<s>[INST] order of $GL(2,\mathbb{Z}_2)$ list all elements show that the group $GL(2,\mathbb{Z}_2)$ has order 6 by listing all its elements

I think that $GL(2,Z_2)$ is 2x2 matrices with $Z_2$ elements that is 
$$\begin{pmatrix} 0 & 0 
               \\ 0 & 0  \end{pmatrix} 
,\begin{pmatrix} 1 & 1 
               \\ 1 & 1  \end{pmatrix} $$ 
$$
\begin{pmatrix} 1 & 0 
               \\ 0 & 0  \end{pmatrix} 
,
\begin{pmatrix} 0 & 1 
               \\ 0 & 0  \end{pmatrix} 
\begin{pmatrix} 0 & 0 
               \\ 1 & 0  \end{pmatrix} 
\begin{pmatrix} 0 & 0 
               \\ 0 & 1  \end{pmatrix} $$
$$
\begin{pmatrix} 1 & 1 
               \\ 0 & 0  \end{pmatrix} 
\begin{pmatrix} 0 & 1 
               \\ 1 & 0  \end{pmatrix} 
\begin{pmatrix} 0 & 0 
               \\ 1 & 1  \end{pmatrix} 
\begin{pmatrix} 0 & 1 
               \\ 0 & 1  \end{pmatrix} 
\begin{pmatrix} 1 & 0 
               \\ 0 & 1  \end{pmatrix} 
$$
$$
\begin{pmatrix} 0 & 1 
               \\ 1 & 1  \end{pmatrix} 
\begin{pmatrix} 1 & 0 
               \\ 1 & 1  \end{pmatrix} 
\begin{pmatrix} 1 & 1 
               \\ 0 & 1  \end{pmatrix} 
\begin{pmatrix} 1 & 1 
               \\ 1 & 0  \end{pmatrix} 
$$
I got that there are 15 elements so that would make it order of 15. but there are suppose to be 6 right?
 [/INST] By definition, elements in $GL(2,Z_2)$ have nonzero determinants, so you have to delete those matrix with determinants equals to 0.
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