Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
194 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | #95 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #344 | C2 | direct (N × H) |
| 13. C2×M16 | C4 | semidirect |
| 14. C2×M16 | C4 | semidirect |
| 15. C2×M16 | C4 | semidirect |
| 19. C22×C8 | C4 | semidirect |
| 50. M16 | C2×C4 | semidirect |
| 56. C2×C8 | C2×C4 | semidirect |
| 132. C2 | #344 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16 | 2 |
| C2×#117 | 2 |
| #344 | 2 |