Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
432 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 139 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×D127 |
| commutator, socle, cosocle | C23×C127 | C24 |
| Fitting | #396×C127 | C2 |
| prime radical | C24×C127 | C23 |
| hypercenter | #396 | D127 |
| hypocenter | C127 | #2401 |
| LCS | G ▷ C23×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C127⋊#401 | C2 | semidirect |
| 2. #22549815 | C2 | semidirect |
| 4. C4⋊C4×Dic127 | C2 | semidirect |
| 6. #22549721 | C2 | semidirect |
| 71. C127 | #2401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127⋊#401 | 2 |
| #22549815 | 2 |
| #22549801 | 2 |
| C4⋊C4×Dic127 | 2 |
| C22⋊C4×Dic127 | 2 |
| #22549721 | 2 |
| #396×C127 | 2 |
| #2401 | 127 |