Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2590 conjugacy classes of subgroups; only the 1515 normal subgroups are shown.
This group has 2590 conjugacy classes of subgroups — too many to draw. Only its 1515 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C16 | D3 |
| commutator, hypocenter | C3 | C24×C16 |
| Frattini | C8 | C24×D3 |
| Fitting | C23×C3×C16 | C2 |
| socle | C24×C3 | C2×C8 |
| prime radical | C24×D3 | C8 |
| cosocle | C3×C8 | C25 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C3×C16 | C2 | semidirect |
| 3. C23×C3⋊C16 | C2 | semidirect |
| 4. C22×D3×C16 | C2 | direct (N × H) |
| 11. C22×C3×C16 | C22 | semidirect |
| 12. C22×C3⋊C16 | C22 | semidirect |
| 13. C2×D3×C16 | C22 | direct (N × H) |
| 17. C23×C16 | D3 | direct (N × H) |
| 28. C2×C3×C16 | C23 | semidirect |
| 29. C2×C3⋊C16 | C23 | semidirect |
| 30. D3×C16 | C23 | direct (N × H) |
| 38. C22×C16 | C2×D3 | direct (N × H) |
| 40. C23×D3 | C16 | direct (N × H) |
| 52. C3×C16 | C24 | semidirect |
| 53. C3⋊C16 | C24 | semidirect |
| 66. C2×C16 | C22×D3 | direct (N × H) |
| 67. C23×C3 | C2×C16 | semidirect |
| 69. C22×D3 | C2×C16 | direct (N × H) |
| 93. C16 | C23×D3 | direct (N × H) |
| 94. C22×C3 | C22×C16 | semidirect |
| 96. C2×D3 | C22×C16 | direct (N × H) |
| 103. C23 | D3×C16 | direct (N × H) |
| 113. C2×C3 | C23×C16 | semidirect |
| 115. D3 | C23×C16 | direct (N × H) |
| 118. C22 | C2×D3×C16 | direct (N × H) |
| 123. C3 | C24×C16 | semidirect |
| 125. C2 | C22×D3×C16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D3×C8 | 2 |
| C23×C3×C16 | 2 |
| C23×C3⋊C16 | 2 |
| C22×D3×C16 | 2 |
| C24×C16 | 3 |