Find generators of finite abelian group, s.t. subgroup embeds diagonally

Question

Let $G$ be a finite abelian group and $T \subseteq G$ a subgroup. Can we find a "minimal set of generators" $g_i \in G$, i.e. $G \cong \bigoplus_{i=1}^n \, \langle g_i\, |\, g_i^{k_i}=0\rangle$, s.t. there are $l_i | k_i$ so that $T = \bigoplus_{i=1}^n \, \langle g_i^{l_i} \rangle$ ?

Answer 1

Not necessarily. A counterexample is $G = C_8 \times C_2 = \langle a \rangle \times \langle b \rangle$ with $T = \langle (a^2,b) \rangle$. Then $T \cong C_4$, but the generators of $T$ are not powers of any element of order $8$.