Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
556 conjugacy classes of subgroups; only the 404 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 404 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #556 | C2 | direct (N × H) |
| 2. C2×C4⋊Q8 | C2 | semidirect |
| 3. C2×#126 | C2 | semidirect |
| 5. C2×C22⋊Q8 | C2 | semidirect |
| 6. C2×C4×Q8 | C2 | semidirect |
| 7. C2×C4×Q8 | C2 | semidirect |
| 9. C2×#117 | C2 | semidirect |
| 14. C4⋊Q8 | C22 | semidirect |
| 15. #126 | C22 | semidirect |
| 17. C22⋊Q8 | C22 | semidirect |
| 18. C4×Q8 | C22 | semidirect |
| 19. C4×Q8 | C22 | semidirect |
| 21. #117 | C22 | semidirect |
| 101. C2 | #556 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #556 | 2 |
| C2×C4⋊Q8 | 2 |
| C2×#126 | 2 |
| C2×#124 | 2 |
| C2×C22⋊Q8 | 2 |
| C2×C4×Q8 | 2 |
| C2×C4×D4 | 2 |
| C2×#117 | 2 |