Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
127 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 26 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×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C8 |
| Frattini | C2×C4 | C2×S4 |
| Fitting | C8×Q8 | D3 |
| socle | C22 | C4×S4 |
| prime radical | C2×GL(2,𝔽3) | C4 |
| cosocle | C4×SL(2,𝔽3) | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C8×SL(2,𝔽3) | C2 | semidirect |
| 3. SL(2,𝔽3)⋊C8 | C2 | semidirect |
| 9. C8×Q8 | D3 | semidirect |
| 18. GL(2,𝔽3) | C8 | direct (N × H) |
| 40. SL(2,𝔽3) | C2×C8 | semidirect |
| 68. Q8 | D3×C8 | semidirect |
| 82. C8 | GL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×GL(2,𝔽3) | 2 |
| C8×SL(2,𝔽3) | 2 |
| SL(2,𝔽3)⋊C8 | 2 |
| C8×SD16 | 3 |
| C2×D3×C8 | 4 |