Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
897 conjugacy classes of subgroups; only the 719 normal subgroups are shown.
This group has 897 conjugacy classes of subgroups — too many to draw. Only its 719 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 | C24 | C22×C4 |
| 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. #2986 | C2 | direct (N × H) |
| 3. C2×#446 | C2 | semidirect |
| 4. C2×C8×Q8 | C2 | semidirect |
| 15. #446 | C22 | semidirect |
| 16. C8×Q8 | C22 | semidirect |
| 116. C2 | #2986 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| #2986 | 2 |
| C2×#446 | 2 |
| C2×C8×Q8 | 2 |
| C2×#433 | 2 |
| C2×#431 | 2 |