Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
915 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 915 conjugacy classes of subgroups — too many to draw. Only its 223 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×D8 | C2 | semidirect |
| 3. D4×C22⋊C4 | C2 | semidirect |
| 4. #1970 | C2 | semidirect |
| 5. #1879 | C2 | semidirect |
| 6. #1808 | C2 | semidirect |
| 7. #1786 | C2 | semidirect |
| 8. C8×C22⋊C4 | C2 | semidirect |
| 11. C22×D8 | C4 | semidirect |
| 16. C2×C4×D4 | C22 | semidirect |
| 19. C2×#107 | C22 | semidirect |
| 20. C2×#102 | C22 | semidirect |
| 21. C2×#102 | C22 | semidirect |
| 23. C2×C4×C8 | C22 | semidirect |
| 24. #390 | C22 | semidirect |
| 29. #340 | C22 | semidirect |
| 30. #336 | C22 | semidirect |
| 36. C22×D4 | C2×C4 | semidirect |
| 45. C2×D8 | C2×C4 | semidirect |
| 46. C2×D8 | C2×C4 | semidirect |
| 49. C22×C8 | C2×C4 | semidirect |
| 106. D8 | C22⋊C4 | direct (N × H) |
| 122. C22⋊C4 | D8 | direct (N × H) |
| 152. D4 | C2×C22⋊C4 | semidirect |
| 167. C2×C4 | C2×D8 | semidirect |
| 174. C8 | C2×C22⋊C4 | semidirect |
| 190. C22 | C4×D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C2×C4×D8 | 2 |
| D4×C22⋊C4 | 2 |
| #1970 | 2 |
| #1879 | 2 |
| #1808 | 2 |
| #1786 | 2 |
| C8×C22⋊C4 | 2 |