Z4 in direct product of Z4 and Z2

Definition
The group $$G$$ is a direct product of Z4 and Z2, which, for convenience, we describe by ordered pairs with the first members from the integers mod 4 (the first direct factor cyclic group:Z4) and the second member from the integers mod 2 (the second direct factor cyclic group:Z2). It has elements:

$$\! \{ (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1) \}$$

We are interested in two subgroups $$H_1$$ and $$H_2$$, both isomorphic to cyclic group:Z4, that are automorphic subgroups (i.e., $$H_1$$ can be sent to $$H_2$$ by an automorphism of $$G$$). We have:

$$H_1 = \{ (0,0), (1,0), (2,0), (3,0) \}$$

$$H_2 = \{ (0,0), (1,1), (2,0), (3,1) \}$$

The corresponding quotient groups in both cases are isomorphic to cyclic group:Z2.

Note that both of these are direct factors of the whole group. They are precisely the non-characteristic subgroups of the whole group of order four. There is also a characteristic subgroup of order four, given by $$\{ (0,0), (0,1), (2,0), (2,1) \}$$, which is described on the page first omega subgroup of direct product of Z4 and Z2.

Cosets
$$H_1$$ is a normal subgroup of $$G$$, so its left cosets coincide with its right cosets. The two cosets are as follows:

$$\! \{ (0,0), (1,0), (2,0), (3,0) \}, \{ (0,1), (1,1), (2,1), (3,1) \}$$

$$H_2$$ is a normal subgroup of $$G$$, so its left cosets coincide with its right cosets. The two cosets are as follows:

$$\! \{ (0,0), (1,1), (2,0), (3,1) \}, \{ (0,1), (1,0), (2,1), (3,0) \}$$

Dual subgroup
We know that subgroup lattice and quotient lattice of finite abelian group are isomorphic, which means that there must exist an automorphism class of subgroups of $$G$$ that plays the role of dual subgroups to $$H_1$$ and $$H_2$$ -- in particular, that is isomorphic to the quotient group $$G/H_1$$ and its quotient group is isomorphic to $$H_1$$. This automorphism class of subgroups is given by non-characteristic order two subgroups of direct product of Z4 and Z2.

GAP implementation
The group-subgroup pair can be defined using the DirectProduct, CyclicGroup, Image, and Embedding as:

G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Image(Embedding(G,1));