Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
579 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 579 conjugacy classes of subgroups — too many to draw. Only its 108 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, hypocenter | A4 | C2×C42 |
| Frattini | C22 | C22×S4 |
| Fitting | C24×C4 | D3 |
| socle | C25 | C2×D3 |
| prime radical | C23×A4 | C22 |
| cosocle | C22×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×SL(2,ℤ/4) | C2 | direct (N × H) |
| 6. C2×C4×A4 | C4 | semidirect |
| 8. C2×SL(2,ℤ/4) | C4 | direct (N × H) |
| 23. C4×A4 | C2×C4 | semidirect |
| 24. SL(2,ℤ/4) | C2×C4 | direct (N × H) |
| 33. C23×C4 | Dic3 | semidirect |
| 47. C2×A4 | C42 | semidirect |
| 82. C22×C4 | C2×Dic3 | semidirect |
| 93. A4 | C2×C42 | semidirect |
| 111. C23 | C4×Dic3 | semidirect |
| 132. C2×C4 | SL(2,ℤ/4) | direct (N × H) |
| 152. C22 | C2×C4×Dic3 | semidirect |
| 159. C4 | C2×SL(2,ℤ/4) | direct (N × H) |
| 165. C2 | C4×SL(2,ℤ/4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×A4 | 2 |
| C22×SL(2,ℤ/4) | 2 |
| C4×SL(2,ℤ/4) | 2 |
| C2×C4×C22⋊C4 | 3 |
| C2×C4×Dic3 | 4 |