Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
360 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C23×D3 |
| commutator, socle | C2×C3 | C23×C8 |
| Fitting | C3×#504 | C2 |
| prime radical | D3×#41 | C4 |
| cosocle | C3×C8 | C24 |
| hypercenter | #504 | D3 |
| hypocenter | C3 | #3407 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ #504 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#504 | C2 | semidirect |
| 3. #5140 | C2 | semidirect |
| 4. #4907 | C2 | semidirect |
| 5. D3×M32 | C2 | semidirect |
| 6. #4901 | C2 | semidirect |
| 7. C2×#798 | C2 | semidirect |
| 16. C3×M32 | C22 | semidirect |
| 17. C2×C3×C16 | C22 | semidirect |
| 18. #812 | C22 | semidirect |
| 19. C2×C3⋊C16 | C22 | semidirect |
| 22. #798 | C22 | semidirect |
| 23. #798 | C22 | semidirect |
| 24. D3×C16 | C22 | semidirect |
| 29. #504 | D3 | semidirect |
| 64. M32 | C2×D3 | semidirect |
| 67. C2×C16 | C2×D3 | semidirect |
| 140. C3 | #3407 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×#131 | 2 |
| C3×#504 | 2 |
| #5140 | 2 |
| #4907 | 2 |
| D3×M32 | 2 |
| #4901 | 2 |
| C2×#798 | 2 |
| #3407 | 3 |