Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1091 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1091 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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#125 | C2 | semidirect |
| 3. #3142 | C2 | direct (N × H) |
| 4. C2×#491 | C2 | semidirect |
| 5. C2×#490 | C2 | semidirect |
| 6. C2×#488 | C2 | semidirect |
| 7. C2×#417 | C2 | semidirect |
| 8. C2×C4×M16 | C2 | semidirect |
| 19. C2×#125 | C22 | semidirect |
| 26. #491 | C22 | semidirect |
| 27. #490 | C22 | semidirect |
| 28. #488 | C22 | semidirect |
| 29. #417 | C22 | semidirect |
| 32. C4×M16 | C22 | semidirect |
| 174. C2 | #3142 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#532 | 2 |
| C22×#125 | 2 |
| #3142 | 2 |
| C2×#491 | 2 |
| C2×#490 | 2 |
| C2×#488 | 2 |
| C2×#417 | 2 |
| C2×C4×M16 | 2 |