Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1492 conjugacy classes of subgroups; only the 1063 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 1063 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator | C2×C3 | C24×C4 |
| Fitting, prime radical | C2×C3×2−1+4 | C2 |
| socle, cosocle | C22×C3 | C25 |
| hypercenter | C2×2−1+4 | D3 |
| hypocenter | C3 | C4×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5827 | C2 | semidirect |
| 3. Dic3×#41 | C2 | semidirect |
| 4. #5812 | C2 | semidirect |
| 5. C2×Q8×Dic3 | C2 | semidirect |
| 7. C3×2−1+4 | C4 | semidirect |
| 23. C3×#41 | C2×C4 | semidirect |
| 24. C2×C3×Q8 | C2×C4 | semidirect |
| 35. 2−1+4 | Dic3 | direct (N × H) |
| 56. #41 | C2×Dic3 | semidirect |
| 57. C2×Q8 | C2×Dic3 | semidirect |
| 75. Dic3 | 2−1+4 | direct (N × H) |
| 98. C3 | C4×2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×2−1+4 | 2 |
| #5827 | 2 |
| Dic3×#41 | 2 |
| #5812 | 2 |
| C2×Q8×Dic3 | 2 |
| C4×2−1+4 | 3 |