Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
222 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 86 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 |
| 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. #532 | C2 | semidirect |
| 2. C2×#124 | C2 | semidirect |
| 3. #496 | C2 | semidirect |
| 4. #489 | C2 | semidirect |
| 5. #463 | C2 | semidirect |
| 6. #448 | C2 | semidirect |
| 7. #447 | C2 | semidirect |
| 8. #432 | C2 | semidirect |
| 12. C2×SD16 | C22 | semidirect |
| 14. C4⋊Q8 | C22 | semidirect |
| 15. #127 | C22 | semidirect |
| 24. C4×D4 | C22 | semidirect |
| 28. C4⋊C8 | C22 | semidirect |
| 29. #103 | C22 | semidirect |
| 32. #97 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #532 | 2 |
| C2×#124 | 2 |
| #496 | 2 |
| #489 | 2 |
| #463 | 2 |
| #448 | 2 |
| #447 | 2 |
| #432 | 2 |