Coexistence of nontrivial topology and flat electronic bands provides a fertile platform for correlated quantum states. The square-octagon lattice hosts Dirac nodes and flat bands at half-filling, yet the effects of intrinsic spin-orbit coupling (SOC) and staggered magnetic flux on its electronic and topological properties remain largely unexplored. Here, using tight-binding models incorporating SOC and staggered magnetic flux, we uncover a rich topological phase diagram in this lattice, comprising a quantum spin Hall phase with spin Chern number $C_s=1$, crossovers to quantum anomalous Hall phases with $C=1$ and $C=2$, and higher-order topological insulator phases with quantized quadrupolar corner charges. The initially dispersionless flat bands evolve into quasiflat topological bands with nearly uniform quantum geometry and large flatness ratios, making them promising candidates for fractional Chern insulator states. We further identify realistic materials, including octagraphene, transition-metal dichalcogenides, synthetic ${\mathrm{MoSi}}_2{\mathrm{N}}_4$, and magnetic $\alpha -{\mathrm{MnO}}_2$, that may realize these tunable topological phases intertwined with flat-band physics, opening new opportunities for correlated topological matter.

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• Flat bands
• Quantum anomalous Hall effect
• Quantum spin Hall effect
• Topological materials
• Tight-binding model