Let $G=C_2^n=\langle x_1,\dots,x_n \rangle$ the elementary abelian group of order $2^n$. There is an isomorphism $(\mathbb{Z}/2\mathbb{Z})^n \to G$, $a=(a_1,\dots,a_n) \mapsto x^a=x_1^{a_1}\cdots x_n^{a_n}$.
$\chi_u : G \to \{\pm 1\} $, $x^a \mapsto (-1)^{ua}$ is a homomorphism for every $u \in (\mathbb{Z}/2\mathbb{Z})^n$, so $\ker \chi_u$ is a subgroup of $G$ of index $2$ for every nonzero $u \in (\mathbb{Z}/2\mathbb{Z})^n$. There are at least $2^{n}-1$ different subgroups of $G$ of index $2$.
How can I see, that there are exactly $2^{n}-1$ subgroups of $C_2^n$ of index $2$?
Perhaps in this particular case there is an easier way to see the statement than here: The number of subgroups of index two in $G$