How can one find all compact connected subgroups of a classical compact Lie group? Suppose I have a compact matrix Lie group $G$.  (Although, really, I'm primarily interested in the cases of $G = \text{SO}(n)$, $\text{SU}(n)$, $\text{U}(n)$, $\text{Sp}(n)$, $\text{G}_2$, $\text{Spin}(n)$, for relatively low values of $n$.)
How does one go about finding all of the compact, connected (positive-dimensional) Lie subgroups of $G$, up to conjugacy, and the inclusions among them?
I would especially appreciate seeing an explicit example worked out, if possible.  For example, how might one determine the entire "tree" of (compact, connected) subgroups of, say, $\text{SU}(3)$?
Edit: I feel like there are probably tables or diagrams of these things somewhere, but I haven't located a source.
 A: Let do $SU(3)$ (which is 8-dimensional). First you want to classify semisimple subgroups. The only simply connected semisimple compact groups of dimension $<8$ are $SU(2)^k$ for $k=0,1,2$.
To classify copies of $SU(2)$, we discuss: if non-irreducible, we get the copy acting on a 2-plane and fixing its orthogonal. It's essentially immediate that this is unique up to automorphism of $SU(3)$. Next, the irreducible case: for this we use that $SU(2)$ has a unique (up to iso) irreducible 3-dimensional representation, which preserves a scalar product (e.g., realized as the adjoint action). Uniqueness says in principle that two such embeddings $SU(2)\to SU(3)$ are conjugate by some element of $GL_3$, but the fact that the scalar product is unique up to rescaling implies that the conjugating element is a scalar multiple of an element of $SU(3)$, and hence can be chosen in $SU(3)$.
Thus we get exactly two copies of $SU(2)$ (the non-irreducible one, and the irreducible one, which is actually a copy of $SO(3)$).
Both copies have a virtually abelian centralizer; in particular there is no copy of $SU(2)^2$. 
Next, we have to consider tori. Maximal tori are 2-dimensional and have a trivial centralizer; they are all conjugate by standard theorems; one is the set $D$ of diagonal matrices with determinant 1 and diagonal entries of modulus 1. 
Then we have the 1-dimensional subtori of the latter; there are infinitely many.
The centralizer $T_1$ of the non-irreducible copy of $SU(2)$ is one of these subtori (the one with a double weight space: in $D$ it corresponds, for instance, to the condition $d_1=d_2$ where $d_i$ are the diagonal coefficient). All 1-dimensional tori not conjugate to this special torus $T_2$ are contained in a unique maximal torus. Two 1-dimensional subtori of $D$ are conjugate if and only if they are in the same orbit for the Weyl group (group of permutations of coordinates). 
There is a second distinguished 1-dimensional torus: the one $T_0$ contained in some (non-unique) conjugate of $SO(3)$. This is the torus given as diagonal matrices $(d,1,d^{-1})$. It is also the only torus embeddable in a (unique) copy of $SU(2)$. 
To summarize, we have, up to conjugation, $\{1\}$, countably many 1-dimensional torus (including two special ones $T_0,T_1$), a unique 2-dimensional torus, an non-irreducible copy of $SU(2)$, an irreducible copy of $SO(3)$; a copy of $S(U(2)\times U(1))$, itself isomorphic to $SU(2)\times U(1)$, and $SU(3)$. 
Up to conjugation: the nontrivial minimal inclusions are: the 1-dimensional tori in the maximal one, $T_0$ in $SU(2)$ and $SO(3)$, $T_1$ in the center of $S(U(2)\times U(1))$, the maximal torus in $S(U(2)\times U(1))$. 
