Let \(L/K\) be a finite separable field extension, and let \(E\) be the normal closure of \(L\) over \(K\). \textit{C. Greither} and \textit{B. Pareigis}, [in J. Algebra 106, 239-258 (1987; Zbl 0615.12026)], describe a procedure for finding \(K\)-Hopf algebras such that \(L/K\) is Hopf Galois. Such Hopf algebras are equivalent to finding subgroups of the permutation group of \(\Gamma/\Gamma'\), where \(\Gamma=\text{Gal}(E/K)\) and \(\Gamma'=\text{Gal}(E/L)\), which are regular (transitive and fixed-point free) as well as normalized by \(\Gamma\) under left regular representation \(\lambda\colon\Gamma\to\text{Perm}(\Gamma/\Gamma')\). By regularity, such a subgroup must have order \(|\Gamma/\Gamma'|=[L:K]\). In the case where \(L=E\), i.e. \(L/K\) is Galois, we have \(\Gamma'\) is trivial, hence we have a \(K\)-Hopf algebra structure for every regular subgroup of \(B:=\text{Perm}(\Gamma)\) normalized by \(\Gamma\). In this work, let \(L/K\) be a Galois extension with Galois group \(\Gamma\) of order \(4p\), \(p>3\) prime. The objective here is to find all the subgroups of \(B\) satisfying the two conditions above. Interestingly, not all such subgroups are isomorphic (as abstract groups) to each other, and hence the author classifies them by their isomorphism class. The groups of order \(4p\) are well-known, in fact there are four or five, depending on \(p\): the two Abelian, the dihedral, the quaternion, and if \(p\equiv 1\bmod 4\) the group \(E_p\) generated by \(x\) and \(t\) subject to the relations \(x^p=t^4=t^{-1}xtx^\zeta=1\), where \(\zeta\in(\mathbb{Z}/p\mathbb{Z})^\times\) has order \(4\). In each case \(\Gamma\) has a unique subgroup \(\mathcal P\) of order \(p\), and it is shown that any regular subgroup \(N\leq B\) is a subgroup of the normalizer of \(\lambda(\mathcal P)\), denoted \(\text{Norm}_B(\mathcal P)\). This subgroup \(\mathcal P\) is generated by a product of four disjoint \(p\)-cycles \(\pi_1\pi_2\pi_3\pi_4\). Let \(P_1=\langle\pi_1\pi_2\pi_3\pi_4\rangle\), \(P_2=\langle\pi_1\pi_2\pi_3^{-1}\pi_4^{-1}\rangle\), \(P_3=\langle\pi_1\pi_2^{-1}\pi_3\pi_4^{-1}\rangle\), \(P_4=\langle\pi_1\pi_2^{-1}\pi_3^{-1}\pi_4\rangle\), and in the case \(p\equiv 1\bmod 4\) let \(P_5=\langle\pi_1\pi_2^{-1}\pi_3^\zeta\pi_4^{\overline\zeta}\rangle\) and \(P_6=\langle\pi_1\pi_2^{-1}\pi_3^{\overline\zeta}\pi_4^\zeta\rangle\) where \(\overline\zeta\neq\zeta\) has order \(4\) in \((\mathbb{Z}/p\mathbb{Z})^\times\). Thus it suffices to find the regular subgroups \(N\) in \(\text{Norm}_B(\mathcal P)\). The author shows that \(\text{Norm}_B(\mathcal P)\) is a semi-direct product \((C_p^4\rtimes U_p)\rtimes S_4\), \(U_p\) cyclic. Using this semi-direct product notation one can find all the regular subgroups normalized by \(\lambda(\Gamma)\) by looking at all possible cases. These are presented in a large table, and as each such \(N\) contains exactly one of \(P_1,\dots,P_6\) above the table counts how many such \(N\) contain each \(P_i\). For example, in the case \(\Gamma=C_{4p}\) and \(p\equiv 1\bmod 4\) there are \(10\) such \(N\): one cyclic containing \(P_1\); one Abelian but not cyclic containing \(P_1\); four isomorphic to \(E_p\), two containing \(P_1\) and the other two containing \(P_5\) or \(P_6\); two dihedral containing \(P_1\) or \(P_2\); and two quaternion containing \(P_1\) or \(P_2\). The table is followed with very explicit computations, realizing each such \(N\) as a subgroup of \((C_p^4\rtimes U_p)\rtimes S_4\). It is interesting to note that in the nonabelian cases the number of such subgroups increases linearly as \(p\) increases (except that the quaternion case fluctuates depending on \(p\bmod 4\)), yet in the Abelian cases the number of such \(N\) is \(10\) or \(6\) if \(\Gamma\) is cyclic (depending again on \(p\bmod 4\)) and \(16\) otherwise.