Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
924 conjugacy classes of subgroups; only the 368 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 368 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| socle | C23×C3 | C24 |
| prime radical | C2×C3×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ 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×#384 | C2 | semidirect |
| 5. C3×C43 | C2 | semidirect |
| 6. C4×#127 | C3 | direct (N × H) |
| 9. C3×#127 | C4 | direct (N × H) |
| 17. C2×C4×D4 | C2×C3 | semidirect |
| 18. #390 | C2×C3 | semidirect |
| 19. #384 | C2×C3 | semidirect |
| 20. C43 | C2×C3 | semidirect |
| 24. C2×C3×D4 | C2×C4 | semidirect |
| 33. C3×C42 | C2×C4 | semidirect |
| 34. C3×C42 | D4 | semidirect |
| 37. #127 | C3×C4 | direct (N × H) |
| 60. C2×D4 | C2×C3×C4 | semidirect |
| 69. C42 | C2×C3×C4 | semidirect |
| 70. C42 | C3×D4 | semidirect |
| 79. C3×C4 | C4×D4 | semidirect |
| 80. C3×C4 | #127 | direct (N × H) |
| 106. C4 | C3×C4×D4 | semidirect |
| 107. C4 | C3×#127 | direct (N × H) |
| 108. C3 | C4×#127 | 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×#384 | 2 |
| C3×C43 | 2 |
| C4×#127 | 3 |