The action of alone in the hyperplane seems to be straightforward. On the contrary, is it true that ?

I think you simply need to draw the subgroup lattice with a single bottom element and the next level elements , and , and the top element . You need to fill the intermediate elements between them.

For instance, every subgroup of fixes .2. For each of the subgroups write down, and justify, its Fixed Point Subspace when acting on the two dimensional surface given by .

What is a Fixed Point Subspace? How do you find it?

This part may need some lengthy computations and take every combination of and into account, and their isotropy subgroups.3. Draw the lattice of isotropy subgroups up to conjugacy.

Same problem as in 1.

If someone could enlighten me on this that would be great.

Thanks!