Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
422 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 422 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 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(5,𝔽2) | C2 | semidirect |
| 2. C2×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C2×#127 | C2 | semidirect |
| 4. #453 | C2 | direct (N × H) |
| 5. C2×#104 | C2 | semidirect |
| 6. C2×#100 | C2 | semidirect |
| 9. H(5,𝔽2) | C22 | semidirect |
| 16. C22×D4 | C22 | semidirect |
| 18. AGL(1,ℤ/8) | C22 | semidirect |
| 21. C2×M16 | C22 | semidirect |
| 23. #127 | C22 | semidirect |
| 24. C2×C42 | C22 | semidirect |
| 25. #104 | C22 | semidirect |
| 26. #100 | C22 | semidirect |
| 34. #41 | D4 | semidirect |
| 36. C2×Q8 | D4 | semidirect |
| 51. C2×D4 | D4 | semidirect |
| 53. C2×D4 | C23 | semidirect |
| 61. M16 | C23 | semidirect |
| 64. C42 | C23 | semidirect |
| 78. Q8 | C2×D4 | semidirect |
| 88. D4 | C2×D4 | semidirect |
| 124. C2 | #453 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2) | 2 |
| C2×AGL(1,ℤ/8) | 2 |
| C2×#127 | 2 |
| #453 | 2 |
| C2×#104 | 2 |
| C2×#100 | 2 |