Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
282 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#137 | C2 | semidirect |
| 2. C2×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C2×#135 | C2 | semidirect |
| 4. C22×SD16 | C2 | semidirect |
| 5. #536 | C2 | semidirect |
| 6. #535 | C2 | semidirect |
| 7. #497 | C2 | semidirect |
| 8. #496 | C2 | semidirect |
| 9. #495 | C2 | semidirect |
| 10. #492 | C2 | semidirect |
| 11. #405 | C2 | semidirect |
| 19. C2×Q16 | C22 | semidirect |
| 20. C2×Q16 | C22 | semidirect |
| 25. C2×SD16 | C22 | semidirect |
| 26. C2×SD16 | C22 | semidirect |
| 27. C2×SD16 | C22 | semidirect |
| 29. C2×D8 | C22 | semidirect |
| 32. C4⋊Q8 | C22 | semidirect |
| 35. #124 | C22 | semidirect |
| 41. #97 | C22 | semidirect |
| 42. C4×C8 | C22 | semidirect |
| 77. M16 | D4 | semidirect |
| 78. C2×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137 | 2 |
| C2×AGL(1,ℤ/8) | 2 |
| C2×#135 | 2 |
| C22×SD16 | 2 |
| #536 | 2 |
| #535 | 2 |
| #497 | 2 |
| #496 | 2 |
| #495 | 2 |
| #492 | 2 |
| #405 | 2 |