Consider the compact group $ F_4 $. What are the maximal closed subgroups of $ F_4 $?
I think the full list is Maximal Subgroups of Type I (normalizer of a maximal connected subgroup): \begin{align*} & Spin_9 \\ & Sp(3) \times SU(2)\\ & (SU(3) \times SU(3)):2\\ & G_2 \times SU(2) \\ & SU(2)_{max} \\ \end{align*}
Here $ (SU(3) \times SU(3)):2 $ is the index $ 2 $ subgroup of $ \Sigma U(3) \times \Sigma U(3) $ given by @DanielSebald in the comments. In particular, recall that $ SU(3) $ has only one outer automorphism and it is of order $ 2 $, given by complex conjugation in the fundamental rep. $ \Sigma U(3) $ denotes $ SU(3) $ extended by its outer automorphism group. $ (\Sigma U(3) \times \Sigma U(3))\frac{1}{2} $ is a group given by extending $ SU(3) \times SU(3) $ by the automorphism of order $ 2 $ which simultaneously applies complex conjugation to both $ SU(3) $ factors.
Maximal Subgroups of Type II (finite maximal closed subgroup): \begin{align*} & N(PSL(2,25))=PSL(2,25).2 \; \text{almost simple}\\ & PSL(2,27) \; \text{simple}\\ & 3^3 \rtimes SL(3,3)=ASL(3,3) \; \text{normalizer of an elementary abelian $ p $ group}\\ &N(^3D_4(2))=^3D_4(2) \cdot 3 \; \text{almost simple} \end{align*}
Maximal Subgroups of Type III (normalizer of subgroup which is connected but not maximal connected) \begin{align*} N(Spin_8)=Spin_8 \rtimes S_3 \\ \end{align*}
Even though $ Spin_8 : 2 $ is a subgroup of $ Spin_9 $ there is no embedding of $ Spin_8 \rtimes S_3 $ into $ Spin_9 $. See
Does $ Spin(2n+1) $ always contain $ Spin(2n) \rtimes Out(Spin(2n)) $?
For the type I maximal I'm using prop 2.3 table 2 row 2 of
The length and depth of compact Lie groups
For the finite subgroups (type II maximal) of $ F_4 $ I'm using theorem 1.1 of this reference
Cohen and Wales - Finite Subgroups of $ F_4(\mathbb{C}) $ and $E_6(\mathbb{C})$
which says that all Lie primitive subgroups (Lie primitive means not contained in any positive dimensional closed subgroup) of $ F_4 $ are contained in either $$ ASL(3,3)=3^3 \rtimes SL(3,3) $$ or are contained in one of the primitive almost simple subgroups . The primitive simple subgroups are further detailed in
table PE $ F_4 $ column entries marked with a P for primitive. Note the Theorem in section 6 of this paper, originally due to Borovik I think, that any finite Lie primitive and thus certainly any finite maximal closed subgroup is either of normalizer type or is almost simple (or is a particular strange $ A_5 \times A_6$ related subgroup of $ E_8 $).
In particular the Lie primitive finite simple subgroups of $ F_4 $ $$ PSL(2,25),PSL(2,27),SL(3,3), ^3D_4(2) $$ The $ SL(3,3) $ is already accounted for in the $ ASL(3,3) $ subgroup of $ F_4 $ confirmed in multiple sources. So $ SL(3,3) $ is not a Socle but rather a normalizer type finite maximal closed subgroup. These are the possible simple Socles of finite maximal closed subgroups. It is possible that these groups are contained in a larger normalizer which is almost simple.
Note that $$ SL(2,8),PSL(2,13), PSL(2,17) $$ are Lie primitive but contained in $ ^3D_4(2) $
The maximality of the subgroups with simple socle $ PSL(2,25),PSL(2,27), ^3D_4(2) $ are shown at the end of section 8 of Cohen and Wales - Finite Subgroups of $ F_4(\mathbb{C}) $ and $E_6(\mathbb{C})$. And the full almost simple subgroups with these simple Socle are described in table 11 of the same reference.
Cohen and Wales - Finite Subgroups of $ F_4(\mathbb{C}) $ and $ E_6(\mathbb{C})$ does a thorough job of classifying the finite maximal closed subgroups of $ F_4 $. So I'm pretty sure about type II.
I am mostly curious about the positive dimensional maximal closed subgroups of $ F_4 $. My current list is almost totally speculative, with the exception of $ Spin(9) $ and $ Spin(8):S_3 $