Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
580 conjugacy classes of subgroups; only the 390 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 390 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #543 | C2 | semidirect |
| 2. #539 | C2 | semidirect |
| 3. #538 | C2 | semidirect |
| 4. #534 | C2 | semidirect |
| 5. #529 | C2 | semidirect |
| 6. #525 | C2 | semidirect |
| 7. C2×C4×D4 | C2 | semidirect |
| 8. C2×#41 | C22 | semidirect |
| 11. C23×C4 | C22 | semidirect |
| 12. #126 | C22 | semidirect |
| 13. #125 | C22 | semidirect |
| 14. #124 | C22 | semidirect |
| 15. #123 | C22 | semidirect |
| 16. #123 | C22 | semidirect |
| 17. C22⋊Q8 | C22 | semidirect |
| 18. C22⋊Q8 | C22 | semidirect |
| 19. C22⋊Q8 | C22 | semidirect |
| 22. #121 | C22 | semidirect |
| 23. #120 | C22 | semidirect |
| 24. C4×Q8 | C22 | semidirect |
| 25. C4×D4 | C22 | semidirect |
| 26. C4×D4 | C22 | semidirect |
| 27. C4×D4 | C22 | semidirect |
| 28. #117 | C22 | semidirect |
| 29. C2×C4⋊C4 | C22 | semidirect |
| 31. C2×C22⋊C4 | C22 | semidirect |
| 32. C2×C42 | C22 | semidirect |
| 67. C23 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #543 | 2 |
| #539 | 2 |
| #538 | 2 |
| #534 | 2 |
| #529 | 2 |
| #525 | 2 |
| C2×C4×D4 | 2 |