Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1093 conjugacy classes of subgroups; only the 703 normal subgroups are shown.
This group has 1093 conjugacy classes of subgroups — too many to draw. Only its 703 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C23×C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2980 | C2 | direct (N × H) |
| 3. C2×#435 | C2 | semidirect |
| 4. C2×#432 | C2 | semidirect |
| 5. C2×#408 | C2 | semidirect |
| 17. #435 | C22 | semidirect |
| 18. #432 | C22 | semidirect |
| 20. #408 | C22 | semidirect |
| 140. C2 | #2980 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#532 | 2 |
| #2980 | 2 |
| C2×#435 | 2 |
| C2×#432 | 2 |
| C2×#408 | 2 |