$G$ be finite abelian group. $H$ be subgroup of $G$. $H$ be direct product of cyclic subgroups $H_1$, $H_2$, .., $H_m$. Do there exist cyclic subgroups $H_{m+1}$, .., $H_n$ such that $G$ is direct product of $H_1$, $H_2$, .., $H_n$?
I'm quite confused.
No, not always. Take $G=C_4$ and $H$ the unique subgroup of order $2$ of $G$. Note the following: you are basically asking whether it is true that if $H$ is a subgroup of the finite abelian group $G$ then $H$ has a complement in $G$. That will certainly not be true if $H \leq \Phi(G)$, as in the example above.