Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
232 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #102 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×M32 | C2 | semidirect |
| 3. C2×C3×#106 | C2 | semidirect |
| 4. C3×#360 | C2 | direct (N × H) |
| 5. C2×#360 | C3 | direct (N × H) |
| 9. C3×#135 | C4 | semidirect |
| 11. C2×C3×Q16 | C4 | semidirect |
| 13. C2×C3×D8 | C4 | semidirect |
| 17. C3×M32 | C22 | semidirect |
| 20. C3×#106 | C22 | semidirect |
| 22. C2×M32 | C2×C3 | semidirect |
| 23. C2×#106 | C2×C3 | semidirect |
| 24. #360 | C2×C3 | direct (N × H) |
| 36. C3×Q16 | C2×C4 | semidirect |
| 40. C3×D8 | C2×C4 | semidirect |
| 53. #135 | C3×C4 | semidirect |
| 55. C2×Q16 | C3×C4 | semidirect |
| 57. C2×D8 | C3×C4 | semidirect |
| 61. M32 | C22×C3 | semidirect |
| 64. #106 | C22×C3 | semidirect |
| 99. Q16 | C2×C3×C4 | semidirect |
| 103. D8 | C2×C3×C4 | semidirect |
| 151. C2×C3 | #360 | direct (N × H) |
| 165. C3 | C2×#360 | direct (N × H) |
| 169. C2 | C3×#360 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#135 | 2 |
| C2×C3×M32 | 2 |
| C2×C3×#106 | 2 |
| C3×#360 | 2 |
| C2×#360 | 3 |