Which invariant rings of finite reflection groups are also subrings of the ring of symmetric polynomials?

Question

The invariant rings of the finite reflection groups corresponding to root systems like $A_{n-1}$, $BC_n$ or $D_n$ are subrings of the ring of symmetric polynomials in $n$ variables. In the paper Polynomial Invariants and Harmonic Functions Related to Exceptional Regular Polytopes by Iwasaki et al., they give generators for the corresponding invariant rings of $H_3$, $H_4$, $F_4$ and they are all (even) symmetric polynomials, so these invariant rings are also subrings of the ring of (even) symmetric polynomials. It seems that for dihedral groups the above is not true (or does this depend on the representation?). (I haven't found papers that compute basic invariants for the groups corresponding to $E_6$, $E_7$ or $E_8$ or at least is not easy for me to see if the obtained generators are symmetric) So, is that true for other irreducible finite reflection groups? Also, how would one prove it without computing a set of generators?