Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
196 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | C22 |
| commutator | C2 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C22×C8 |
| prime radical | C23×C3 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#346 | C2 | semidirect |
| 4. C3×C4×C16 | C2 | semidirect |
| 5. C4×M32 | C3 | direct (N × H) |
| 9. C3×M32 | C4 | direct (N × H) |
| 17. #346 | C2×C3 | semidirect |
| 18. C4×C16 | C2×C3 | semidirect |
| 30. C3×C16 | C2×C4 | semidirect |
| 34. M32 | C3×C4 | direct (N × H) |
| 63. C16 | C2×C3×C4 | semidirect |
| 69. C3×C4 | M32 | direct (N × H) |
| 92. C4 | C3×M32 | direct (N × H) |
| 95. C3 | C4×M32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×M32 | 2 |
| C2×C3×C4×C8 | 2 |
| C3×#346 | 2 |
| C3×C4×C16 | 2 |
| C4×M32 | 3 |