Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
258 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×#157 |
| commutator, hypocenter | C33 | C2×C3×C8 |
| Frattini | C2 | C3×D3×#157 |
| Fitting, socle | C2×C34 | C2×C4 |
| prime radical | C2×C3×C3⋊(C3⋊D3) | C4 |
| cosocle | C3×C3⋊Dic3 | C22×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C3⋊#622 | C2 | semidirect |
| 3. C32×#622 | C2 | semidirect |
| 5. #88924 | C3 | direct (N × H) |
| 12. C3⋊#622 | C2×C3 | semidirect |
| 15. C3×#622 | D3 | semidirect |
| 16. C3×#622 | C2×C3 | semidirect |
| 18. C3×C3⋊(C3⋊D3) | C8 | semidirect |
| 33. C34 | C2×C8 | semidirect |
| 39. #622 | C3×D3 | semidirect |
| 49. C3⋊(C3⋊D3) | C3×C8 | semidirect |
| 77. C33 | D3×C8 | semidirect |
| 78. C33 | C2×C3×C8 | semidirect |
| 130. C32 | C3×D3×C8 | semidirect |
| 131. C32 | #3787 | semidirect |
| 154. C3 | #88924 | direct (N × H) |
| 155. C3 | C3×#3787 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6289 | 2 |
| C3×C3⋊#622 | 2 |
| C32×#622 | 2 |
| C3×#3787 | 3 |
| #88924 | 3 |
| C3×D3×C8 | 9 |