Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
803 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 803 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, hypercenter | C22×C4 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C2×S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ 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×#985 | C2 | direct (N × H) |
| 3. C22×GL(2,𝔽3) | C2 | semidirect |
| 4. C22×#226 | C2 | semidirect |
| 7. C2×#41⋊C3 | C22 | semidirect |
| 9. #985 | C22 | direct (N × H) |
| 10. C2×GL(2,𝔽3) | C22 | semidirect |
| 11. C2×#226 | C22 | semidirect |
| 13. C22×#41 | D3 | semidirect |
| 25. #41⋊C3 | C23 | semidirect |
| 27. GL(2,𝔽3) | C23 | semidirect |
| 28. #226 | C23 | semidirect |
| 32. C2×#41 | C2×D3 | semidirect |
| 64. #41 | C22×D3 | semidirect |
| 126. C22 | #985 | direct (N × H) |
| 135. C2 | C2×#985 | 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×#985 | 2 |
| C22×GL(2,𝔽3) | 2 |
| C22×#226 | 2 |
| C22×#135 | 3 |
| C22×C4×D3 | 4 |