Form Hilbert spaces are constructed over fields that are complete in a non-archimedean valuation. They share with classical Hilbert spaces the basic property expressed by the Projection Theorem. However, there appear some
remarkable geometric features which are unknown in Euclidean geometry. In fact, due to the so-called type condition
there are only a few orthogonal straight lines containing vectors of the same length, so these non-archimedean spaces
are utmost inhomogeneous.
In the paper we consider a typical Form Hilbert space $(E, < , >)$ and we show that this geometric feature
has a strong impact on the group $\mathcal{O}(E)$ of all isometries $T:E\longrightarrow E$ and on the lattice
$\mathcal{L}$ of all normal subgroups of $\mathcal{O}$. In particular, we describe some remarkable sublattices of
$\mathcal{L}$ which have no analogue in the classical orthogonal groups.