Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3607 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 3607 conjugacy classes of subgroups — too many to draw. Only its 1039 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×M16 | C2 | semidirect |
| 3. C22×#117 | C2 | semidirect |
| 4. C2×#418 | C2 | direct (N × H) |
| 5. C22×#102 | C2 | semidirect |
| 10. C23×D4 | C4 | semidirect |
| 15. C22×M16 | C22 | semidirect |
| 16. C23×C8 | C22 | semidirect |
| 18. C2×#117 | C22 | semidirect |
| 19. C22×C4⋊C4 | C22 | semidirect |
| 22. #418 | C22 | direct (N × H) |
| 23. C2×#102 | C22 | semidirect |
| 37. C22×D4 | C2×C4 | semidirect |
| 48. C2×M16 | C23 | semidirect |
| 50. C22×C8 | C23 | semidirect |
| 52. #117 | C23 | semidirect |
| 54. C2×C4⋊C4 | C23 | semidirect |
| 57. #102 | C23 | semidirect |
| 77. C2×D4 | C22×C4 | semidirect |
| 94. C2×C8 | C24 | semidirect |
| 96. C4⋊C4 | C24 | semidirect |
| 136. C22 | #418 | direct (N × H) |
| 148. C2 | C2×#418 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D4 | 2 |
| C23×M16 | 2 |
| C22×#117 | 2 |
| C2×#418 | 2 |
| C22×#102 | 2 |