Supermassive black hole binaries (SMBHB) are expected to be detected by the future space-based gravitational-wave detector LISA with a large signal-to-noise ratio (SNR). This prospect enhances the possibility of differentiating higher harmonics in the inspiral-merger-ringdown (IMR) waveform. In this study, we test the ability of LISA to identify the presence of different modes in the IMR waveform from a SMBHB. We analyze the contribution of each mode to the total SNR for different sources. We show that higher modes, in particular, the mode $(3, 3)$ and $(4, 4)$, can dominate the signal observed through the LISA detector for SMBHB of the order of $10^8 M_\odot$. With Bayesian analysis, we can discriminate models with different IMR modes. While higher modes are often considered to be orthogonal, it is no longer the case in the merger-ringdown phase. Therefore, omitting harmonics not only diminishes the SNR but can also lead to biases in the parameter estimation. We analyze the bias for each model for our example system and quantify the threshold SNR where we can expect the parameter bias to be comparable to the statistical error. Our work highlights the importance of higher modes to describe the gravitational waveform of events detected by LISA.