Hate to necro a thread, but I have an answer. Before the Toseiha and the Kodoha, there was the Baden-Baden clique, which was started after WWI when all the Japanese military attaches in Europe got together at a spa in Germany. They decided that the Japanese army needed desperately to modernize, and that the domination of top jobs by samurai from Yamaguchi lineages had to end. The leading lights of both Kodoha and Toseiha were there before the split. Aside from the Toseiha and the Kodoha, there was also the Seigun-ha, or purification faction. This was officers from those Yamaguchi Prefecture samurai families, organized in reaction to Sadao Araki's construction of a new Saga-Tosa hanbatsu that was taking their place as the domineering regional clique. Very traditionalist, as none had been welcome at Baden-Baden tables. Their main deal was to stamp out insubordination and disobedience in the army by harshly punishing all perpetrators, which would incidentally remove their rivals. They were not effective advocates because of how obviously self-interested they were, but they are an important part of how Nagata Tetsuzan and the Toseiha kicked the Kodoha to the side.
Next, I should clarify something about the Kodoha. There was an Araki clique, and a movement of young officers. They were not the same thing. The young officers were big fans of Kita Ikki and had a tendency to latch on to big important men who seemed to agree with them (like Kazushige Ugaki). Araki was interested in creating a new dominant regional clique and maneuvering the country into another war with Russia in order to make himself the greatest military hero of Japanese history. When push came to shove, as it did in 1936, Araki rejected the young officers entreaties and supported the government. His actual base was high-up generals who were from either Saga or Tosa prefectures, like Tomoyuki Yamashita, Jinzaburo Masaki, and Toshiro Obata. The highest real (rather than imaginary) backers for the young officers were the army's imperial princes, specifically Asaka, Chichibu, and perhaps Higashikuni.
The Toseiha was more of an actual thing, but had crab-bucket tendencies. After Tetsuzan Nagata, the erstwhile leader, got whacked by a Young Officers colonel, the whole clique started to divide between supporters of Kanji Ishiwara and the many, many officers he'd insulted or offended. Kanji Ishiwara might have been a supporter of the Tohokai: of course he pretended to being apolitical, but Nakano Seigo was his main mouthpiece for things he wanted to get past the Diet, and he may have backed him financially. He was eventually ran out of the army on a rail. Later Hideki Tojo rose to prominence after becoming Prime Minister, and he became a lightning rod for all criticism of government policy. All military cliques collapsed into a pro-war, pro-Tojo movement and an anti-war, anti-Tojo movement. Seigo Nakano, Kanji Ishiwara, Prince Higashikuni, Prince Asaka, Jinzaburo Masaki, and Toshiro Obata all participated in the latter, despite prior disagreements.
Also worth noting is that support for all of the above were strictly limited to the home islands. Every area army outside of Japan was a clique of its own. The best example would be the Tianjin clique, which conducted the Marco Polo Incident by itself as part of a plan to increase its prominence vis-a-vis the Kwantung Army. They became the North China Clique when the war expanded, primarily opposed to the Central China Clique which had a very different plan for China. The North China clique wanted to secure its provinces as independent Japanese satellites to secure resources and ensure China would not rise again, while Central China just wanted to set up a government that could weaken Chiang's legitimacy. They set up rival Chinese governments, one in Beijing and one in Nanjing. Eventually Central China got the upper hand because they coaxed Wang Jingwei to their side and North China was forced to recognize his government to play into the new propaganda strategy Central China persuaded the higher-ups of(though North China still ran their territory as autonomous satellites in practice). North China might have won if they had gotten Yan Xishan to publicly switch sides before that point, as was their goal for most of the war. As it happens, they got Yan to privately collaborate with their control over most of Shanxi, but a formal accord remained elusive.
As for Konoe, he ruled past the time when the Kodoha was really relevant, but he supported them anyway as a counterweight to the dominant Toseiha. He was a good friend of Seigo Nakano as well, but rather cynically kept him out of government as he knew he could count on the Tohokai's support without giving them any ministries.
