Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
197 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C42 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C2×C4×Q8 | D3 |
| prime radical, cosocle | C22×SL(2,𝔽3) | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C4×SL(2,𝔽3) | C4 | semidirect |
| 7. #859 | C4 | direct (N × H) |
| 21. C4×Q8 | Dic3 | semidirect |
| 37. SL(2,𝔽3) | C42 | semidirect |
| 67. Q8 | C4×Dic3 | semidirect |
| 96. C4 | #859 | direct (N × H) |
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 |
| C4×#103 | 3 |
| C2×C4×Dic3 | 4 |