Weimar politics and political parties can frankly get a little bewildering. Here is an attempt to take a stab at the German Democratic Party's place in that history.

The first thing to note is that the German Democratic Party was ideologically liberal (in the classical liberal sense), but was not the only liberal party in Weimar Germany. To its right there was a second liberal party, the German People's Party. Without getting too deep into Wilhelmine politics, it should be noted that these two liberal parties descended from splits among liberals that began back in Bismarck's chancellorship: the German People's Party was more or less the descendant of the Progressive People's Party, which itself was formed from various left-liberal splits from the National Liberal Party, which had gone on to form the German People's Party (with some members of the Free Conservatives, while the rest of the Free Conservatives had joined with the German Conservative Party in 1918 to form the right-wing German National People's Party).

The German People's Party was in general skeptical of the Weimar republic, but under the leadership of Gustav Stresemann grudgingly participated in Weimar governments from 1923 to 1929 (with Stresemann acting as foreign minister). At his death in 1929 that party moved to the right, but also lost much of its influence.

The German Democratic Party might not have been explicitly confessional, but was heavily based in the German Protestant middle classes. A Protestant theologian, Friedrich Naumann, was its first leader in 1919 (although other founders like Theodor Wolff and Hugo Preuss were from Jewish middle class backgrounds). Other major members were Hjalmar Schacht (who left in 1926 and would ultimately serve as Hitler's Economics Minister), sociologist Max Weber, and Walther Rathenau (who would serve as Foreign Minister until his assassination in 1922). While a lot of these figures were liberals, many came from previous careers working in the Wilhelmine administration to one degree or another, or advocating for nationalist causes, and so the party and its leaders were perhaps first and foremost anti-Bolshevik, but otherwise susceptible to a general rightward drift in German politics (Schacht being perhaps only the most notable example).

Anyway, specifically to the "Weimar Coalition" of the Social Democrats, Democratic Party and Center Party. First, a table:

Election	SDP Vote	SDP Seats	ZP Vote	ZP Seats	DDP Vote	DDP Seats	Total Vote	Total Seats
January 1919	37.86	163	19.67	91	18.56	75	76.09	329
June 1920	21.92	103	13.64	64	8.28	39	43.84	206
May 1924	20.52	100	13.37	65	5.65	28	39.54	193
Dec 1924	26.02	131	13.6	69	6.34	32	45.96	232
May 1928	29.76	153	12.07	61	8.71	45	50.54	259
Sept 1930	24.53	143	11.81	68	3.78	20	40.12	231

What we can see are a few trends. First, 1919 was unique, and after this election, the "coalition" never really had a majority of votes or seats in the Reichstag, and relied on other parties (like the German People's Party) for forming governments. Part of the reason that 1919 was unique was that many middle class voters tactically supported the coalition as the best chance to stop "Bolshevism" in the form of more radical parties to the left. Once this immediate danger dissipated, many of those voters moved to more conservative options (again, the German People's Party for liberal voters, and the Bavarian People's Party for Bavarian Center voters, although this party later tactically supported the Center Party in the Reichstag).

The coalition members were themselves pulled in different directions. The Social Democrats found themselves in the strange position of being the largest party supporting the Weimar Republic, and largely responsible for its governance, while also still being a Marxist party committed to the overthrow of capitalism by the proletariat. They faced competition from the left by the Independent Social Democrats and the Communists, who had split from the main Social Democrats over issues related to World War I and the 1918-1919 German Revolution, and who did support the republic. The party was largely based in the state of Prussia, and its political opponents were very much focused on ousting it from power there, which finally happened in the Preussenschlag or "Prussian Coup" of June 1932, when Hindenberg's Chancellor Franz von Papen ousted the Weimar coalition running the state and instituted direct federal rule.

While it participated in governing coalitions throughout the entire Weimar era, and was instrumental in working with Social Democrats to craft social welfare provisions for the republic, the Center Party was largely committed to Catholic issues (fighting pornography and contraception, maintaining Catholic educational institutions), above all relations between Germany and the Vatican. This left the Party and its leaders open to both the general rightward trend in German politics, and open to the idea that its goals might best be met by negotiating with more authoritarian elements in politics (as it seemed to have worked out well with the Mussolini-Vatican Concordat in 1929).

Finally we get to the German Democratic Party. Unlike the Social Democrats (who permanently lost a section of middle class voters but otherwise bounced back and maintained a decent base) or the Center Party (which largely kept its numbers consistent), it had the most precipitate fall. A major reason for this, again, was that it was torn between defending political centrism, defending the Weimar Republic (which it enthusiastically defended), and following its putative base of middle class voters, who began to shift rightwards. Its vote share in the 1928 elections finally prompted its leaders to pursue the latter strategy, and the party actually merged with a paramilitary organization called Young German Order in 1930 to form the German State Party. The hope was that this would create a strong centrist party that could effectively combat the rise of the NSDAP, but the results were disastrous. The Young German Order was (like many other paramilitaries in Germany at the time) very skeptical of Weimar parliamentary politics, and its outlook was not immune from anti-Semitism. Likewise, this merger alienated many left-wing members of the party, who resigned, while also closing off potential mergers with larger parties. The party shifted sharply to the right, and while it continued to participate in coalitions until 1932, it declared (via historian Friedrich Meinecke) a goal of shifting power from the Reichstag to a unitary presidential government. The German Democratic Party found itself responding to a dwindling voting base by pushing itself further to the right, but ironically this just made it less competitive compared to the stronger, far-right and fascist parties, notably the NSDAP.

Source: Richard J. Evans. The Coming of the Third Reich