Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
9999 conjugacy classes of subgroups; only the 5839 normal subgroups are shown.
This group has 9999 conjugacy classes of subgroups — too many to draw. Only its 5839 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C22 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| socle | C25 | C23 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C25×C4 | C2 | semidirect |
| 3. C22×C4×D4 | C2 | direct (N × H) |
| 4. C23×C4⋊C4 | C2 | semidirect |
| 5. C23×C22⋊C4 | C2 | semidirect |
| 6. C23×C42 | C2 | semidirect |
| 8. C23×D4 | C4 | direct (N × H) |
| 15. C24×C4 | C22 | semidirect |
| 16. C2×C4×D4 | C22 | direct (N × H) |
| 17. C22×C4⋊C4 | C22 | semidirect |
| 18. C22×C22⋊C4 | C22 | semidirect |
| 19. C22×C42 | C22 | semidirect |
| 22. C25 | C2×C4 | semidirect |
| 25. C22×D4 | C2×C4 | direct (N × H) |
| 30. C23×C4 | C2×C4 | semidirect |
| 31. C23×C4 | D4 | direct (N × H) |
| 36. C23×C4 | C23 | semidirect |
| 37. C4×D4 | C23 | direct (N × H) |
| 38. C2×C4⋊C4 | C23 | semidirect |
| 39. C2×C22⋊C4 | C23 | semidirect |
| 40. C2×C42 | C23 | semidirect |
| 45. C24 | C22×C4 | semidirect |
| 50. C2×D4 | C22×C4 | direct (N × H) |
| 55. C22×C4 | C22×C4 | semidirect |
| 56. C22×C4 | C2×D4 | direct (N × H) |
| 61. C22×C4 | C24 | semidirect |
| 62. C4⋊C4 | C24 | semidirect |
| 63. C22⋊C4 | C24 | semidirect |
| 64. C42 | C24 | semidirect |
| 68. C23 | C4×D4 | direct (N × H) |
| 70. C23 | C23×C4 | semidirect |
| 75. D4 | C23×C4 | direct (N × H) |
| 79. C2×C4 | C23×C4 | semidirect |
| 80. C2×C4 | C22×D4 | direct (N × H) |
| 88. C22 | C2×C4×D4 | direct (N × H) |
| 90. C22 | C24×C4 | semidirect |
| 96. C4 | C24×C4 | semidirect |
| 97. C4 | C23×D4 | direct (N × H) |
| 99. C2 | C22×C4×D4 | 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 |
| C25×C4 | 2 |
| C22×C4×D4 | 2 |
| C23×C4⋊C4 | 2 |
| C23×C22⋊C4 | 2 |
| C23×C42 | 2 |