We study the existence of families of periodic orbits near a symmetric equilibrium point in different classes of Hamiltonian systems with symmetry. We centre our attention to special types of symmetry less-studied in the literature, such as systems with (semi-)invariant Hamiltonian and reversible equivariant Hamiltonian systems, when the linearisation has two pairs of purely imaginary eigenvalues.In each case, we provide normal forms for the symmetries, the linear structure map and the linearisation. Moreover, the existence of symmetric and non-symmetric periodic orbits is proved. Another result we found is the classification of Hamiltonian systems with dihedral symmetry, of order eight, with all different possible combinations of time-reversing and symplectic-reversing actions.The method used in finding periodic orbits is the Liapunov-Schmidt reduction. The symmetry plays a vital role in determining the set of (semi-)invariants, in order to write the reduced problem and then to distinguish the solutions according to their symmetry type.