Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
247 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 117 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 | C2×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×Q16 | C2 | semidirect |
| 3. #2254 | C2 | semidirect |
| 5. #2225 | C2 | semidirect |
| 6. Q8×C16 | C2 | semidirect |
| 35. SD32 | Q8 | direct (N × H) |
| 54. Q16 | C2×Q8 | semidirect |
| 69. C16 | C2×Q8 | semidirect |
| 75. Q8 | SD32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Q16 | 2 |
| Q8×D8 | 2 |
| #2254 | 2 |
| #2226 | 2 |
| #2225 | 2 |
| Q8×C16 | 2 |
| C4×SD32 | 2 |