Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
313 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | C2×H(5,𝔽2) | D3 |
| prime radical | C2×#994 | C2 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C2×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5426 | C2 | semidirect |
| 3. #5424 | C2 | semidirect |
| 5. #994 | C4 | semidirect |
| 8. #859 | C22 | semidirect |
| 9. #859 | C22 | semidirect |
| 18. #41⋊C3 | C2×C4 | semidirect |
| 23. H(5,𝔽2) | Dic3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#994 | 2 |
| #5426 | 2 |
| #5424 | 2 |
| #2897 | 3 |
| Q8×Dic3 | 4 |