Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
355 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 355 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, socle | C23 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini | C22 | C22×S4 |
| Fitting | C23×Q8 | D3 |
| prime radical | C23×SL(2,𝔽3) | C2 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C24 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5420 | C2 | direct (N × H) |
| 3. C2×#859 | C2 | semidirect |
| 5. C22×SL(2,𝔽3) | C4 | semidirect |
| 9. #859 | C22 | semidirect |
| 20. C2×SL(2,𝔽3) | C2×C4 | semidirect |
| 26. C22×Q8 | Dic3 | semidirect |
| 59. C2×Q8 | C2×Dic3 | semidirect |
| 122. C2 | #5420 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SL(2,𝔽3) | 2 |
| #5420 | 2 |
| C2×#859 | 2 |
| C2×#419 | 3 |
| C2×#217 | 4 |