Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
288 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 74 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 |
|---|---|---|
| 4. C3×#3756 | C3 | semidirect |
| 6. #3756⋊C3 | C3 | direct (N × H) |
| 23. #3756 | C32 | semidirect |
| 40. C3×H(3,𝔽3) | Q16 | semidirect |
| 99. C33 | #216 | semidirect |
| 100. C33 | C3×Q16 | semidirect |
| 102. H(3,𝔽3) | C3×Q16 | semidirect |
| 157. C32 | C3×#216 | semidirect |
| 158. C32 | C32×Q16 | semidirect |
| 177. C3 | #3756⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q8×H(3,𝔽3) | 2 |
| C3×#634⋊C3 | 2 |
| C3×(C3⋊(C3⋊C8))⋊C3 | 2 |
| C3×#3756 | 3 |
| C32×#216 | 3 |
| #3756⋊C3 | 3 |