Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
640 conjugacy classes of subgroups; only the 520 normal subgroups are shown.
This group has 640 conjugacy classes of subgroups — too many to draw. Only its 520 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C251 | C22 |
| commutator | C2 | C23×C8×C251 |
| Frattini, cosocle | C2×C4 | C24×C251 |
| socle, prime radical | C24×C251 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4⋊C8×C251 | C2 | direct (N × H) |
| 11. C4⋊C8×C251 | C22 | direct (N × H) |
| 14. C22×C4×C251 | C8 | semidirect |
| 31. C2×C4×C251 | C2×C8 | semidirect |
| 39. C22×C251 | C4⋊C8 | direct (N × H) |
| 46. C4×C251 | C22×C8 | semidirect |
| 49. C2×C251 | C2×C4⋊C8 | direct (N × H) |
| 53. C251 | C22×C4⋊C8 | direct (N × H) |
| 54. C22×C4⋊C8 | C251 | direct (N × H) |
| 57. C2×C4⋊C8 | C2×C251 | direct (N × H) |
| 65. C4⋊C8 | C22×C251 | direct (N × H) |
| 68. C22×C4 | C8×C251 | semidirect |
| 85. C2×C4 | C2×C8×C251 | semidirect |
| 93. C22 | C4⋊C8×C251 | direct (N × H) |
| 100. C4 | C22×C8×C251 | semidirect |
| 103. C2 | C2×C4⋊C8×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C8×C251 | 2 |
| C22×C42×C251 | 2 |
| C2×C4⋊C8×C251 | 2 |
| C22×C4⋊C8 | 251 |