$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin} $Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $\frac{\SU(3)\times\SU(2)\times \SU(2)\times \U(1)}{\mathbb{Z}_6} \subset \U(16)$:
First we can embed the spin group $\Spin(10)\subset \U(16)$. Here we choose the $\mathbf{16}$-dimensional spinor representation of $\Spin(10)$ to be also the $\mathbf{16}$-dimensional fundamental representation of $\U(16)$. Thus, the data for the representation: $\text{$\mathbf{16}$ in $\Spin(10)$ as $\mathbf{16}$ in $\U(16)$}$ gives us an action of $\Spin(10)$ and $\U(16)$ on the complex vector space $\mathbb{C}^{16}$.
Then, we can embed $\frac{\SU(3)\times\SU(2)\times \SU(2)\times \U(1)}{\mathbb{Z}_6}$ into $\Spin(10)$. See the discussion of this embedding of $\SU(n)$ into $\Spin(2n)$. Here we choose:
the complex fundamental representation ${\mathbf 3}$ of SU(3),
the complex fundamental representation ${\mathbf 2}$ of (any of) SU(2).
the 1-dimensional representation (some number $k$) of $\U(1)$.
For example, we would denote the representation of ${\SU(3)\times\SU(2)\times \SU(2)\times \U(1)}$ as: $ ({\mathbf 3}, {\mathbf 2}, {\mathbf 1}, k) $ or $ ({\mathbf 3}, {\mathbf 1}, {\mathbf 2}, k) $, or others.
$$\text{$({\mathbf 3}, {\mathbf 2}, {\mathbf 1}, 1) \oplus ({\mathbf 1}, {\mathbf 2}, {\mathbf 1}, -3) \oplus ({\mathbf 3}, {\mathbf 1}, {\mathbf 2}, -1) \oplus ({\mathbf 1}, {\mathbf 1}, {\mathbf 2}, 3)$ in $\frac{\SU(3)\times\SU(2)\times \SU(2)\times \U(1)}{\mathbb{Z}_6}$ as a decomposition of $\mathbf{16}$ in $\U(16)$}.$$ The data for this above representation gives us an action of $\frac{\SU(3)\times\SU(2)\times \SU(2)\times \U(1)}{\mathbb{Z}_6}$ on $\mathbb{C}^{16}$, which further gives its embedding into $\U(16)$.
Question
Then my question is about the centralizer and normalizer of this $\frac{\SU(3)\times\SU(2)\times \SU(2)\times \U(1)}{\mathbb{Z}_6}$ inside $\U(16)$? Which certainly depends on the embedding that I provided above. So what are the centralizer and normalizer inside $\U(16)$? My suspicion is that other than some U(1) factors, there could be additional discrete finite groups.
