Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
6135 conjugacy classes of subgroups; only the 5327 normal subgroups are shown.
This group has 6135 conjugacy classes of subgroups — too many to draw. Only its 5327 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 | C24 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#520 | C2 | direct (N × H) |
| 3. C22×C4×Q8 | C2 | semidirect |
| 4. C22×C4×D4 | C2 | semidirect |
| 5. C22×#117 | C2 | semidirect |
| 6. C23×C4⋊C4 | C2 | semidirect |
| 7. C22×#41 | C4 | semidirect |
| 13. #520 | C22 | direct (N × H) |
| 14. C2×C4×Q8 | C22 | semidirect |
| 15. C2×C4×D4 | C22 | semidirect |
| 16. C2×#117 | C22 | semidirect |
| 20. C22×C4⋊C4 | C22 | semidirect |
| 24. C2×#41 | C2×C4 | semidirect |
| 26. C22×Q8 | C2×C4 | semidirect |
| 28. C22×D4 | C2×C4 | semidirect |
| 31. C23×C4 | C2×C4 | semidirect |
| 38. C4×Q8 | C23 | semidirect |
| 39. C4×D4 | C23 | semidirect |
| 40. #117 | C23 | semidirect |
| 44. C2×C4⋊C4 | C23 | semidirect |
| 52. #41 | C22×C4 | semidirect |
| 53. C2×Q8 | C22×C4 | semidirect |
| 55. C2×D4 | C22×C4 | semidirect |
| 62. C22×C4 | C22×C4 | semidirect |
| 83. Q8 | C23×C4 | semidirect |
| 84. D4 | C23×C4 | semidirect |
| 90. C2×C4 | C23×C4 | semidirect |
| 98. C22 | #520 | direct (N × H) |
| 109. C2 | C2×#520 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C2×#520 | 2 |
| C22×C4×Q8 | 2 |
| C22×C4×D4 | 2 |
| C22×#117 | 2 |
| C23×C4⋊C4 | 2 |