Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
286 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 286 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 ◁ C2×C42 ◁ 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. #532 | C2 | semidirect |
| 4. C2×#124 | C2 | semidirect |
| 5. #497 | C2 | semidirect |
| 6. #496 | C2 | semidirect |
| 7. #495 | C2 | semidirect |
| 8. C4×M16 | C2 | semidirect |
| 14. C2×Q16 | C22 | semidirect |
| 16. C2×SD16 | C22 | semidirect |
| 19. C4⋊Q8 | C22 | semidirect |
| 20. #127 | C22 | semidirect |
| 24. #124 | C22 | semidirect |
| 29. #97 | C22 | semidirect |
| 30. C4×C8 | C22 | semidirect |
| 52. M16 | 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 |
| #532 | 2 |
| C2×#124 | 2 |
| #497 | 2 |
| #496 | 2 |
| #495 | 2 |
| C4×M16 | 2 |