Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
866 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 866 conjugacy classes of subgroups — too many to draw. Only its 189 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×D3×H(3,𝔽3) | C2 | semidirect |
| 3. #67×H(3,𝔽3) | C2 | direct (N × H) |
| 4. C2×Dic3×H(3,𝔽3) | C2 | semidirect |
| 5. C2×C32×#67 | C3 | semidirect |
| 9. C22×C3×H(3,𝔽3) | C22 | semidirect |
| 11. C2×D3×H(3,𝔽3) | C22 | semidirect |
| 12. Dic3×H(3,𝔽3) | C22 | semidirect |
| 13. C23×C33 | C2×C3 | semidirect |
| 14. C22×C32×D3 | C2×C3 | semidirect |
| 15. C32×#67 | C2×C3 | semidirect |
| 16. C2×C32×Dic3 | C2×C3 | semidirect |
| 19. C23×H(3,𝔽3) | D3 | semidirect |
| 23. C2×C3×H(3,𝔽3) | D4 | semidirect |
| 31. C22×C33 | C22×C3 | semidirect |
| 33. C2×C32×D3 | C22×C3 | semidirect |
| 34. C32×Dic3 | C22×C3 | semidirect |
| 42. C22×H(3,𝔽3) | C2×D3 | semidirect |
| 44. C3×H(3,𝔽3) | C2×D4 | semidirect |
| 49. C23×C32 | C3×D3 | semidirect |
| 60. C2×C33 | C3×D4 | semidirect |
| 68. C2×H(3,𝔽3) | #67 | direct (N × H) |
| 72. C2×#67 | H(3,𝔽3) | direct (N × H) |
| 87. C22×C32 | C2×C3×D3 | semidirect |
| 97. C33 | C2×C3×D4 | semidirect |
| 99. H(3,𝔽3) | C2×#67 | direct (N × H) |
| 105. C23×C3 | C2×H(3,𝔽3) | semidirect |
| 107. C22×D3 | C2×H(3,𝔽3) | semidirect |
| 112. #67 | C2×H(3,𝔽3) | direct (N × H) |
| 113. C2×Dic3 | C2×H(3,𝔽3) | semidirect |
| 126. C2×C32 | C3×#67 | semidirect |
| 149. C22×C3 | C22×H(3,𝔽3) | semidirect |
| 153. C2×D3 | C22×H(3,𝔽3) | semidirect |
| 156. Dic3 | C22×H(3,𝔽3) | semidirect |
| 160. C32 | C2×C3×#67 | semidirect |
| 163. C23 | D3×H(3,𝔽3) | semidirect |
| 179. C2×C3 | D4×H(3,𝔽3) | semidirect |
| 188. C22 | C2×D3×H(3,𝔽3) | semidirect |
| 193. C3 | C2×D4×H(3,𝔽3) | semidirect |
| 197. C2 | #67×H(3,𝔽3) | 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×D3×H(3,𝔽3) | 2 |
| #67×H(3,𝔽3) | 2 |
| C2×Dic3×H(3,𝔽3) | 2 |
| C2×C32×#67 | 3 |
| C2×D4×H(3,𝔽3) | 3 |