Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
180 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C251 | C2×D4 |
| commutator | C4 | C22×C8×C251 |
| Frattini, cosocle | C42 | C23×C251 |
| socle | C22×C251 | C2×C4⋊C4 |
| prime radical | C23×C251 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C251 ◁ C2×C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #335×C251 | C2 | semidirect |
| 5. #334×C251 | C2 | semidirect |
| 22. M16×C251 | C8 | semidirect |
| 49. C8×C251 | C2×C8 | semidirect |
| 66. C251 | #1565 | direct (N × H) |
| 67. #1565 | C251 | direct (N × H) |
| 71. #335 | C2×C251 | semidirect |
| 72. #334 | C2×C251 | semidirect |
| 89. M16 | C8×C251 | semidirect |
| 116. C8 | C2×C8×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #431×C251 | 2 |
| C2×C4⋊C8×C251 | 2 |
| C4×M16×C251 | 2 |
| #335×C251 | 2 |
| #334×C251 | 2 |
| #1565 | 251 |