In a typical introductory programming course, grading of student submitted programs typically involves an autograder that tests functionality of the programs with predefined testcases. However, in an educational setting, it is desirable for the educator to examine the source code prior to assigning the final grade for reasons such as checking for compliance to some criteria (e.g. ‘Use iteration, not recursion’, or ‘do not use additional arrays’). A rubric is often used by graders to grade according such criteria. However, manual grading of source code can be labor-intensive and impractical for large-scale online courses. Therefore, in this paper, we propose techniques based on Large Language Models for code to automatically grade student programs according to instructor-specified rubrics. Leveraging a graded dataset comprising 27 problems, each with approximately 140 submissions, to be graded over 210 criteria, we develop methodologies within the frameworks of Zero-Shot prompting, Few-Shot prompting, Supervised Fine Tuning, and Direct Preference Optimization (DPO), and compare their effectiveness on nine different LLMs for code. We also investigate approaches of code scrambling and code augmentation to make our models more robust. Our findings indicate that fine-tuning can result in 10-15% improvement in accuracy and F1-scores for smaller models such as Phi-3B, however larger models such as Codestral 22B gave accuracy of 86% and F1 score of 81% with and without fine-tuning, even for new, unique criteria specific to new problem statements, which were not a part of the fine-tuning. With these results, we believe we have a promising methodology and sufficient knowledge of SOTA model for actual deployment in autograder applications used in courses with large enrollments.