Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1655 conjugacy classes of subgroups; only the 252 normal subgroups are shown.
This group has 1655 conjugacy classes of subgroups — too many to draw. Only its 252 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | S4 |
| commutator, hypocenter | A4 | C23×C4 |
| Frattini | C2 | C23×S4 |
| Fitting, socle | C26 | D3 |
| prime radical | C24×A4 | C2 |
| cosocle | C2×A4 | C24 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×SL(2,ℤ/4) | C2 | direct (N × H) |
| 4. C23×A4 | C4 | semidirect |
| 7. C2×SL(2,ℤ/4) | C22 | direct (N × H) |
| 14. C22×A4 | C2×C4 | semidirect |
| 17. SL(2,ℤ/4) | C23 | direct (N × H) |
| 21. C25 | Dic3 | semidirect |
| 32. C2×A4 | C22×C4 | semidirect |
| 44. C24 | C2×Dic3 | semidirect |
| 56. A4 | C23×C4 | semidirect |
| 69. C23 | SL(2,ℤ/4) | direct (N × H) |
| 70. C23 | C22×Dic3 | semidirect |
| 89. C22 | C2×SL(2,ℤ/4) | direct (N × H) |
| 90. C22 | C23×Dic3 | semidirect |
| 98. C2 | C22×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 |
|---|---|
| C24×A4 | 2 |
| C22×SL(2,ℤ/4) | 2 |
| C23×C22⋊C4 | 3 |
| C23×Dic3 | 4 |