Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
4291 conjugacy classes of subgroups; only the 1007 normal subgroups are shown.
This group has 4291 conjugacy classes of subgroups — too many to draw. Only its 1007 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 | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C22×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×C4 | C2 | semidirect |
| 3. C22×C4×D4 | C2 | semidirect |
| 4. C4×#120 | C2 | direct (N × H) |
| 5. C2×#386 | C2 | semidirect |
| 6. C2×#385 | C2 | semidirect |
| 7. C2×#379 | C2 | semidirect |
| 8. C2×C4×C22⋊C4 | C2 | semidirect |
| 14. C24×C4 | C22 | semidirect |
| 15. C24×C4 | C22 | semidirect |
| 16. C2×#120 | C4 | direct (N × H) |
| 19. C2×C4×D4 | C22 | semidirect |
| 20. C22×C4⋊C4 | C22 | semidirect |
| 24. C22×C22⋊C4 | C22 | semidirect |
| 25. C22×C42 | C22 | semidirect |
| 26. #386 | C22 | semidirect |
| 27. #385 | C22 | semidirect |
| 28. #379 | C22 | semidirect |
| 29. C4×C22⋊C4 | C22 | semidirect |
| 30. C2×#95 | C22 | semidirect |
| 33. C25 | C2×C4 | semidirect |
| 39. C22×D4 | C2×C4 | semidirect |
| 47. C23×C4 | D4 | semidirect |
| 55. C23×C4 | C23 | semidirect |
| 56. #120 | C2×C4 | direct (N × H) |
| 59. C2×C4⋊C4 | C23 | semidirect |
| 63. C2×C22⋊C4 | C2×C4 | semidirect |
| 67. C2×C22⋊C4 | C23 | semidirect |
| 69. C2×C42 | C23 | semidirect |
| 70. #95 | C23 | semidirect |
| 81. C24 | C22×C4 | semidirect |
| 82. C24 | C22×C4 | semidirect |
| 91. C2×D4 | C22×C4 | semidirect |
| 106. C22×C4 | C22×C4 | semidirect |
| 109. C22×C4 | C2×D4 | semidirect |
| 119. C22⋊C4 | C22×C4 | semidirect |
| 136. C23 | C4×D4 | semidirect |
| 139. C23 | C23×C4 | semidirect |
| 159. C2×C4 | #120 | direct (N × H) |
| 160. C2×C4 | C23×C4 | semidirect |
| 174. C22 | C2×C4×D4 | semidirect |
| 184. C4 | C2×#120 | direct (N × H) |
| 187. C2 | C4×#120 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C4 | 2 |
| C22×#120 | 2 |
| C22×C4×D4 | 2 |
| C4×#120 | 2 |
| C2×#386 | 2 |
| C2×#385 | 2 |
| C2×#379 | 2 |
| C2×C4×C22⋊C4 | 2 |