣL(1,𝔽27) | semidirect |
| 130. C22 | #66363 | direct (N × H) |
| 131. C22 | C22×AΣL(1,𝔽27) | semidirect |
| 138. C2 | C2×#66363 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66363 | 2 |
| C22×C32×A4 | 3 |
| C24×H(3,𝔽3) | 3 |
| C22×#1094 | 3 |
| C22×AΣL(1,𝔽27) | 4 |