Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
706 conjugacy classes of subgroups; only the 426 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 426 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C25 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #585 | C2 | semidirect |
| 2. C2×2−1+4 | C2 | semidirect |
| 3. #578 | C2 | semidirect |
| 4. #577 | C2 | semidirect |
| 5. #575 | C2 | semidirect |
| 6. C2×#137 | C2 | semidirect |
| 7. #568 | C2 | semidirect |
| 10. 2−1+4 | C22 | semidirect |
| 11. 2−1+4 | C22 | semidirect |
| 13. H(5,𝔽2) | C22 | semidirect |
| 18. C2×#41 | C22 | semidirect |
| 20. C22×Q8 | C22 | semidirect |
| 21. #137 | C22 | semidirect |
| 22. #137 | C22 | semidirect |
| 23. #137 | C22 | semidirect |
| 24. AGL(1,ℤ/8) | C22 | semidirect |
| 25. AGL(1,ℤ/8) | C22 | semidirect |
| 26. #135 | C22 | semidirect |
| 27. C2×Q16 | C22 | semidirect |
| 28. C2×SD16 | C22 | semidirect |
| 29. #131 | C22 | semidirect |
| 30. C2×M16 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #585 | 2 |
| C2×2−1+4 | 2 |
| #578 | 2 |
| #577 | 2 |
| #575 | 2 |
| C2×#137 | 2 |
| #568 | 2 |