Direct product of Z8 and Z4 and V4

Definition
This group is a finite abelian group of order $$2^7 = 128$$ given as the direct product of cyclic group:Z8, cyclic group:Z4, and Klein four-group.

As an abelian group of prime power order
This group is the abelian group of prime power order corresponding to the unordered integer partition:

$$\! 7 = 3 + 2 + 1 + 1$$

and the prime $$p = 2$$. In other words, it is the group $$\mathbb{Z}_{p^3} \times \mathbb{Z}_{p^2} \times \mathbb{Z}_p \times \mathbb{Z}_p$$.

Note
This group is the smallest example of a finite abelian group within which we can find series-equivalent subgroups that are not automorphic subgroups. For more, see series-equivalent not implies automorphic in finite abelian group.