Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
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×D127 |
| commutator, socle, cosocle | C23×C127 | C24 |
| Fitting | #384×C127 | C2 |
| prime radical | C23×D127 | C23 |
| hypercenter | #384 | D127 |
| hypocenter | C127 | #2857 |
| LCS | G ▷ C23×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16861209 | C2 | semidirect |
| 4. #22549781 | C2 | semidirect |
| 5. #22549775 | C2 | semidirect |
| 7. #384×C127 | C2 | semidirect |
| 35. C2×C4×D127 | Q8 | semidirect |
| 82. C2×D127 | C4⋊Q8 | semidirect |
| 85. C4×C127 | C22⋊Q8 | semidirect |
| 96. C127 | #2857 | semidirect |
| 102. #384 | D127 | semidirect |
| 184. C4 | #16861189 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16861209 | 2 |
| C2×#16861189 | 2 |
| C2×C42×D127 | 2 |
| #22549781 | 2 |
| #22549775 | 2 |
| #22549733 | 2 |
| #384×C127 | 2 |
| #2857 | 127 |