Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1056 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 150 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. C23×C3×H(3,𝔽3) | C2 | semidirect |
| 2. C22×C3×#272 | C2 | semidirect |
| 3. C3×#6235 | C2 | direct (N × H) |
| 4. C2×C3×#1084 | C2 | semidirect |
| 6. C2×#6235 | C3 | direct (N × H) |
| 9. C22×C3×H(3,𝔽3) | C22 | semidirect |
| 11. C2×C3×#272 | C22 | semidirect |
| 12. C3×#1084 | C22 | semidirect |
| 13. C23×C33 | D3 | semidirect |
| 18. C23×H(3,𝔽3) | C2×C3 | semidirect |
| 19. C22×#272 | C2×C3 | semidirect |
| 20. #6235 | C2×C3 | direct (N × H) |
| 21. C2×#1084 | C2×C3 | semidirect |
| 23. C2×C3×H(3,𝔽3) | D4 | semidirect |
| 31. C22×C33 | C2×D3 | semidirect |
| 40. C22×H(3,𝔽3) | C22×C3 | semidirect |
| 42. C2×#272 | C22×C3 | semidirect |
| 43. #1084 | C22×C3 | semidirect |
| 44. C3×H(3,𝔽3) | C2×D4 | semidirect |
| 48. C23×C32 | C3×D3 | semidirect |
| 59. C2×C33 | #67 | semidirect |
| 66. C2×H(3,𝔽3) | C3×D4 | semidirect |
| 83. C22×C32 | C2×C3×D3 | semidirect |
| 93. C33 | C2×#67 | semidirect |
| 95. H(3,𝔽3) | C2×C3×D4 | semidirect |
| 100. C23×C3 | #272 | semidirect |
| 118. C2×C32 | C3×#67 | semidirect |
| 138. C22×C3 | C2×#272 | semidirect |
| 148. C32 | C2×C3×#67 | semidirect |
| 151. C23 | C3×#272 | semidirect |
| 164. C2×C3 | #6235 | direct (N × H) |
| 173. C22 | C2×C3×#272 | semidirect |
| 177. C3 | C2×#6235 | direct (N × H) |
| 181. C2 | C3×#6235 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×H(3,𝔽3) | 2 |
| C22×C3×#272 | 2 |
| C3×#6235 | 2 |
| C2×C3×#1084 | 2 |
| C2×C32×#67 | 3 |
| C2×#6235 | 3 |