Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6997 conjugacy classes of subgroups; only the 2928 normal subgroups are shown.
This group has 6997 conjugacy classes of subgroups — too many to draw. Only its 2928 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #53015 | C2 | semidirect |
| 2. #52307 | C2 | semidirect |
| 3. #45996 | C2 | semidirect |
| 4. #36457 | C2 | semidirect |
| 5. #36091 | C2 | semidirect |
| 6. C22×H(5,𝔽2) | C22 | semidirect |
| 7. C2×H(3,𝔽4) | C22 | semidirect |
| 8. C2×Q82 | C22 | semidirect |
| 10. #2858 | C22 | semidirect |
| 11. #2840 | C22 | semidirect |
| 12. #2819 | C22 | semidirect |
| 13. #2811 | C22 | semidirect |
| 14. #2785 | C22 | semidirect |
| 15. #2733 | C22 | semidirect |
| 16. #2679 | C22 | semidirect |
| 17. #2670 | C22 | semidirect |
| 18. #2615 | C22 | semidirect |
| 19. #2563 | C22 | semidirect |
| 20. #2548 | C22 | semidirect |
| 21. #2491 | C22 | semidirect |
| 22. #2433 | C22 | semidirect |
| 23. #2408 | C22 | semidirect |
| 24. #2388 | C22 | semidirect |
| 25. #2341 | C22 | semidirect |
| 26. C4×#124 | C22 | semidirect |
| 27. C42×Q8 | C22 | semidirect |
| 109. C2×Q8 | H(5,𝔽2) | semidirect |
| 142. Q8 | H(3,𝔽4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #53015 | 2 |
| #52307 | 2 |
| #45996 | 2 |
| #36457 | 2 |
| #36091 | 2 |