Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
386 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 386 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, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3229 | C2 | semidirect |
| 4. #3165 | C2 | semidirect |
| 6. C2×#487 | C2 | semidirect |
| 7. #1716 | C2 | semidirect |
| 8. #1714 | C2 | semidirect |
| 9. #1708 | C2 | semidirect |
| 10. #1707 | C2 | semidirect |
| 11. C8×M16 | C2 | semidirect |
| 126. M16 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3229 | 2 |
| #3227 | 2 |
| #3167 | 2 |
| #3165 | 2 |
| #3152 | 2 |
| C2×#487 | 2 |
| #1716 | 2 |
| #1714 | 2 |
| #1708 | 2 |
| #1707 | 2 |
| C8×M16 | 2 |