Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
386 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×M16 | C2 | semidirect |
| 3. C2×#117 | C2 | semidirect |
| 4. #421 | C2 | direct (N × H) |
| 5. C2×#104 | C2 | semidirect |
| 10. C2×#41 | C4 | semidirect |
| 11. C2×#41 | C4 | semidirect |
| 14. C22×Q8 | C4 | semidirect |
| 16. C22×D4 | C4 | semidirect |
| 21. C2×M16 | C22 | semidirect |
| 22. C2×M16 | C22 | semidirect |
| 25. #117 | C22 | semidirect |
| 28. C2×C42 | C22 | semidirect |
| 29. #104 | C22 | semidirect |
| 40. #41 | C2×C4 | semidirect |
| 45. C2×Q8 | C2×C4 | semidirect |
| 53. C2×D4 | C2×C4 | semidirect |
| 72. M16 | C23 | semidirect |
| 78. C42 | C23 | semidirect |
| 149. C2 | #421 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41 | 2 |
| C22×M16 | 2 |
| C2×#117 | 2 |
| #421 | 2 |
| C2×#104 | 2 |