Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
730 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#390 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2911 | C2 | semidirect |
| 3. C2×#421 | C2 | semidirect |
| 4. C2×#410 | C2 | semidirect |
| 5. #2873 | C2 | semidirect |
| 6. #1976 | C2 | semidirect |
| 11. C2×#137 | C4 | semidirect |
| 12. C2×AGL(1,ℤ/8) | C4 | semidirect |
| 19. #427 | C22 | semidirect |
| 21. #421 | C22 | semidirect |
| 24. #410 | C22 | semidirect |
| 27. #405 | C22 | semidirect |
| 52. C2×Q16 | C2×C4 | semidirect |
| 53. C2×SD16 | C2×C4 | semidirect |
| 54. C2×D8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#575 | 2 |
| #2911 | 2 |
| C2×#421 | 2 |
| C2×#410 | 2 |
| #2873 | 2 |
| #1976 | 2 |