The move is the latest in a crackdown on voices critical of the authorities in Egypt.
At the same time, a fifth member of the group known as Street Children arrested on Saturday was ordered released on bail.
Their latest production appears to have touched a nerve as police round up activists involved in April protests against President Abdel Fattah al-Sisi for handing over two islands to Saudi Arabia.
Rights groups accuse Sisi of running an ultra-authoritarian and repressive regime since he deposed in 2013 his democratically elected Islamist predecessor Mohamed Morsi.
Mahmud Ottman, a lawyer for the four, said they were arrested late Monday while visiting a friend's home in central Cairo.
On Tuesday, Mohammed Adel, Mohammed Gabr, Mohammed al-Dessouki and Mohammed Yehya were remanded in custody for 15 days, their lawyer and a judiciary official said.
In the group's latest video, Street Children mock the devaluation of the Egyptian pound as well as the return of the islands to Saudi Arabia.
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The four are accused of "promoting ideas calling for terrorist acts by posting a video on social networks and YouTube," Ottman said.
They are also suspected of "incitement to take part in demonstrations disturbing the public order" and "inciting mobs to commit hostile actions against state institutions," he added.
A Cairo court on Tuesday ordered the release on bail of 10,000 Egyptian pounds (about 990 euros) for a fifth member of the group, Ezzedine Khaled.
He is accused of "inciting protests and publishing a video that insulted state institutions".
Ottman said the bail had been paid and he expected Khaled to be released.
Attempts to protest last month against the handover of the two islands to Saudi Arabia were nipped in the bud by the authorities.
Since then, they have also cracked down on activists, bloggers, lawyers and journalists.
The overthrow of Morsi, who was deeply unpopular, unleashed a police crackdown on his supporters that has killed hundreds of protesters and imprisoned thousands of people.
Hundreds of people including Morsi have also been sentenced to death in speedy mass trials denounced by the United Nations as "unprecedented in recent history".