There are $5$ non-cyclic groups of order $128$ with an element of order $64$, namely
$$C64 \times C2\ ,\ C64 : C2\ ,\ D128\ ,\ QD128\ ,\ Q128$$
Similar, there are $5$ non-cyclic groups of order $256$ with an element of order $128$, namely
$$C128\times C2\ ,\ C128:C2\ ,\ D256\ ,\ QD256\ ,\ Q256$$
Are there exactly $5$ non-cyclic groups of order $2^n $ with an element of order $2^{n-1}$ for every $n\ge 7$ , and are the groups analogue to the groups listed above ?
I also counted the groups of order $128$ with an element of order $32$, but no element with a larger order, and similar the groups of order $256$ with an element of order $64$ , but no element with a larger order. I got $27$ in both cases. Is this a coincide, or does it hold for every $2^n$ with $n\ge 7$ ?