Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
359 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 70 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) | C22×C4 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | #41⋊C3 | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×GU(2,𝔽3) | C2 | direct (N × H) |
| 4. C2×#41⋊C3 | C4 | semidirect |
| 6. C22×SL(2,𝔽3) | C4 | semidirect |
| 8. GU(2,𝔽3) | C22 | direct (N × H) |
| 15. #41⋊C3 | C2×C4 | semidirect |
| 17. C2×SL(2,𝔽3) | C2×C4 | semidirect |
| 37. SL(2,𝔽3) | C22×C4 | semidirect |
| 98. C22 | GU(2,𝔽3) | direct (N × H) |
| 108. C2 | C2×GU(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41⋊C3 | 2 |
| C2×GU(2,𝔽3) | 2 |
| C22×#104 | 3 |
| C22×C3⋊C8 | 4 |