I'm looking for a reference describing the closed connected Lie subgroups of $SU(3)$. I know they are $SU(2)\times U(1)$, $SU(2)$, $SO(3)$ and several abelian subgroups based on this mathoverflow post. However the references given there only point to general information about Lie groups, and don't directly list the Lie subgroups of $SU(3)$.
Does anyone know of any direct reference for this fact?
The three maximal (proper) closed subgroups of $ SU_2 $ are known. The first is $ N(T) $ the normalizer of the maximal torus. The two other maximal closed subgroups are finite: the binary icosahedral group $ 2.A_5 \cong SL(2,5) $ of order $ 120 $ and the binary octahedral group $ 2.S_4 \cong SL(2,3):2 \cong SmallGroup(48,28) $ of order $ 48 $. For details see
What are the finite subgroups of $SU_2(C)$?
What are the maximal closed subgroups of $ SU_3 $? https://arxiv.org/pdf/math/0605784.pdf classifies all maximal closed subgroups of $ SU_n $ whose identity component is not simple (here trivial counts as simple). According to this paper, pages 1013-1018, the maximal closed subgroups of $ SU_3 $ of this type are the normalizer of the maximal torus $$ N(T)=S(U_1 \times U_1 \times U_1)\rtimes S_3 $$ As well as $$ S(U_2 \times U_1 )\cong U_2 $$ The maximal closed subgroups with trivial identity component are the finite groups: $$ 3.A_6 $$ of order $ 3(360)=1080 $ (known as the Valentiner group https://en.wikipedia.org/wiki/Valentiner_group) and $$ <\zeta_3I> \times GL_3(\mathbb{F}_2) $$ of order $ 3(168)=504 $. Both these two groups are central extensions by $ \zeta_3 I $ of a finite simple group. A third finite maximal closed subgroup is the complex reflection group with Shephard-Todd number 25
https://en.wikipedia.org/wiki/Complex_reflection_group
and order $ 3(216)=648 $ which happens to be a a central extension again by $ \zeta_3 I $ of the Hessian group https://en.wikipedia.org/wiki/Hessian_group of order 216. (since you are from quantum computing this last subgroup would be known to you as the (determinant 1 subgroup of the) qutrit Clifford group, which I denote $ S(Cl_1(3)) $.)
The only maximal closed subgroup of $ SU_3 $ with nontrivial simple identity component is the direct product $$ <\zeta_3I> \times SO_3(\mathbb{R}) $$
To summarize, the maximal closed subgroups of $ SU_3 $ are Type I (normalizer of a maximal connected subgroup) \begin{align*} & U_2 \cong S(U_2\times U_1)\\ & \zeta_3 \times SO_3(\mathbb{R}) \\ \end{align*} Type II (finite maximal closed subgroup) \begin{align*} & \zeta_3 \times GL(3,2) \\ & 3.A_6\\ & S(Cl_1(3)) \end{align*} Type III (normalizer of a positive dimensional non maximal connected subgroup) $$ N(T)=S(U_1 \times U_1 \times U_1)\rtimes S_3 $$ the 1 component is not maximal connected because it is contained in the rank 2 subgroup $ U_2 $ above.
Again, $ S(Cl_1(3)) $ is the determinant one subgroup of the single qutrit Clifford group i.e. Shephard-Todd-25.
Also since unitary $ t $ designs are popular in quantum computing it may be of interest to you that $ 3.A_6 $ is a unitary 3-design and $ \zeta_3 \times GL_3(\mathbb{F}_2) $ and $ S(CL_1(3)) $ are both unitary 2-designs.
This is consistent with claim 3 of https://math.stackexchange.com/a/4477296/758507 that all maximal $ 2 $-design subgroups of $ SU_n $ ( all $ 3 $ designs are also $ 2 $ designs) are finite maximal closed subgroups of $ SU_n $.
The only other closed subgroups of $ SU_3 $ which are unitary $ t $-designs for $ t \geq 2 $ are the $ GL_3(\mathbb{F}_2) $ subgroup of $ \zeta_3 \times GL_3(\mathbb{F}_2) $ and the commutator subgroup of the qutrit Clifford group, which has size $ 3(72)=216 $. These are again unitary $ t $-designs for $ t=2 $.
