Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
244 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 96 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×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #533 | C2 | semidirect |
| 3. #492 | C2 | semidirect |
| 4. #486 | C2 | semidirect |
| 5. #460 | C2 | semidirect |
| 6. #450 | C2 | semidirect |
| 7. #431 | C2 | semidirect |
| 11. C2×SD16 | C22 | semidirect |
| 12. C4⋊Q8 | C22 | semidirect |
| 13. C4⋊Q8 | C22 | semidirect |
| 22. C4×Q8 | C22 | semidirect |
| 25. C4⋊C8 | C22 | semidirect |
| 26. #102 | C22 | semidirect |
| 28. C4×C8 | C22 | semidirect |
| 75. C2×C4 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533 | 2 |
| C2×#127 | 2 |
| #492 | 2 |
| #486 | 2 |
| #460 | 2 |
| #450 | 2 |
| #431 | 2 |