In
Bump, Daniel, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics. 55. Cambridge: Cambridge University Press. xiv, 574 p. (1997). ZBL0868.11022.,
Exercise 4.5.1 on p. 487 it is claimed that every maximal compact subgroup of $\mathrm{GL}(2,\mathbb Q_p),\ p$ prime, is conjugate to a subgroup of $K:=\mathrm{GL}(2,\mathbb Z_p)$. However, I am confused by this result since there seems to be a maximal compact subgroup which is not conjugate to $K$, namely the subgroup
$$K_1:=\langle B,\omega\rangle_{group}\leq \mathrm{GL}(2,\mathbb Q_p),$$
where $B:=\left\{\begin{pmatrix} a&b\\ c&d \end{pmatrix}\in K\mid c\equiv 0\text{ mod }p\right\}$ and $\omega:=\begin{pmatrix} 0&p\\ 1&0 \end{pmatrix}$. On the Bruhat-Tits building $K$ corresponds to the stabilizer of the vertex given by the standard lattice $L_0:=e_1\mathbb Z_p+e_2\mathbb Z_p$ whereas $K_1$ stabilizes the midpoint between the vertices $L_0$ and $L_1:=e_1\mathbb Z_p+e_2p\mathbb Z_p$. In the overview article at https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.217.9089&rep=rep1&type=pdf it is also claimed that $K_1$ is maximal compact, but here for the case of the projective general linear group $\mathrm{PGL}(2,\Bbb Q_p)$.
Any ideas which detail I am overseeing here? Thanks for any help in advance!
As Aphelli pointed out in the comments, with $\omega$ also $\omega^{-2}=\begin{pmatrix}p^{-1}&0\\ 0&p^{-1} \end{pmatrix}$ is in $K_1$ and $p^{-n}$ diverges to $\infty$ in the $p$-adic norm. So $K_1$ is only compact in $\mathrm{PGL}(2,\Bbb Q_p)$ where $\omega^{-2}$ is the identity.