•The thermal conductivities κ of three carbon allotropes were investigated by HNEMD, EMD, and NEMD methods.•The κ of biphenylene network is only about one-thirteenth of the value of graphene.•The great reduction in κ of biphenylene network and pentaheptite arises from the decline of structural symmetry.•Phonon MFP, phonon group velocity, elastic modulus and ELF are analyzed to reveal the mechanism. Display omitted The biphenylene network with periodically arranged four-, six-, and eight-membered rings has been successfully synthesized in very recent experiments. This novel two-dimensional (2D) carbon allotrope has potentials in applications of lithium storage and carbon-based circuitry. Understanding the thermal transport properties of biphenylene network is of critical importance for the performance and reliability of its practical applications. To this end, the thermal transport in biphenylene network is comprehensively investigated in this paper with the aid of molecular dynamics simulations together with first-principles calculations. For the sake of comparison, the thermal conductivities of other 2D sp2-hybridized carbon allotropes including graphene and pentaheptite are also investigated using the same method. It is found that the thermal conductivities of biphenylene network and pentaheptite are, respectively, only about one-thirteenth and one-eighth of graphene. Through the analysis of phonon property, mechanical property and electron density distribution, it is demonstrated that the great reduction in the thermal conductivity of biphenylene network and pentaheptite arises from the decline in their structural symmetry, which leads to the decrease of phonon group velocity and the reduction of phonon mean free path.