Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1102 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 1102 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, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C2×C3×#120 | C2 |
| 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. C24×C3×C4 | C2 | semidirect |
| 2. C2×C3×#123 | C2 | semidirect |
| 4. C3×#386 | C2 | semidirect |
| 5. C3×#385 | C2 | semidirect |
| 6. C3×#381 | C2 | semidirect |
| 7. C3×#379 | C2 | semidirect |
| 8. #2849 | C3 | direct (N × H) |
| 18. C2×C3×C4⋊C4 | C22 | semidirect |
| 23. C2×C3×C22⋊C4 | C22 | semidirect |
| 24. C3×#95 | C22 | semidirect |
| 25. C24×C4 | C2×C3 | semidirect |
| 26. C2×#123 | C2×C3 | semidirect |
| 28. #386 | C2×C3 | semidirect |
| 29. #385 | C2×C3 | semidirect |
| 30. #381 | C2×C3 | semidirect |
| 31. #379 | C2×C3 | semidirect |
| 47. C22×C3×C4 | D4 | semidirect |
| 67. C2×C4⋊C4 | C22×C3 | semidirect |
| 72. C2×C22⋊C4 | C22×C3 | semidirect |
| 73. #95 | C22×C3 | semidirect |
| 116. C22×C4 | C3×D4 | semidirect |
| 137. C22×C3 | #123 | semidirect |
| 188. C22 | C3×#123 | semidirect |
| 196. C3 | #2849 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C4 | 2 |
| C2×C3×#123 | 2 |
| C2×C3×#120 | 2 |
| C3×#386 | 2 |
| C3×#385 | 2 |
| C3×#381 | 2 |
| C3×#379 | 2 |
| #2849 | 3 |