Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1304 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×A4 |
| commutator, cosocle | C23 | C24×C3 |
| Frattini | C2 | C24×A4 |
| Fitting | C24×Q8 | C3 |
| socle | C25 | C22×C3 |
| prime radical | C23×A4 | C22 |
| hypercenter | C22×Q8 | A4 |
| hypocenter | C22 | C22×C3×Q8 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×A4 | C2 | direct (N × H) |
| 3. C24×Q8 | C3 | semidirect |
| 6. Q8×A4 | C22 | direct (N × H) |
| 11. C23×Q8 | C2×C3 | semidirect |
| 13. C22×A4 | Q8 | direct (N × H) |
| 27. C22×Q8 | A4 | direct (N × H) |
| 28. C22×Q8 | C22×C3 | semidirect |
| 34. C2×A4 | C2×Q8 | direct (N × H) |
| 41. C24 | C3×Q8 | semidirect |
| 49. C2×Q8 | C2×A4 | direct (N × H) |
| 61. A4 | C22×Q8 | direct (N × H) |
| 72. C23 | C2×C3×Q8 | semidirect |
| 77. Q8 | C22×A4 | direct (N × H) |
| 95. C22 | Q8×A4 | direct (N × H) |
| 96. C22 | C22×C3×Q8 | semidirect |
| 104. C2 | C2×Q8×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×A4 | 2 |
| C22×C4×A4 | 2 |
| C24×Q8 | 3 |
| C22×C3×Q8 | 4 |