You can appeal the rejection of your asylum application as "inadmissible" ("Unzulässig"). You must file the appeal at the responsible administrative court, the name and address of which is specified at the end of your rejection notice under the headline "legal remedies" ("Rechtsmittelbelehrung"). You have two weeks to file the appeal, starting from the date indicated on your rejection notice's envelope. If you intend to file an "urgent appeal" ("Eilantrag"), you must do so within a week. A successful urgent appeal prevents you from being deported during the legal process. Without an urgent appeal, you may be deported before the court decides your case. Please note, however, that an urgent appeal usually means that the above mentioned six-month transfer deadline will start over again. Make sure you consult a lawyer in this regard.
The chances of success are usually not significant in Dublin cases. Appealing the decision is especially worthwhile if "systemic deficiencies" ("Systematische Mängel") have been identified in the Dublin country which is responsible for your asylum application; i.e. when the responsible state cannot guarantee sufficient support or a fair asylum procedure.
In recent years, various courts have repeatedly identified systemic deficiencies in multiple European countries (including Greece, Italy, Bulgaria and Hungary) and banned the transfer of refugees to these countries as a result.
If the lawsuit is not successful, you can try to submit a petition to the state parliament ("Landtag") or the German parliament ("Bundestag") or seek refuge in a church to prevent your transfer to the other European country. If you want to opt for the petition, however, keep in mind that the relevant process is often lengthy, and the submission of a petition does not suspend your deportation/transfer, i.e. you may still be deported during the process. Church asylum ("Kirchenasyl") can be your last option in the face of imminent deportation, as it usually means that you spend your six-month-long transfer period in the church asylum and then Germany will be responsible for your asylum application. However, the original six-month deadline may be extended to 18 months as a result of church asylum. This is the case, namely when the authorities assume that you have taken refuge in a church to "hide" yourself from them. For more information seek help from a counselling centre or a lawyer.
To find a lawyer you can check, for instance, the Pro Asyl website. Their staff speak German and English and can be reached at +49 (0) 69-242 314 20 and email@example.com. To learn more about church asylum visit http://www.kirchenasyl.de.