Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
908 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 223 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 | #395×C127 | C2 |
| prime radical | C23×D127 | C23 |
| hypercenter | #395 | D127 |
| hypocenter | C127 | C2×#395 |
| LCS | G ▷ C23×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ #395 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #22549797 | C2 | semidirect |
| 4. #22549796 | C2 | semidirect |
| 5. #22549795 | C2 | semidirect |
| 7. #395×C127 | C2 | semidirect |
| 34. C2×C4×D127 | Q8 | semidirect |
| 52. C2×Dic127 | C2×Q8 | semidirect |
| 63. C2×C4×C127 | C2×Q8 | semidirect |
| 79. D127 | #395 | direct (N × H) |
| 81. C127 | C2×#395 | semidirect |
| 87. #395 | D127 | direct (N × H) |
| 144. C2×C4 | Q8×D127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×D127 | 2 |
| C2×C4⋊C4×D127 | 2 |
| #22549797 | 2 |
| #22549796 | 2 |
| #22549795 | 2 |
| #95×D127 | 2 |
| #395×C127 | 2 |
| C2×#395 | 127 |