Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
244 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 100 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 | #121 |
| 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×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C22×D8 | C2 | semidirect |
| 3. C2×#131 | C2 | semidirect |
| 4. #472 | C2 | direct (N × H) |
| 5. C2×#108 | C2 | semidirect |
| 6. C2×#100 | C2 | semidirect |
| 10. AGL(1,ℤ/8) | C22 | semidirect |
| 16. C2×D8 | C22 | semidirect |
| 18. #131 | C22 | semidirect |
| 24. #108 | C22 | semidirect |
| 25. #100 | C22 | semidirect |
| 106. C2 | #472 | direct (N × H) |
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 |
| C22×D8 | 2 |
| C2×#131 | 2 |
| #472 | 2 |
| C2×#108 | 2 |
| C2×#100 | 2 |