Let \(S\) be a semigroup. If \(a=axa\) for \(a,x\in A\) then \(x\) is called an associate of \(a\). A subgroup \(G\) of \(S\) is an associate subgroup of \(S\) if it contains exactly one associate of each element of \(S\). Every maximal subgroup of a regular semigroup \(S\) is an associate subgroup if and only if \(S\) is completely simple, or, equivalently, if \(S\) admits a unary operation \(*\) satisfying certain axioms. Any two associate subgroups of a semigroup are isomorphic; however, \(*\)-semigroups corresponding to two different associate subgroups need not be \(*\)-isomorphic. A homomorphism \(\chi\colon S\to T\) of regular semigroups is called \(\mathcal H\)-surjective if \(a\in S\), \(b\in T\), \(a\chi\mathcal Hb\) implies the existence of \(x\in S\) such that \(x\mathcal Ha\), \(x\chi=b\). The article presents (pairwise equivalent) descriptions of associate subgroups of normal cryptogroups. For example, a normal cryptogroup \(S\) has an associate subgroup if and only if \(S\) is an \(\mathcal H\)-surjective subdirect product of a normal band \(B\) such that \(B/\mathcal D\) is a monoid and a completely simple semigroup \(M\). Structure theorem of \(\mathcal H\)-surjective \(*\)-semigroups that are a subdirect product of a normal \(*\)-band and a completely simple \(*\)-semigroup is given as well.