After posting the previous answer (and going to bed), I realized you probably are looking for something more concrete (i.e. geometric). I don't know any systematic way of solving this type of problem geometrically but here's the answer for $D_4$.
I will use the labels for rotations and reflections that appear in Gallian's Contemporary Abstract Algebra textbook.

$D_4$ acts on the various points, lines, and arrows in this picture. By picking the right objects (or groups of objects) you can get any subgroup to appear as a stabilizer (i.e. isotropy) subgroup.
$D_4$ itself stabilizes the red dot (the origin).
Pick one of the green dots (just off of a diagonal) and you'll find that only the identity, $\{ R_0 \}$, stabilizes that dot.
Pick an orange dot and either $\{R_0, D \}$ or $\{R_0, D' \}$ will stabilize it.
Pick a blue dot and either $\{R_0, V \}$ or $\{ R_0, H \}$ will stabilize it.
If you pick a diagonal line, $\{R_0, R_{180}, D, D' \}$ stabilizes.
If you pick a horizontal or vertical line, $\{R_0, R_{180}, H, V \}$ stabilizes.
Getting rotations alone is a little trickier. We have to break the reflective symmetry. That's why I've included some arrows to act on.

Acting on pairs of arrows (like for example the pair circled on the left), you'll get the isotropy subgroup $\{R_0, R_{180} \}$ (which is missing from your list above).
Acting on groups of four arrows (like those circled on the right), you'll get the isotropy group $\{R_0, R_{90}, R_{180}, R_{270} \}$ (all rotations).
I think that does it.
In general, the orbit-stabilizer theorem will help guide you to find the right objects to act on. It says the size of an orbit times the size of a stabilizer is equal to the size of the group. Thus when looking to find something whose isotropy group is all of $D_4$, I need to find something that is in an orbit of size $8/8=1$ -- it needs to be in an orbit by itself (thus the origin!). When I'm looking for a point only fixed by the identity, I'll need it to have an orbit with $|D_4|=8$ things in it (the green dots!).
The other thing to keep in mind is to keep certain things out of an orbit, you need to pick objects which violate that symmetry (the arrows don't play nice with the reflective symmetries).
Finally, as I said above, I'm not sure how to do this in general. Given a subgroup, I can probably cook up some collection of geometric objects fixed by it. But knowing I haven't missed a subgroup is a completely different story.