Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
911 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 911 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C22×#41 | D3 |
| 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 |
|---|---|---|
| 1. C22×#41⋊C3 | C2 | semidirect |
| 2. C2×#986 | C2 | direct (N × H) |
| 3. C22×GL(2,𝔽3) | C2 | semidirect |
| 6. C2×#41⋊C3 | C22 | semidirect |
| 7. C22×SL(2,𝔽3) | C22 | semidirect |
| 8. #986 | C22 | direct (N × H) |
| 9. C2×GL(2,𝔽3) | C22 | semidirect |
| 10. C22×#41 | D3 | semidirect |
| 20. #41⋊C3 | C23 | semidirect |
| 21. C2×SL(2,𝔽3) | C23 | semidirect |
| 22. GL(2,𝔽3) | C23 | semidirect |
| 25. C2×#41 | C2×D3 | semidirect |
| 45. SL(2,𝔽3) | C24 | semidirect |
| 52. #41 | C22×D3 | semidirect |
| 105. C22 | #986 | direct (N × H) |
| 113. C2 | C2×#986 | 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×#986 | 2 |
| C22×GL(2,𝔽3) | 2 |
| C22×AGL(1,ℤ/8) | 3 |
| C22×D12 | 4 |