Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
403 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 110 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#41 | C2 | semidirect |
| 2. C2×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C2×#121 | C2 | semidirect |
| 4. #449 | C2 | semidirect |
| 5. #448 | C2 | semidirect |
| 6. #447 | C2 | semidirect |
| 7. #417 | C2 | semidirect |
| 8. #407 | C2 | semidirect |
| 13. C2×#41 | C22 | semidirect |
| 14. C22×Q8 | C22 | semidirect |
| 17. C22×D4 | C22 | semidirect |
| 18. C22×D4 | C22 | semidirect |
| 22. C2×SD16 | C22 | semidirect |
| 23. C2×D8 | C22 | semidirect |
| 24. C2×M16 | C22 | semidirect |
| 28. #121 | C22 | semidirect |
| 29. C2×C4⋊C4 | C22 | semidirect |
| 31. #103 | C22 | semidirect |
| 32. #102 | C22 | semidirect |
| 33. C22⋊C8 | C22 | semidirect |
| 34. C22⋊C8 | C22 | semidirect |
| 43. #41 | D4 | semidirect |
| 95. Q8 | C2×D4 | semidirect |
| 104. D4 | C2×D4 | semidirect |
| 139. C22 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41 | 2 |
| C2×AGL(1,ℤ/8) | 2 |
| C2×#121 | 2 |
| #449 | 2 |
| #448 | 2 |
| #447 | 2 |
| #417 | 2 |
| #407 | 2 |