You can appeal against the decision to refuse your child a place at any of the schools you applied for to an independent appeal panel. You will find details of how to make an appeal in the local authority's admission letter outlining your offer. The letter will also specify a deadline that you must meet if you want to appeal. Further details about the appeal process can be found on the GOV.UK website.
You may also wish to speak to the admissions team at your local council about the availability of places at other schools in your area.