What are the maximal closed subgroups of $ \operatorname{SU}_3 $?

Question

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$What are the maximal closed subgroups of $ \SU_3 $?

This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that question, treating only the case of connected subgroups, are given by José Figueroa-O'Farrill and Neil Strickland.

Easier example: The three maximal closed subgroups of $ \SU_2 $ are the binary octahedral group, the binary icosahedral group and the normalizer of the maximal torus. Details given here: What are the finite subgroups of $SU_2(C)$?

My attempt: The maximal closed subgroups of $ \SU_3 $ are $$ U_2\cong S(U_2\times U_1) $$ and $$ \operatorname{SO}_3(\mathbb{R}) \times C_3 \cong \langle\operatorname{SO}_3(\mathbb{R}),\zeta_3 I\rangle $$ (I think this is the full normalizer, maybe there is some extra finite order stuff? Weirdly this type of subgroup doesn't seem to show up in table 5 of Antoneli, Forger, and Gaviria - Maximal Subgroups of Compact Lie Groups) and the normalizer of the maximal torus $$ N(T)=T^2 \rtimes S_3 $$ where here the symmetric group $ S_3=W $ is the Weyl group of $ \SU_3 $. The finite maximal closed subgroups are $$ 3 \times \Sigma_{168} \;, \; 3.\Sigma_{216},\;, \; 3.\Sigma_{360} $$ where $ \Sigma_{168} \cong \SL_3(2) $ is a the simple group of order $ 168 $, $ \Sigma_{360} \cong A_6 $ is the simple group of order $ 360 $ and $ \Sigma_{216} \cong \mathbb{F}_3^2 \rtimes \SL_2(3) $ is an affine transformation group, sometimes called the Hessian group. I believe that $ 3.\Sigma_{216} $ is the full automorphism group of an extra special $ 3 $-group of order $ 27 $ of $ + $ type, $ \operatorname{Aut}(3^{2+1}_+) $.

Answer 1

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Cl{Cl}$Yes the the above is the correct list of maximal closed subgroups of $ \SU_3 $.

Antoneli, Forger, and Gaviria - Maximal Subgroups of Compact Lie Groups 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 $$ N(T)=S(U_1 \times U_1 \times U_1): S_3 $$ 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) and $$ \langle \zeta_3I\rangle \times \GL_3(\mathbb{F}_2) $$ of order $ 3(168)=504 $. Both these two groups are central extensions by $\langle\zeta_3 I\rangle$ of a finite simple group. But the first is perfect central extension, indeed a Schur cover. While the second is just a direct product. A third finite maximal closed subgroup is the complex reflection group with Shephard–Todd number 25 (see Complex reflection group) and order $ 3(216)=648 $ which happens to be a central extension again by $\langle\zeta_3 I\rangle$ of the 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.)

The only maximal closed subgroup of $ \SU_3 $ with nontrivial simple identity component is the direct product $$ \langle\zeta_3I\rangle \times \SO_3(\mathbb{R}). $$

To summarize, the maximal closed subgroups of $ \SU_3 $ are \begin{align*} &N(T)\\ & S(U_2\times U_1)\cong U_2\\ & \langle\zeta_3 I\rangle \times \SO_3(\mathbb{R}) \\ & \langle\zeta_3 I\rangle \times \GL_3(\mathbb{F}_2) = 3 \times \text{PerfectGroup}(168) \\ & 3.A_6=\text{PerfectGroup}(1080)\\ & S(\Cl_1(3))=\text{SmallGroup}(648,532) \end{align*} where $ S(\Cl_1(3)) $ is the determinant-1 subgroup of the single qutrit Clifford group. Every closed subgroup of $ \SU_3 $ is contained in one of these $ 6 $ maximal groups.

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 my answer to Finite maximal closed subgroup of connected Lie group 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 $ \langle\zeta_3 I\rangle \times \GL_3(\mathbb{F}_2) $ and the commutator subgroup of the qutrit Clifford group, SmallGroup(216,88), which has size $ 3(72)=216 $. These are again unitary $ t $-designs for $ t=2 $.

In addition to all 5 of these 2-designs there is one other Lie primitive subgroup (i.e. not contained in any proper positive dimensional closed subgroup): it is a subgroup of the qutrit Clifford group of size $ 6(36)=216 $.