Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
40620 conjugacy classes of subgroups; only the 29370 normal subgroups are shown.
This group has 40620 conjugacy classes of subgroups — too many to draw. Only its 29370 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #63040 | C2 | direct (N × H) |
| 2. C2×#3498 | C2 | semidirect |
| 3. C2×#3549 | C2 | semidirect |
| 4. C2×#3483 | C2 | semidirect |
| 5. C22×H(5,𝔽2) | C22 | semidirect |
| 6. #3549 | C22 | semidirect |
| 7. C2×H(3,𝔽4) | C22 | semidirect |
| 8. #3498 | C22 | semidirect |
| 9. #3483 | C22 | semidirect |
| 10. C2×Q82 | C22 | semidirect |
| 11. C2×#555 | C22 | semidirect |
| 12. C2×#550 | C22 | semidirect |
| 13. C2×D42 | C22 | semidirect |
| 14. C2×#535 | C22 | semidirect |
| 15. C2×#532 | C22 | semidirect |
| 16. C22×#127 | C22 | semidirect |
| 17. C2×C4×#41 | C22 | semidirect |
| 19. C2×H(5,𝔽2) | C23 | semidirect |
| 23. H(3,𝔽4) | C23 | semidirect |
| 24. Q82 | C23 | semidirect |
| 25. #555 | C23 | semidirect |
| 26. #550 | C23 | semidirect |
| 27. D42 | C23 | semidirect |
| 28. #535 | C23 | semidirect |
| 29. #532 | C23 | semidirect |
| 33. C2×#127 | C23 | semidirect |
| 37. C4×#41 | C23 | semidirect |
| 75. C2×Q8 | H(5,𝔽2) | semidirect |
| 97. Q8 | C2×H(5,𝔽2) | semidirect |
| 113. C2 | #63040 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63040 | 2 |
| C2×#3498 | 2 |
| C2×#3549 | 2 |
| C2×#3483 | 2 |