Recently, silicon-based photonic integrated circuits (PICs) have attracted considerable interest due to the advantages of high index contrast and complementary metal oxide semiconductor compatible process. Furthermore, to meet the ever-growing bandwidth requirements for data center and supercomputing, several multiplexing on-chip technologies by using silicon PICs are proposed. Among them, the mode division multiplexing (MDM) is widely recognized to be important, where mode-order converters (MOCs) are fundamental building blocks. In addition, slot waveguides can efficiently confine the light in low-index regions, thus forming various kinds of novel photonic devices. In this paper, a compact 12 multimode interference (MMI) mode-order converter (MOC) for silicon-based slot nanowires is proposed, where straight waveguides, as the input/output channels, are connected to a quadratic-tapered multimode waveguide via linear-tapered waveguides. A full-vectorial finite-difference frequency-domain method is used to analyze the modal characteristics of the used silicon-based vertical slot waveguides; from this, quasi-TM mode is chosen as an input optical signal since its field distribution is strongly confined in the slot, i. e., the electric field strength is greatly enhanced in the vertical slot, and with the increase of the width of the slot waveguide, it can support higher-order quasi-TM modes. Compared with the beating length of rectangular MMI structure, the beating length of quadratic-tapered MMI structure can be effectively reduced with transmission loss lowering. From the imaging position of the guided-mode in MMI region via self-imagining effect, the length of quadratic-tapered MMI structure can be determined accurately where first-order and fundamental quasi-TM modes are outputs, respectively, from wider and narrower channels. A three-dimensional finite-difference time-domain method is utilized to assess the performance of the proposed MOC, where the insertion loss and crosstalk are analyzed in detail. The results show that an MOC with an MMI section of 35 m2 is achieved to be an insertion loss and a crosstalk of~0.35 dB and~-16.9 dB, respectively, at a wavelength of 1.55 m by carefully optimizing the key structural parameters. Moreover, the fabrication deviation of the proposed device is also analyzed in detail and the performance is evaluated, where insertion loss and the contrast are considered. To demonstrate the transmission characteristics of the proposed MOC, the evolution of the excited fundamental quasi-TM mode through the MOC is also presented. Numerical results show that the presented MOC realizes the desired function, converting the fundamental quasi-TM mode into first-order one with reasonable performance. We remark that the present MOC has a good potential application in MDM system to improve the capacity of the silicon-based on-chip transmission system.