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


This question is inspired by [Lie subgroups of SU(3)](https://mathoverflow.net/questions/65522/lie-subgroups-of-su3). Interesting partial answers to that question, treating only the case of connected subgroups, are given by [José Figueroa-O'Farrill](https://mathoverflow.net/a/65530) and [Neil Strickland](https://mathoverflow.net/a/65834).


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 $SU\_2(C)$?](https://math.stackexchange.com/q/40351)


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](https://arxiv.org/abs/math/0605784)) 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}\_+) $.



What are the maximal closed subgroups of $ \operatorname{SU}_3 $?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Cl{Cl}$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](https://arxiv.org/abs/math/0605784)
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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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](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.)


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 PerfectGroup(168) \\
& 3.A\_6=PerfectGroup(1080)\\
& S(\Cl\_1(3))=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](https://math.stackexchange.com/a/4477296/758507) to [Finite maximal closed subgroup of connected Lie group](https://math.stackexchange.com/questions/4272017/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 $.


