Multiplicative groups modulo $p$, for certain classes of primes, can be constructed to my knowledge from the following sets with straightforward group operations:
$\mathbb{Z}_p$, with standard multiplication $\mod p$ as the operation,
$a+b\sqrt{2}, a,b\in\mathbb{Z}_p$, with $(a+b\sqrt{2})(c+d\sqrt{2})=(ac+{2}bd \bmod p)+\sqrt{2}(ad+bc \bmod p)$ as the operation,
$a+b\sqrt[3]{2}+c\sqrt[3]{2^2},a,b,c\in\mathbb{Z}_p$, with such an operation,
and so on. Since $0$, $(0,0)$, $(0,0,0)$, and so on, respectively, are not elements of the group for the common group operations, the sizes of the groups are $p-1$, $p^2-1$, $p^3-1$, and so on.
Obviously, one can make groups of multiples of these orders, e.g. $p(p+1)(p-1)^2$. Additionally, some clear subgroup orders are present, such as the group of quadratic nonresidues in $\mathbb{Z}_p$ having order $(p-1)/2$. Apart from such cases, is it possible to construct other multiplicative groups modulo $p$ with different sizes? Specifically, what are the group operations that make this possible if so? Examples would be appreciated if possible. If not possible, why not?
E.g. The size of a group given by the set of unordered pairs $\{a,b\},$ $a,b\in\mathbb{Z}_p,a\neq b$ would have size $1/2(n)(n-1)$, but is there a possible group operation for this set?
Edit: I mean "multiplicative groups" in the sense that there is a "$0$" element of the set that is not included in the group.
Edit2: This question is not on a specific group operation, but rather whether there exists ANY group operation that allows one to construct a "$\mod p$" group of a different size.