Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
177 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C42 |
| Fitting | C2×C4×Q8 | D3 |
| prime radical, cosocle | C22×SL(2,𝔽3) | C22 |
| hypercenter | C22×C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C4×SL(2,𝔽3) | C4 | semidirect |
| 6. #859 | C4 | semidirect |
| 20. C4×Q8 | Dic3 | semidirect |
| 35. SL(2,𝔽3) | C42 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×SL(2,𝔽3) | 2 |
| C2×#859 | 2 |
| #1763 | 3 |
| C3⋊#95 | 4 |