Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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 | #120 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #379 |
| prime radical | C3×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#413 | C2 | semidirect |
| 3. C3×#408 | C2 | semidirect |
| 4. #1791 | C3 | direct (N × H) |
| 10. C2×C3×M16 | C22 | semidirect |
| 11. C22×C3×C8 | C22 | semidirect |
| 16. #413 | C2×C3 | semidirect |
| 17. #408 | C2×C3 | semidirect |
| 40. C2×M16 | C22×C3 | semidirect |
| 41. C22×C8 | C22×C3 | semidirect |
| 113. C3 | #1791 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585 | 2 |
| C3×#413 | 2 |
| C3×#408 | 2 |
| #1791 | 3 |