Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
144 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 39 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 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q64 | C2 | semidirect |
| 3. C2×SD64 | C2 | semidirect |
| 4. #2216 | C2 | semidirect |
| 5. #1416 | C2 | semidirect |
| 6. #1415 | C2 | semidirect |
| 7. C22⋊C32 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#508 | 2 |
| C2×Q64 | 2 |
| C2×SD64 | 2 |
| #2216 | 2 |
| #1416 | 2 |
| #1415 | 2 |
| C22⋊C32 | 2 |