Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
456 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 456 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 | C22 | D4×A4 |
| commutator, cosocle | C22×C4 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | C23×D8 | C3 |
| socle | C24 | C3×D4 |
| hypercenter | C2×D8 | A4 |
| hypocenter | C22 | C2×C3×D8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×A4 | C2 | semidirect |
| 2. A4×D8 | C2 | direct (N × H) |
| 3. C2×C8×A4 | C2 | semidirect |
| 4. C23×D8 | C3 | semidirect |
| 7. D4×A4 | C22 | semidirect |
| 10. C8×A4 | C22 | semidirect |
| 11. C23×D4 | C2×C3 | semidirect |
| 16. C22×D8 | C2×C3 | semidirect |
| 17. C23×C8 | C2×C3 | semidirect |
| 33. C22×D4 | C22×C3 | semidirect |
| 44. C2×D8 | A4 | direct (N × H) |
| 48. C22×C8 | C22×C3 | semidirect |
| 51. C2×A4 | D8 | direct (N × H) |
| 78. C2×D4 | C2×A4 | semidirect |
| 88. D8 | C2×A4 | direct (N × H) |
| 94. C2×C8 | C2×A4 | semidirect |
| 98. A4 | C2×D8 | direct (N × H) |
| 118. C23 | C3×D8 | semidirect |
| 127. D4 | C22×A4 | semidirect |
| 138. C8 | C22×A4 | semidirect |
| 159. C22 | C2×C3×D8 | semidirect |
| 171. C2 | A4×D8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×A4 | 2 |
| A4×D8 | 2 |
| C2×C8×A4 | 2 |
| C23×D8 | 3 |
| C2×C3×D8 | 4 |