Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
165 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×S4 |
| commutator | C2×A4 | C2×C8 |
| Fitting | C22×C42 | D3 |
| socle | C24 | C2×Dic3 |
| prime radical | C22×A4 | C2×C4 |
| cosocle | C2×C4×A4 | C22 |
| hypercenter | C42 | S4 |
| hypocenter | A4 | C4⋊C8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 16. C4×A4 | C8 | semidirect |
| 55. C22×C4 | C3⋊C8 | semidirect |
| 70. A4 | C4⋊C8 | semidirect |
| 118. C22 | C4⋊(C3⋊C8) | semidirect |
| 125. C4 | A4⋊C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C42×A4 | 2 |
| C2×A4⋊C8 | 2 |
| #1797 | 3 |
| C4⋊(C3⋊C8) | 4 |