Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
162 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 19 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 | C2×C3 | (C3⋊D3)⋊A4 |
| commutator, Fitting, cosocle, hypocenter | Q8×H(3,𝔽3) | C2×C3 |
| Frattini | C2×C32 | D3×A4 |
| LCS | G ▷ Q8×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10619580 | C2 | semidirect |
| 3. Q8×#272 | C3 | semidirect |
| 5. C32×SL(2,𝔽3) | D3 | semidirect |
| 6. Q8×H(3,𝔽3) | C2×C3 | semidirect |
| 34. #272 | SL(2,𝔽3) | semidirect |
| 49. H(3,𝔽3) | C2×SL(2,𝔽3) | semidirect |
| 86. Q8 | #4174 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10619580 | 2 |
| C3×D3×SL(2,𝔽3) | 3 |
| Q8×#272 | 3 |
| C2×#4174 | 4 |