Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
688 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C2×C3×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#390 | C2 | semidirect |
| 3. C3×#388 | C2 | semidirect |
| 4. C3×#386 | C2 | semidirect |
| 5. C3×#379 | C2 | semidirect |
| 6. #2375 | C3 | direct (N × H) |
| 9. C23×C3×C4 | C22 | semidirect |
| 10. C3×#120 | C4 | semidirect |
| 11. C2×C3×C4⋊C4 | C22 | semidirect |
| 15. C2×C3×C22⋊C4 | C22 | semidirect |
| 16. C2×C3×C42 | C22 | semidirect |
| 17. C3×#95 | C22 | semidirect |
| 19. #390 | C2×C3 | semidirect |
| 20. #388 | C2×C3 | semidirect |
| 21. #386 | C2×C3 | semidirect |
| 22. #379 | C2×C3 | semidirect |
| 25. C24×C3 | C2×C4 | semidirect |
| 39. C3×C22⋊C4 | C2×C4 | semidirect |
| 43. C23×C4 | C22×C3 | semidirect |
| 44. #120 | C3×C4 | semidirect |
| 45. C2×C4⋊C4 | C22×C3 | semidirect |
| 49. C2×C22⋊C4 | C22×C3 | semidirect |
| 50. C2×C42 | C22×C3 | semidirect |
| 51. #95 | C22×C3 | semidirect |
| 75. C24 | C2×C3×C4 | semidirect |
| 89. C22⋊C4 | C2×C3×C4 | semidirect |
| 147. C3 | #2375 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#120 | 2 |
| C3×#390 | 2 |
| C3×#388 | 2 |
| C3×#386 | 2 |
| C3×#379 | 2 |
| #2375 | 3 |