Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
155 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #120 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #386 |
| prime radical | #575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #568 | C2 | semidirect |
| 3. #421 | C2 | semidirect |
| 4. #416 | C2 | semidirect |
| 5. #408 | C2 | semidirect |
| 6. #344 | C2 | semidirect |
| 7. #341 | C2 | semidirect |
| 11. #137 | C4 | semidirect |
| 13. AGL(1,ℤ/8) | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #575 | 2 |
| #568 | 2 |
| #421 | 2 |
| #416 | 2 |
| #408 | 2 |
| #344 | 2 |
| #341 | 2 |