We consider the problem of calculating the Hurewicz image of Mahowald's
family $\eta_i\in{_2\pi_{2^i}^S}$. This allows us to identify specific
spherical classes in $H_*\Omega_0^{2^{i+1}-8+k}S^{2^i-2}$ for $0\leqslant
k\leqslant 6$. We then identify the type of the subalgebras that these classes
give rise to, and calculate the $A$-module and $R$-module structure of these
subalgebras. We shall the discuss the relation of these calculations to the
Curtis conjecture on spherical classes in $H_*Q_0S^0$, and relations with
spherical classes in $H_*Q_0S^{-n}$.