Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
226 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C2 | C2×C4×D4 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #568 | C2 | semidirect |
| 3. #444 | C2 | semidirect |
| 4. #443 | C2 | semidirect |
| 5. C2×#108 | C2 | semidirect |
| 6. #421 | C2 | semidirect |
| 7. #405 | C2 | semidirect |
| 9. #135 | C4 | semidirect |
| 10. #135 | C4 | semidirect |
| 13. C2×Q16 | C4 | semidirect |
| 14. C2×SD16 | C4 | semidirect |
| 15. C2×D8 | C4 | semidirect |
| 17. #131 | C22 | semidirect |
| 23. #108 | C22 | semidirect |
| 24. #108 | C22 | semidirect |
| 25. #104 | C22 | semidirect |
| 26. #97 | C22 | semidirect |
| 27. C4×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135 | 2 |
| #568 | 2 |
| #444 | 2 |
| #443 | 2 |
| C2×#108 | 2 |
| #421 | 2 |
| #405 | 2 |