Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
831 conjugacy classes of subgroups; only the 483 normal subgroups are shown.
This group has 831 conjugacy classes of subgroups — too many to draw. Only its 483 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 | C23 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #3395 | C2 | semidirect |
| 3. Q8×Q16 | C2 | semidirect |
| 4. #3363 | C2 | semidirect |
| 57. C2×Q8 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3502 | 2 |
| #3395 | 2 |
| Q8×Q16 | 2 |
| #3363 | 2 |
| #3249 | 2 |
| #3170 | 2 |
| C2×C8×Q8 | 2 |