Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
166 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 166 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 | #121 |
| 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. C4×#41 | C2 | semidirect |
| 3. #421 | C2 | semidirect |
| 4. C2×#104 | C2 | semidirect |
| 5. #410 | C2 | semidirect |
| 6. #344 | C2 | semidirect |
| 7. #339 | C2 | semidirect |
| 11. #137 | C4 | semidirect |
| 13. AGL(1,ℤ/8) | C4 | semidirect |
| 37. C2×Q8 | C2×C4 | semidirect |
| 41. C2×D4 | C2×C4 | semidirect |
| 55. M16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #575 | 2 |
| C4×#41 | 2 |
| #421 | 2 |
| C2×#104 | 2 |
| #410 | 2 |
| #344 | 2 |
| #339 | 2 |