Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
760 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 549 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×C8 | D251 |
| commutator, hypocenter | C251 | C22×C4×C8 |
| Frattini | C2×C4 | C23×D251 |
| Fitting | C23×C8×C251 | C2 |
| socle, prime radical | C24×C251 | C2×C4 |
| cosocle | C2×C4×C251 | C24 |
| LCS | G ▷ C251 | |
| UCS | C1 ◁ C23×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C8×Dic251 | C2 | direct (N × H) |
| 6. C22×C251⋊C8 | C4 | semidirect |
| 10. C8×Dic251 | C22 | direct (N × H) |
| 12. C22×C8×C251 | C4 | semidirect |
| 14. C22×Dic251 | C8 | direct (N × H) |
| 16. C2×C251⋊C8 | C2×C4 | semidirect |
| 25. C2×C8×C251 | C2×C4 | semidirect |
| 27. C2×Dic251 | C2×C8 | direct (N × H) |
| 29. C251⋊C8 | C22×C4 | semidirect |
| 38. C8×C251 | C22×C4 | semidirect |
| 39. C22×C251 | C4×C8 | semidirect |
| 46. Dic251 | C22×C8 | direct (N × H) |
| 47. C2×C251 | C2×C4×C8 | semidirect |
| 50. C251 | C22×C4×C8 | semidirect |
| 60. C22×C8 | Dic251 | direct (N × H) |
| 74. C2×C8 | C2×Dic251 | direct (N × H) |
| 89. C8 | C22×Dic251 | direct (N × H) |
| 90. C22 | C8×Dic251 | direct (N × H) |
| 98. C2 | C2×C8×Dic251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C251⋊C8 | 2 |
| C22×C4×Dic251 | 2 |
| C2×C8×Dic251 | 2 |
| C23×C8×C251 | 2 |
| C22×C4×C8 | 251 |