Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
227 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 57 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 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2862 | C2 | semidirect |
| 4. #1286 | C2 | semidirect |
| 5. #1276 | C2 | semidirect |
| 22. C4⋊Q8 | Q8 | semidirect |
| 23. #127 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2862 | 2 |
| #1962 | 2 |
| #1517 | 2 |
| #1286 | 2 |
| #1276 | 2 |