Recently, it was proposed that a nontrivial $Z_8$ topological index could induce double topological insulator (TI) surface states on the (001) surface of ${\mathrm{LaBaFe}}_4{\mathrm{As}}_4$ (LBFA). Here, we carry out a comprehensive investigation of the electronic structure, topological index, and topological surface states of other 1144-type iron-based compounds. We find that the coexistence of double band inversions in ${\mathrm{LaSrFe}}_4{\mathrm{As}}_4$ leads to a nontrivial $Z_8$ topological index and two sets of TI surface states on the (001) surface, similar to LBFA. On the other hand, when Ba is substituted by Ca, K, Rb, or Cs, the system is characterized by a nontrivial $Z_2$ topological index and exhibits only one set of TI surface states on the (001) surface. Further analysis shows that both the lattice parameter $c$ and the nominal valence of the alkali or alkaline earth metal atom are crucial in determining the bandwidth of La $5d_{x^2-y^2}$, topological index, and topological surface states. Moreover, we substitute La with Lu and Hf in LBFA and find that they also have a nontrivial $Z_8$ topological index. However, only one set of the double TI surface states is clearly visible on the (001) surface, whereas the other set is difficult to identify due to a different mechanism of the band inversion between As $4p_z$ and Fe $3d_{z^2}$ bands. Our findings extend previous studies on ${\mathrm{LaBaFe}}_4{\mathrm{As}}_4$ and provide insights for manipulating the topological index $Z_8$ or $Z_2$ and topological surface states in iron-based compounds.

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• Electronic structure
• Surface states
• Iron-based superconductors
• Topological materials
• Density functional theory
• Dynamical mean field theory
• First-principles calculations