Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
871 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 871 conjugacy classes of subgroups — too many to draw. Only its 487 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 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×Q16 | C2 | semidirect |
| 3. #3338 | C2 | semidirect |
| 4. #3230 | C2 | semidirect |
| 5. #3170 | C2 | semidirect |
| 6. C2×#500 | C2 | semidirect |
| 67. #41 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3503 | 2 |
| Q8×Q16 | 2 |
| #3338 | 2 |
| #3230 | 2 |
| #3170 | 2 |
| C2×#500 | 2 |
| C8×#41 | 2 |