Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
231 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 88 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×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C22⋊Q8 | C2 | semidirect |
| 3. #484 | C2 | semidirect |
| 4. #480 | C2 | semidirect |
| 5. #468 | C2 | semidirect |
| 6. #450 | C2 | semidirect |
| 7. #448 | C2 | semidirect |
| 8. #407 | C2 | semidirect |
| 9. C22×Q8 | C22 | semidirect |
| 13. C2×SD16 | C22 | semidirect |
| 14. C2×M16 | C22 | semidirect |
| 18. C22⋊Q8 | C22 | semidirect |
| 25. C2×C4⋊C4 | C22 | semidirect |
| 28. #107 | C22 | semidirect |
| 29. #103 | C22 | semidirect |
| 30. #102 | C22 | semidirect |
| 32. C22⋊C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊Q8 | 2 |
| C2×#121 | 2 |
| #484 | 2 |
| #480 | 2 |
| #468 | 2 |
| #450 | 2 |
| #448 | 2 |
| #407 | 2 |