Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
376 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 108 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#137 | C2 | semidirect |
| 3. C22×SD16 | C2 | semidirect |
| 4. #536 | C2 | semidirect |
| 5. #452 | C2 | semidirect |
| 6. #450 | C2 | semidirect |
| 7. #418 | C2 | semidirect |
| 8. #416 | C2 | semidirect |
| 9. #408 | C2 | semidirect |
| 16. C22×Q8 | C22 | semidirect |
| 22. C2×Q16 | C22 | semidirect |
| 24. C2×SD16 | C22 | semidirect |
| 25. C2×SD16 | C22 | semidirect |
| 26. C2×M16 | C22 | semidirect |
| 27. C22×C8 | C22 | semidirect |
| 31. C22⋊Q8 | C22 | semidirect |
| 32. #117 | C22 | semidirect |
| 33. #103 | C22 | semidirect |
| 34. #102 | C22 | semidirect |
| 35. #102 | C22 | semidirect |
| 36. C22⋊C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2) | 2 |
| C2×#137 | 2 |
| C22×SD16 | 2 |
| #536 | 2 |
| #452 | 2 |
| #450 | 2 |
| #418 | 2 |
| #416 | 2 |
| #408 | 2 |