State troops are, technically speaking, soldiers of the army of a State, not under the jurisdiction of the United States.
The Constitution says the Congress can raise and support armies, and make rules for their government (Art. I, sec. 8, clause 12, 14). It also says the States can not keep troops or ships of war during peacetime, or engage in war, etc., without the permission of Congress unless actually invaded, or in such imminent Danger as will not admit of delay. (Art. 1, sec. X, clause 3).
These forces (armies/troops) are distinct from the militia. The Congress may provide (by laws) for the arming, organization and discipline of the militia of the United States, and for governing such as may be employed in the service of the United States, but may not appoint the officers, or train them according to the discipline prescribed by Congress (Art. I, sec. 8, clause 16). Ordinarily, the US can call them forth only to repel invasions, resist insurrections, and execute the laws of the Union. (Art. I, sec. 8, clause 15). Among the States severally, a well-regulated militia "being necessary" to the security of each free State, the people's right to keep and bear arms is not infringed. (2nd Amdmnt.)
The President of the United States is the commander in chief of the army and navy, and of the militia of the several States when in the actual service of the United States. (Art. II. sec. 2). There is frequently alot of unnecessary confusion on the subject as militia in active service are sometimes informally called "troops" and federal Volunteer troops are frequently called "State troops." In any case the important thing is the law under which the various troops are authorized to be raised and supported, either under one of the federal Constitution's provisions noted above, or by the provisions of a State's law and Constitution.
During the War of 1812, for example, besides providing militia and volunteer troops to the US service, several States raised their own troops/armies for their own defense external to US authorities. Connecticut among them:
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Late in that war, as noted above, Congress passed a law providing for the raising of an army of 40,000 "State Troops" to be absorbed into federal service, for local defense. These were besides the hundreds of thousands of militia and volunteer troops already employed by calling forth or by quotas for volunteers, etc.
The only Army in the US in 1860 was the regular army of the United States.
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In contrast, in 1860 the general return of the militia of the United States showed
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During the secession crisis, several Southern States "seceded" from the Union, and immediately raised "troops" from among their militia, particularly their uniformed volunteer-militia companies that volunteered for this service, among others raised specifically for it (the States claiming in part that they were no longer bound by the US Constitution, etc.). Among them Louisiana, Alabama, etc. After the organization of the "Provisional" government of the Confederate States in Feb., 1861 (with similar constitutional provisions as in that of the United States), it allowed for a Confederate regular army, and a "Provisional army" composed of the State Troops (State regular troops) turned over to it (the Confederacy then flipping the bill for their maintenance, etc. thereafter). The Confederate government also called for the recruiting of large numbers of Volunteer troops from the several States, but as in the US service these were considered distinct from the State troops.
The State of Virginia meanwhile, which had not seceded, raised an army, under "imminent" threat of invasion, but only "seceded" on May 23, and turned its army/troops etc. over to the Provisional Army of the Confederate States, etc. Besides the State troops absorbed into the Provisional Army, the Confederacy allowed for a large force of "volunteer" troops from the several States. The Confederate constitution also allowed the CSA to employ the militia of the States in the same measure as previously, but the massive mobilization generally suppressed the militia by drawing forth its manpower into the armies, etc. especially after the inception of conscription in April, 1862.
Both sides employed Volunteer troops for the most part, derived by quotas levied upon the States. These were not militia, nor regular army, but a separate "army" if you will. The States were only employed in their basic organization, and according to the federal laws. They were generally named in the order of their organization, viz. 10th Pennsylvania Volunteer Infantry Regiment, or 32nd Viginia Volunteer infantry, etc. etc. An example of the separation of the Regulars and Volunteers can be seen in the ranks of the various commanders. George A. Custer in the United States service, at the close of the war, was a Major General of Volunteers, but only a captain in the regular army. When the volunteers were disbanded at wars' end, he returned to the ranks of the regular army, and died in 1876 as a lieutenant-colonel.
Besides these Confederate armies/troops, considering the state of war which the Confederacy recognized, several Confederate States maintained their own troops for their own defense external to Confederate authority. Georgia for example did so. For example in 1864, one brigade of Georgia State troops provided for Confederate service included:
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An excellent book on the subject is this one:
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Joe Brown's Army: Georgia State Line
Same author, wrote a similar history of Georgia's wartime militia:
Joe Browns' Pets book
Although the lions share of the Virginia troops were turned over to the Confederacy in mid-1861, and the State's militia subject to calling forth by the Confederate government, the State maintained a separate "State Line" formed in 1862, commanded by Major General John B. Floyd, and some partisan ranger corps until early, 1863, when its passed acts turning these over to the Confederacy for continued service, or otherwise disbanding them.
Here is a return of that State force in early 1863, just over 3,000 strong:
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This left Virginia with only the barest remnants of its militia, exempt from the Confederate draft, which eventually included men up to 50 years (militia ages were 18 to 45). Some States formed "Home Guards" to include any able bodied men of any age to replace their militia, etc. Virginia noted by 1863:
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On the part of the United States, besides the Regular Army, President Lincoln after Fort Sumter called for 75,000 Militia for three months of active service to overthrow the "combinations" in the South obstructing the enforcement of the laws of the Union. The majority of those units provided were from among the uniformed volunteer-militia of the States. On May 4, he called for 500,000 "volunteers" (troops) from the several States, and over 700,000 were subsequently recruited. The Regular Army of the United States, and the number of volunteers, was expanded during the war, and a national draft instituted in 1863. As noted, many regular army officers, served with the volunteers during the war. Lincoln also continued to call forth militia from the several States, including 300,000 for nine months service in August, 1863, etc.
Here's the US totals regarding volunteers etc. provided under quotas called for by Lincoln's administration:
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Here's an interesting, and brief, discussion of the current United States "Armies" etc.
Youtube: Current United States Armies...