Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
743 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 743 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 |
| prime radical | C22×#41⋊C3 | C2 |
| 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 |
|---|---|---|
| 2. C2×#984 | C2 | direct (N × H) |
| 3. C22×#226 | C2 | semidirect |
| 8. #984 | C22 | direct (N × H) |
| 9. C2×#226 | C22 | semidirect |
| 22. #226 | C23 | semidirect |
| 96. C22 | #984 | direct (N × H) |
| 104. C2 | C2×#984 | 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×#984 | 2 |
| C22×#226 | 2 |
| C22×#137 | 3 |
| C22×Dic6 | 4 |