Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
460 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 197 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 | C22×C127 | C23×C4 |
| Fitting | #382×C127 | C2 |
| socle, prime radical, cosocle | C23×C127 | C24 |
| hypercenter | #382 | D127 |
| hypocenter | C127 | #2331 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ #382 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #16861209 | C4 | semidirect |
| 34. C4×Dic127 | Q8 | semidirect |
| 80. C127 | #2331 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16861209 | 2 |
| #22549780 | 2 |
| #22549772 | 2 |
| #22549761 | 2 |
| C4⋊C4×Dic127 | 2 |
| #22549722 | 2 |
| #382×C127 | 2 |
| #2331 | 127 |