Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1631 conjugacy classes of subgroups; only the 863 normal subgroups are shown.
This group has 1631 conjugacy classes of subgroups — too many to draw. Only its 863 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C22×C4 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×M16 | C2 | semidirect |
| 3. C2×#436 | C2 | direct (N × H) |
| 4. C22×C4⋊C8 | C2 | semidirect |
| 5. C22×C22⋊C8 | C2 | semidirect |
| 6. C22×C4×C8 | C2 | semidirect |
| 10. C22×M16 | C22 | semidirect |
| 19. #436 | C22 | direct (N × H) |
| 20. C2×C4⋊C8 | C22 | semidirect |
| 21. C2×C22⋊C8 | C22 | semidirect |
| 22. C2×C4×C8 | C22 | semidirect |
| 35. C2×M16 | C23 | semidirect |
| 38. C22×C8 | D4 | semidirect |
| 48. C4⋊C8 | C23 | semidirect |
| 49. C22⋊C8 | C23 | semidirect |
| 50. C4×C8 | C23 | semidirect |
| 63. C22×C4 | M16 | semidirect |
| 76. C2×C8 | C2×D4 | semidirect |
| 100. C2×C4 | C2×M16 | semidirect |
| 110. C8 | C22×D4 | semidirect |
| 115. C22 | #436 | direct (N × H) |
| 123. C4 | C22×M16 | semidirect |
| 127. C2 | C2×#436 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C22×C4×D4 | 2 |
| C2×#436 | 2 |
| C22×C4⋊C8 | 2 |
| C22×C22⋊C8 | 2 |
| C22×C4×C8 | 2 |