On translation and dilation invariant subspaces of $L^p(\mathbb R^n)$, $0<p<1$

We prove that each translation and dilation invariant subspace $X\subset L^p(\mathbb R^n)$, $X\ne L^p(\mathbb R^n)$, is contained in a maximal translation and dilation invariant subspace of $L^p(\mathbb R^n)$. Moreover, we prove that the set of all maximal translation and dilation invariant subspaces of $L^p(\mathbb R^n)$ has the power of the continuum.