Let $G := \langle a, b, c \mid ab = ba, c^2 = 1 \rangle$. By considering non-trivial homomorphisms $\phi : G \rightarrow \mathbb Z/2\mathbb Z$, there are exactly 7 subgroups of $G$ given by kernels of $\phi$.
Let $H_{(n, m, k)}$ be the kernel of $\phi: G \rightarrow \mathbb Z/2\mathbb Z$ with $\phi(a) = n$, $\phi(b) = m$, and $\phi(c) = k$.
Clearly, $H_{(0, 1, 0)} \cong H_{(1, 0, 0)}$ and $H_{(0, 1, 1)} \cong H_{(1, 0, 1)}$. Moreover, $H_{(n, m, 0)} \not\cong H_{(n', m', 1)}$ as $H_{(n, m, 0)}$ has an order 2 element, but $H_{(n', m', 1)}$ doesn't.
Now, I am not sure how to complete the classification. In particular, I need to prove whether the following pairs are isomorphic/non-isomorphic to each others: $(H_{(0, 1, 0)}, H_{(1, 1, 0)})$, $(H_{(0, 1, 1)}, H_{(1, 1, 1)})$, $(H_{(0, 1, 1)}, H_{(0, 0, 1)})$, and $(H_{(0, 0, 1)}, H_{(1, 1, 1)})$.
I would appreciate any hint or reference.