Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
644 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 209 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, cosocle | C23×C251 | C24 |
| Fitting | #384×C251 | C2 |
| prime radical | C23×D251 | C23 |
| hypercenter | #384 | D251 |
| hypocenter | C251 | #2857 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22548715 | C2 | semidirect |
| 4. #30631592 | C2 | semidirect |
| 5. #30631586 | C2 | semidirect |
| 7. #384×C251 | C2 | semidirect |
| 35. C2×C4×D251 | Q8 | semidirect |
| 82. C2×D251 | C4⋊Q8 | semidirect |
| 85. C4×C251 | C22⋊Q8 | semidirect |
| 95. C251 | #2857 | semidirect |
| 102. #384 | D251 | semidirect |
| 184. C4 | #22548695 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548715 | 2 |
| C2×#22548695 | 2 |
| C2×C42×D251 | 2 |
| #30631592 | 2 |
| #30631586 | 2 |
| #30631544 | 2 |
| #384×C251 | 2 |
| #2857 | 251 |