First omega subgroup of 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:

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

The subgroup $$H$$ is given as:

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

Cosets
$$H$$ is a subgroup of index two and hence a normal subgroup, so its left cosets and its right cosets coincide. The following are its two cosets:

$$\! H = \{ (0,0), (2,0), (0,1), (2,1) \}, G \setminus H = \{ (1,0), (1,1), (3,0), (3,1) \}$$

Dual subgroup
We know that subgroup lattice and quotient lattice of finite abelian group are isomorphic, which means that there must exist a subgroup of $$G$$ that plays the role of a dual subgroup to $$H$$ -- in particular, that is isomorphic to the quotient group $$G/H$$ and its quotient group is isomorphic to $$H$$. The subgroup is first agemo subgroup of direct product of Z4 and Z2.

GAP implementation
The group-subgroup pair can be constructed using the DirectProduct, CyclicGroup, and Filtered functions as follows:

G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Group(Filtered(G, x -> IsOne(x^2)));