Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1564 conjugacy classes of subgroups; only the 280 normal subgroups are shown.
This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 280 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, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×D4 | C2 | semidirect |
| 2. C2×C3×#120 | C2 | semidirect |
| 3. C3×#392 | C2 | semidirect |
| 4. C3×#386 | C2 | semidirect |
| 5. C3×#379 | C2 | semidirect |
| 6. #2403 | C3 | direct (N × H) |
| 8. C25×C3 | C22 | semidirect |
| 12. C22×C3×D4 | C22 | semidirect |
| 13. C22×C3×D4 | C22 | semidirect |
| 14. C23×C3×C4 | C22 | semidirect |
| 16. C2×C3×C22⋊C4 | C22 | semidirect |
| 17. C2×C3×C22⋊C4 | C22 | semidirect |
| 18. C2×C3×C22⋊C4 | C22 | semidirect |
| 19. C3×#95 | C22 | semidirect |
| 20. C23×D4 | C2×C3 | semidirect |
| 21. C2×#120 | C2×C3 | semidirect |
| 22. #392 | C2×C3 | semidirect |
| 23. #386 | C2×C3 | semidirect |
| 24. #379 | C2×C3 | semidirect |
| 31. C24×C3 | D4 | semidirect |
| 43. C2×C3×D4 | D4 | semidirect |
| 52. C25 | C22×C3 | semidirect |
| 56. C22×D4 | C22×C3 | semidirect |
| 57. C22×D4 | C22×C3 | semidirect |
| 58. C23×C4 | C22×C3 | semidirect |
| 60. C2×C22⋊C4 | C22×C3 | semidirect |
| 61. C2×C22⋊C4 | C22×C3 | semidirect |
| 62. C2×C22⋊C4 | C22×C3 | semidirect |
| 63. #95 | C22×C3 | semidirect |
| 82. C23×C3 | C2×D4 | semidirect |
| 92. C2×C3×C4 | C2×D4 | semidirect |
| 99. C24 | C3×D4 | semidirect |
| 111. C2×D4 | C3×D4 | semidirect |
| 132. C22×C3 | #120 | semidirect |
| 156. C23 | C2×C3×D4 | semidirect |
| 166. C2×C4 | C2×C3×D4 | semidirect |
| 186. C22 | C3×#120 | semidirect |
| 192. C3 | #2403 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D4 | 2 |
| C2×C3×#120 | 2 |
| C3×#392 | 2 |
| C3×#386 | 2 |
| C3×#379 | 2 |
| #2403 | 3 |