Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
188 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 100 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×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #544 | C2 | semidirect |
| 3. #479 | C2 | semidirect |
| 4. #478 | C2 | semidirect |
| 5. #477 | C2 | semidirect |
| 6. #474 | C2 | semidirect |
| 7. #428 | C2 | semidirect |
| 8. C2×#107 | C2 | semidirect |
| 9. #424 | C2 | semidirect |
| 42. #41 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #544 | 2 |
| #520 | 2 |
| #479 | 2 |
| #478 | 2 |
| #477 | 2 |
| #474 | 2 |
| #428 | 2 |
| C2×#107 | 2 |
| #424 | 2 |
| #417 | 2 |
| #416 | 2 |