Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
351 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×S4 |
| commutator | C2×A4 | C22×C4 |
| Fitting | C23×Q8 | D3 |
| socle | C24 | C22×D3 |
| prime radical, cosocle | C22×A4 | C23 |
| hypercenter | C2×Q8 | S4 |
| hypocenter | A4 | C4×Q8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. Q8×A4 | C4 | semidirect |
| 21. SL(2,ℤ/4) | Q8 | direct (N × H) |
| 27. C22×Q8 | Dic3 | semidirect |
| 75. A4 | C4×Q8 | semidirect |
| 88. Q8 | SL(2,ℤ/4) | direct (N × H) |
| 120. C22 | Q8×Dic3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×A4 | 2 |
| C4⋊SL(2,ℤ/4) | 2 |
| C4×SL(2,ℤ/4) | 2 |
| Q8×C22⋊C4 | 3 |
| Q8×Dic3 | 4 |