Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1160 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 360 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 | C23×C3×D4 | 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. C22×C3×C22⋊C4 | C2 | semidirect |
| 3. C3×#386 | C2 | semidirect |
| 4. C3×#381 | C2 | semidirect |
| 5. C3×#379 | C2 | semidirect |
| 6. #2308 | C3 | direct (N × H) |
| 10. C22×C3×D4 | C4 | semidirect |
| 13. C23×C3×C4 | C22 | semidirect |
| 16. C2×C3×C22⋊C4 | C22 | semidirect |
| 17. C2×C3×C22⋊C4 | C22 | semidirect |
| 18. C3×#95 | C22 | semidirect |
| 20. C22×C22⋊C4 | C2×C3 | semidirect |
| 21. #386 | C2×C3 | semidirect |
| 22. #381 | C2×C3 | semidirect |
| 23. #379 | C2×C3 | semidirect |
| 28. C24×C3 | C2×C4 | semidirect |
| 36. C2×C3×D4 | C2×C4 | semidirect |
| 40. C22×C3×C4 | C2×C4 | semidirect |
| 49. C22×D4 | C3×C4 | semidirect |
| 52. C23×C4 | C22×C3 | semidirect |
| 55. C2×C22⋊C4 | C22×C3 | semidirect |
| 56. C2×C22⋊C4 | C22×C3 | semidirect |
| 57. #95 | C22×C3 | semidirect |
| 67. C23×C3 | C22⋊C4 | semidirect |
| 86. C24 | C2×C3×C4 | semidirect |
| 94. C2×D4 | C2×C3×C4 | semidirect |
| 98. C22×C4 | C2×C3×C4 | semidirect |
| 130. C23 | C3×C22⋊C4 | semidirect |
| 168. C3 | #2308 | 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 |
| C22×C3×C22⋊C4 | 2 |
| C3×#386 | 2 |
| C3×#381 | 2 |
| C3×#379 | 2 |
| #2308 | 3 |