Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
556 conjugacy classes of subgroups; only the 504 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 504 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator | C2 | C22×C3×C42 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C23×C4 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#405 | C2 | semidirect |
| 4. C3×C4×M16 | C2 | semidirect |
| 5. C2×C3×#97 | C2 | semidirect |
| 6. #2875 | C3 | direct (N × H) |
| 8. C3×#131 | C4 | semidirect |
| 21. #405 | C2×C3 | semidirect |
| 22. C4×M16 | C2×C3 | semidirect |
| 23. C2×#97 | C2×C3 | semidirect |
| 29. C3×M16 | C2×C4 | semidirect |
| 30. C2×C3×C8 | C2×C4 | semidirect |
| 39. #131 | C3×C4 | semidirect |
| 66. M16 | C2×C3×C4 | semidirect |
| 67. C2×C8 | C2×C3×C4 | semidirect |
| 103. C3 | #2875 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#131 | 2 |
| C3×C4×#41 | 2 |
| C3×#405 | 2 |
| C3×C4×M16 | 2 |
| C2×C3×#97 | 2 |
| #2875 | 3 |