Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
251 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C22 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| prime radical | C3×D3×SL(2,𝔽3) | C2 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3⋊(C3⋊#226) | C2 | semidirect |
| 3. C3×C3⋊#226 | C2 | semidirect |
| 17. C3⋊#226 | D3 | direct (N × H) |
| 112. C3×D3 | #226 | semidirect |
| 140. C32 | C2×#226 | semidirect |
| 157. D3 | C3⋊#226 | direct (N × H) |
| 166. C3 | D3×#226 | semidirect |
| 167. C3 | C2×C3⋊#226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D3×SL(2,𝔽3) | 2 |
| C3⋊(C3⋊#226) | 2 |
| C3×C3⋊#226 | 2 |
| C2×C3⋊#226 | 3 |
| D3×#226 | 3 |
| D3×#216 | 3 |
| D3×C3⋊Dic3 | 4 |