Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
200 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | A4 |
| commutator, hypocenter | C22 | C3×C4×C8 |
| Frattini | C2×C4 | C22×A4 |
| Fitting | C22×C4×C8 | C3 |
| socle | C24 | C2×C3×C4 |
| prime radical | C22×A4 | C2×C4 |
| cosocle | C23×C4 | C22×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C4×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C4×C8 | C3 | semidirect |
| 6. C8×A4 | C4 | direct (N × H) |
| 14. C4×A4 | C8 | direct (N × H) |
| 23. C22×C8 | C3×C4 | semidirect |
| 30. C4×C8 | A4 | direct (N × H) |
| 43. C22×C4 | C3×C8 | semidirect |
| 56. A4 | C4×C8 | direct (N × H) |
| 77. C8 | C4×A4 | direct (N × H) |
| 87. C22 | C3×C4×C8 | semidirect |
| 91. C4 | C8×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×A4 | 2 |
| C42×A4 | 2 |
| C22×C4×C8 | 3 |
| C3×C4×C8 | 4 |