Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
318 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C3 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | H(3,𝔽3) | C22×C3×C4 |
| Frattini | C2×C32 | C22×C3×D3 |
| Fitting, prime radical | C23×#732 | C2 |
| cosocle | C2×H(3,𝔽3) | C23×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#66325 | C2 | direct (N × H) |
| 3. C22×#1084 | C3 | semidirect |
| 5. C22×#732 | C4 | semidirect |
| 7. #66325 | C22 | direct (N × H) |
| 11. C2×#1084 | C2×C3 | semidirect |
| 13. C2×#732 | C2×C4 | semidirect |
| 22. C22×C3×C9 | Dic3 | semidirect |
| 24. #1084 | C22×C3 | semidirect |
| 26. #732 | C22×C4 | semidirect |
| 38. C2×C3×C9 | C2×Dic3 | semidirect |
| 56. C3×C9 | C22×Dic3 | semidirect |
| 98. C22 | #66325 | direct (N × H) |
| 105. C2 | C2×#66325 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#732 | 2 |
| C2×#66325 | 2 |
| C22×#1084 | 3 |
| C22×C9×Dic3 | 3 |