Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
648 conjugacy classes of subgroups; only the 245 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 245 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D251 |
| commutator, socle, prime radical, cosocle | C23×C251 | C24 |
| Fitting | C2×C4⋊Q8×C251 | C2 |
| hypercenter | C2×C4⋊Q8 | D251 |
| hypocenter | C251 | #2868 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 24. C4×Dic251 | Q8 | semidirect |
| 50. C4×C251 | C4⋊Q8 | semidirect |
| 52. Dic251 | C4⋊Q8 | semidirect |
| 56. C251 | #2868 | semidirect |
| 109. C4 | #22548713 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548715 | 2 |
| C2×#22548713 | 2 |
| #30631556 | 2 |
| C42×Dic251 | 2 |
| C2×C4⋊Q8×C251 | 2 |
| #2868 | 251 |