Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
740 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 296 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×#127 | 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. C2×C3×C4×D4 | C2 | semidirect |
| 3. C3×#390 | C2 | semidirect |
| 4. C3×#386 | C2 | semidirect |
| 5. C3×#382 | C2 | semidirect |
| 6. C3×#376 | C2 | semidirect |
| 7. #2328 | C3 | direct (N × H) |
| 11. C3×#127 | C4 | semidirect |
| 22. C2×C4×D4 | C2×C3 | semidirect |
| 23. #390 | C2×C3 | semidirect |
| 24. #386 | C2×C3 | semidirect |
| 25. #382 | C2×C3 | semidirect |
| 26. #376 | C2×C3 | semidirect |
| 31. C2×C3×D4 | C2×C4 | semidirect |
| 32. C2×C3×D4 | C2×C4 | semidirect |
| 44. C3×C42 | C2×C4 | semidirect |
| 45. C3×C42 | D4 | semidirect |
| 49. #127 | C3×C4 | semidirect |
| 83. C2×D4 | C2×C3×C4 | semidirect |
| 84. C2×D4 | C2×C3×C4 | semidirect |
| 96. C42 | C2×C3×C4 | semidirect |
| 97. C42 | C3×D4 | semidirect |
| 151. C3 | #2328 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#127 | 2 |
| C2×C3×C4×D4 | 2 |
| C3×#390 | 2 |
| C3×#386 | 2 |
| C3×#382 | 2 |
| C3×#376 | 2 |
| #2328 | 3 |