Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1504 conjugacy classes of subgroups; only the 1384 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 1384 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×C8 | C22 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C4 | C25×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#131 | C2 | direct (N × H) |
| 3. C22×C3×M16 | C2 | semidirect |
| 4. C23×C3×C8 | C2 | semidirect |
| 5. C22×#131 | C3 | direct (N × H) |
| 11. C3×#131 | C22 | direct (N × H) |
| 12. C2×C3×M16 | C22 | semidirect |
| 15. C22×C3×C8 | C22 | semidirect |
| 17. C2×#131 | C2×C3 | direct (N × H) |
| 18. C22×M16 | C2×C3 | semidirect |
| 19. C23×C8 | C2×C3 | semidirect |
| 29. C3×M16 | C23 | semidirect |
| 32. C2×C3×C8 | C23 | semidirect |
| 38. #131 | C22×C3 | direct (N × H) |
| 39. C2×M16 | C22×C3 | semidirect |
| 42. C22×C8 | C22×C3 | semidirect |
| 64. M16 | C23×C3 | semidirect |
| 67. C2×C8 | C23×C3 | semidirect |
| 69. C22×C3 | #131 | direct (N × H) |
| 88. C2×C3 | C2×#131 | direct (N × H) |
| 91. C22 | C3×#131 | direct (N × H) |
| 97. C3 | C22×#131 | direct (N × H) |
| 99. C2 | C2×C3×#131 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#41 | 2 |
| C2×C3×#131 | 2 |
| C22×C3×M16 | 2 |
| C23×C3×C8 | 2 |
| C22×#131 | 3 |