Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1488 conjugacy classes of subgroups; only the 813 normal subgroups are shown.
This group has 1488 conjugacy classes of subgroups — too many to draw. Only its 813 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 | C3×#519 | C2 |
| socle, cosocle | C22×C3 | C25 |
| prime radical | C23×D3 | C23 |
| hypercenter | #519 | D3 |
| hypocenter | C3 | C2×#519 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#519 | C2 | semidirect |
| 3. #5812 | C2 | semidirect |
| 4. #5572 | C2 | semidirect |
| 5. C4×D3×Q8 | C2 | semidirect |
| 7. #5516 | C2 | semidirect |
| 10. C2×D3×Q8 | C4 | semidirect |
| 15. C3×C4×Q8 | C22 | semidirect |
| 17. Q8×Dic3 | C22 | semidirect |
| 21. #887 | C22 | semidirect |
| 26. C4×Dic6 | C22 | semidirect |
| 31. #519 | D3 | direct (N × H) |
| 35. C2×C3×Q8 | C2×C4 | semidirect |
| 40. D3×Q8 | C2×C4 | semidirect |
| 45. C2×Dic6 | C2×C4 | semidirect |
| 65. C4×Q8 | C2×D3 | semidirect |
| 95. C2×Q8 | C4×D3 | semidirect |
| 144. D3 | #519 | direct (N × H) |
| 157. C3 | C2×#519 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D3×Q8 | 2 |
| C3×#519 | 2 |
| #5812 | 2 |
| #5572 | 2 |
| C4×D3×Q8 | 2 |
| D3×#117 | 2 |
| #5516 | 2 |
| C2×#519 | 3 |