Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
8373525 conjugacy classes of subgroups; only the 8283765 normal subgroups are shown.
This group has 8373525 conjugacy classes of subgroups — too many to draw. Only its 8283765 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×C4 | C24 |
| commutator, Frattini, cosocle | C2 | C29 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#585 | C2 | direct (N × H) |
| 2. C24×2−1+4 | C2 | semidirect |
| 3. C24×H(5,𝔽2) | C2 | semidirect |
| 4. C25×#41 | C2 | semidirect |
| 5. C22×#585 | C22 | direct (N × H) |
| 6. C23×2−1+4 | C22 | semidirect |
| 7. C23×H(5,𝔽2) | C22 | semidirect |
| 8. C24×#41 | C22 | semidirect |
| 9. C24×#41 | C22 | semidirect |
| 10. C24×#41 | C22 | semidirect |
| 11. C25×Q8 | C22 | semidirect |
| 12. C25×D4 | C22 | semidirect |
| 13. C26×C4 | C22 | semidirect |
| 15. C2×#585 | C23 | direct (N × H) |
| 16. C22×2−1+4 | C23 | semidirect |
| 17. C22×H(5,𝔽2) | C23 | semidirect |
| 18. C23×#41 | C23 | semidirect |
| 19. C23×#41 | C23 | semidirect |
| 20. C23×#41 | C23 | semidirect |
| 22. C24×Q8 | C23 | semidirect |
| 24. C24×D4 | C23 | semidirect |
| 27. C25×C4 | C23 | semidirect |
| 31. #585 | C24 | direct (N × H) |
| 32. C2×2−1+4 | C24 | semidirect |
| 33. C2×H(5,𝔽2) | C24 | semidirect |
| 34. C22×#41 | C24 | semidirect |
| 35. C22×#41 | C24 | semidirect |
| 36. C22×#41 | C24 | semidirect |
| 38. C23×Q8 | C24 | semidirect |
| 40. C23×D4 | C24 | semidirect |
| 45. C24×C4 | C24 | semidirect |
| 51. 2−1+4 | C25 | semidirect |
| 52. H(5,𝔽2) | C25 | semidirect |
| 53. C2×#41 | C25 | semidirect |
| 54. C2×#41 | C25 | semidirect |
| 55. C2×#41 | C25 | semidirect |
| 57. C22×Q8 | C25 | semidirect |
| 59. C22×D4 | C25 | semidirect |
| 64. C23×C4 | C25 | semidirect |
| 67. C24 | #585 | direct (N × H) |
| 71. #41 | C26 | semidirect |
| 72. #41 | C26 | semidirect |
| 74. C2×Q8 | C26 | semidirect |
| 76. C2×D4 | C26 | semidirect |
| 81. C22×C4 | C26 | semidirect |
| 84. C23 | C2×#585 | direct (N × H) |
| 96. C22 | C22×#585 | direct (N × H) |
| 102. C2 | C23×#585 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#585 | 2 |
| C24×2−1+4 | 2 |
| C24×H(5,𝔽2) | 2 |
| C25×#41 | 2 |