Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
403 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 92 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 | C22×C8 |
| Frattini | C4 | C22×S4 |
| Fitting | C24×C4 | D3 |
| socle | C25 | Dic3 |
| prime radical | C23×A4 | C4 |
| cosocle | C4×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×A4⋊C8 | C2 | direct (N × H) |
| 8. A4⋊C8 | C22 | direct (N × H) |
| 14. C22×A4 | C8 | semidirect |
| 36. C2×A4 | C2×C8 | semidirect |
| 44. C24 | C3⋊C8 | semidirect |
| 70. A4 | C22×C8 | semidirect |
| 82. C23 | C2×C3⋊C8 | semidirect |
| 111. C22 | A4⋊C8 | direct (N × H) |
| 112. C22 | C22×C3⋊C8 | semidirect |
| 122. C2 | C2×A4⋊C8 | 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 |
| C2×A4⋊C8 | 2 |
| C22×C22⋊C8 | 3 |
| C22×C3⋊C8 | 4 |