Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
253 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 140 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 | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ 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 |
|---|---|---|
| 2. #520 | C2 | semidirect |
| 3. #518 | C2 | semidirect |
| 4. #442 | C2 | semidirect |
| 5. #440 | C2 | semidirect |
| 6. C4×SD16 | C2 | semidirect |
| 7. C4×D8 | C2 | semidirect |
| 8. #428 | C2 | semidirect |
| 9. #416 | C2 | semidirect |
| 10. C2×#102 | C2 | semidirect |
| 11. #405 | C2 | semidirect |
| 14. AGL(1,ℤ/8) | C4 | semidirect |
| 19. C4×Q8 | C22 | semidirect |
| 25. C4×D4 | C22 | semidirect |
| 32. #107 | C22 | semidirect |
| 33. #106 | C22 | semidirect |
| 34. #103 | C22 | semidirect |
| 35. #103 | C22 | semidirect |
| 38. #102 | C22 | semidirect |
| 39. #102 | C22 | semidirect |
| 40. #97 | C22 | semidirect |
| 41. C4×C8 | C22 | semidirect |
| 44. #41 | C2×C4 | semidirect |
| 49. C2×D4 | C2×C4 | semidirect |
| 58. SD16 | C2×C4 | semidirect |
| 59. D8 | C2×C4 | semidirect |
| 60. M16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AGL(1,ℤ/8) | 2 |
| #520 | 2 |
| #518 | 2 |
| #442 | 2 |
| #440 | 2 |
| C4×SD16 | 2 |
| C4×D8 | 2 |
| #428 | 2 |
| #416 | 2 |
| C2×#102 | 2 |
| #405 | 2 |