Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
171 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 40 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 | C42 |
| Fitting | C23×C8 | D3 |
| socle | C24 | C4×D3 |
| prime radical | C22×A4 | C2×C4 |
| cosocle | C2×C4×A4 | C22 |
| hypercenter | C2×C8 | S4 |
| hypocenter | A4 | #97 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C8×A4 | C4 | semidirect |
| 8. A4⋊C8 | C4 | semidirect |
| 30. C22×C8 | Dic3 | semidirect |
| 74. A4 | #97 | semidirect |
| 101. C8 | SL(2,ℤ/4) | semidirect |
| 111. C22 | #815 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×A4 | 2 |
| C4×SL(2,ℤ/4) | 2 |
| C2×A4⋊C8 | 2 |
| #1752 | 3 |
| #815 | 4 |