Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C251 | C2×D4 |
| commutator | C2×C4 | C22×C4×C251 |
| Frattini, cosocle | C22×C4 | C23×C251 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C251 ◁ C2×C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C2×Q16×C251 | C4 | semidirect |
| 51. C2×C4×C251 | Q16 | semidirect |
| 77. C251 | #1969 | direct (N × H) |
| 78. #1969 | C251 | direct (N × H) |
| 87. C2×Q16 | C4×C251 | semidirect |
| 129. C2×C4 | Q16×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q16×C251 | 2 |
| C2×#107×C251 | 2 |
| C2×#103×C251 | 2 |
| C2×C4×C8×C251 | 2 |
| #391×C251 | 2 |
| #1969 | 251 |