$F_2$ is free group with 2 generators. Now I want to know how many subgroup with index 2 it has. And what about index 3. For the latter case, how can I judge whether the subgroup is normal?
I am trying to solve this by considering covering space. For the index 2 case, I actually only get the wedge sum of 3 circles as covering space(or homotopy equivalent to this). I believe there should be others. For index 3 case, I find two kinds of covering space: wedge sum of 4 circles and 3 vertex with 3 circles. So which should be normal?