Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
308 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2283 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1841 | C2 | semidirect |
| 3. #1833 | C2 | semidirect |
| 7. #1772 | C2 | semidirect |
| 9. #1747 | C2 | semidirect |
| 17. #405 | C4 | semidirect |
| 18. #405 | C4 | semidirect |
| 22. C4×M16 | C4 | semidirect |
| 24. C2×C4×C8 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#131 | 2 |
| #1841 | 2 |
| #1833 | 2 |
| #1827 | 2 |
| #1813 | 2 |
| #1812 | 2 |
| #1772 | 2 |
| #1757 | 2 |
| #1747 | 2 |