= Solution
A <subgroup> $N\leq G$ is a <normal subgroup> if $gNg^{-1}=N$ for every $g\in G$.
Place the cube with its opposite-face axes along the coordinate axes. Consider
$$
N=\left\{I,\ \operatorname{diag}(1,-1,-1),\ \operatorname{diag}(-1,1,-1),\ \operatorname{diag}(-1,-1,1)\right\}.
$$
The three nonidentity elements are the half-turns about those axes. Multiplying two different ones gives the third, and each squares to the identity, so this is a <Klein four-group> of order four.
Every cube rotation permutes the three pairs of opposite faces. The associated <group homomorphism> $G\to S_3$ has exactly $N$ as its kernel: fixing all three face-axis pairs allows only independent sign changes of the three coordinate directions, with <determinant> one. Hence the <kernel of the cube face-axis action> is a <normal subgroup>, and
$$
\boxed{N\trianglelefteq G,\qquad |N|=4.}
$$
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