Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
680 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | A4 |
| commutator, hypocenter | C22 | C2×C3×C42 |
| Frattini | C22 | C23×A4 |
| Fitting | C23×C42 | C3 |
| socle | C25 | C22×C3 |
| prime radical | C23×A4 | C22 |
| cosocle | C24 | C23×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C42×A4 | C2 | direct (N × H) |
| 3. C23×C42 | C3 | semidirect |
| 5. C2×C4×A4 | C4 | direct (N × H) |
| 12. C22×C42 | C2×C3 | semidirect |
| 16. C4×A4 | C2×C4 | direct (N × H) |
| 23. C23×C4 | C3×C4 | semidirect |
| 31. C2×C42 | A4 | direct (N × H) |
| 33. C2×A4 | C42 | direct (N × H) |
| 54. C22×C4 | C2×C3×C4 | semidirect |
| 58. C42 | C2×A4 | direct (N × H) |
| 61. A4 | C2×C42 | direct (N × H) |
| 72. C23 | C3×C42 | semidirect |
| 83. C2×C4 | C4×A4 | direct (N × H) |
| 96. C22 | C2×C3×C42 | semidirect |
| 100. C4 | C2×C4×A4 | direct (N × H) |
| 105. C2 | C42×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×A4 | 2 |
| C42×A4 | 2 |
| C23×C42 | 3 |
| C2×C3×C42 | 4 |