Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
112 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C22×D3 |
| commutator, socle | C2×C3 | C22×C16 |
| Fitting | C3×M64 | C2 |
| prime radical | C22×D3 | C16 |
| cosocle | C3×C16 | C23 |
| hypercenter | M64 | D3 |
| hypocenter | C3 | C2×M64 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C16 ◁ M64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×M64 | C2 | semidirect |
| 3. #4499 | C2 | semidirect |
| 4. #4447 | C2 | semidirect |
| 5. D3×C32 | C2 | semidirect |
| 14. C3×C32 | C22 | semidirect |
| 15. C3⋊C32 | C22 | semidirect |
| 19. M64 | D3 | direct (N × H) |
| 40. C32 | C2×D3 | semidirect |
| 85. D3 | M64 | direct (N × H) |
| 94. C3 | C2×M64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×C16 | 2 |
| C3×M64 | 2 |
| #4499 | 2 |
| #4447 | 2 |
| D3×C32 | 2 |
| C2×M64 | 3 |