Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
257 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 139 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #2255 | C2 | semidirect |
| 4. #2254 | C2 | semidirect |
| 5. #2252 | C2 | semidirect |
| 47. M32 | Q8 | semidirect |
| 94. C16 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3160 | 2 |
| C2×#500 | 2 |
| #2255 | 2 |
| #2254 | 2 |
| #2252 | 2 |
| #2159 | 2 |
| C4×M32 | 2 |