The processes in and around mitotic division in eukaryotes are very interesting. The short answer is their organelles do not replicate when the cell does. Some of these organelles have lost their own distinctive cycles however mitochondria (and chloroplasts in plants) have retained some independence. They still have their own DNA.This DNA is one long circular strand much like you would see in a prokaryotic cell. The mitochondrion has its own replication cycle, completely separate from the cell in which it resides. The mitochondria are dispersed throughout the cell so that when the cell divides some mitochondria wind up in one daughter cell and some in the other. This process to the best of our knowledge is not regulated so that a very unlucky cell could actually wind up with no mitochondria at all. (As an aside: This is one of the reasons scientists think that eukaryotic cells evolved from prokaryotic cells.)

Many of the other organelles do divide at the same time as the cell divides (especially organelles that do not have their own DNA). One example is illustrated in the endoplasmic reticulum. This structure divides into many pieces contained in vesicles which are then separated into the two daughter cells. This is a common way for organelles which only have one copy in the cell to segregate into the two daughter cells. These organelles do not appear to replicate before cell division the way DNA does.

During the gap phases (G1 and G2) the cell increases the amount of protein and organelles it contains in preparation for cytokinesis. How exactly the cell partitions its organelles during division is not well understood. It might be a stochastic process or perhaps there is some direction to it (via microtubules???) It is important to note that some organelles (i.e. mitochondria and chloroplast) have their own DNA and replicate themselves under the control of the cell cycle. Mitochondria divide by binary fission like bacteria.

It seems that detailed information about organelle divisions during mitosis is elusive. I was able to find the following cellbiology

"During the cell division process there is a reorganization of nearly all cell organelles and the cell cytoskeleton.

Interestingly, of the cell organelles, mitochondria appear to undergo there own cycles of division (similar to bacterial division) independent of the cell."

Mitochondrial division is in fact closely associated with cell division and is regulated at "distinct checkpoints" during mitosis, while mitochondrial morphology and segregation is controlled by microtubules in the cell.

A basic college biology textbook, "Life -- The Science of Biology" (Purves, Sadava, Orians, and Heller, 6th edition) states on page 164 "Followingcytokineses, both daughter cells contain all the components of a complete cell.... Organelles such as ribosome, mitochondria, and chloroplasts need not be distributed equally between daughter cells as long as some of each are present in both cells; accordingly, there is no mechanism with a precision comparable to that of mitosis to provide for their equal allocation to daughter cells."

Another website suggested that organelle division and synthesis occurred primarily during cytokineses cohmetrix

This is not an "edu" website so I can't vouch for its accuracy:

"Cytokineses, the second stage of cell division, begins to occur before mitosisis complete (usually during telophase) and continues after the nuclei of the daughter cells are completely formed. The preliminary steps of cytokineses occur during the growth interphases (called the G phases) of the cell cycle.In the G phases, various membrane structures and organelles, such as the endoplasmic reticulum and Golgi bodies, are produced out of components in the cytoplasm. Therefore, before cytokineses begins, there is growth in the size of the cytoplasm and in the number of its organelles. During the G phases there is also reproduction of the mitochondria and chloroplasts. These organelles contain their own DNA, called organelle DNA, and the organelles' reproduction includes the replication of the organelle DNA.

During cytokineses, the cytoplasm and its contents divide. In animal cells, the cytoplasm divides by pinching inward, whereas in plant cells, a partition, called the cell plate, begins to grow and divide the cytoplasm.Cytokineses is not as precise a process as mitosis because the amount of cytoplasm in a daughter cell will be about half, but not exactly half, the amount of cytoplasm in the parent cell. In addition, each daughter cell will have about half of the organelles from the cytoplasm of the parent cell. In contrast to mitosis, there is no precise mechanism working during cytokinesesto guarantee that each daughter cell receives exactly half of the parent cell's cytoplasm and its organelles.

Cytokineses does not always occur when mitosis occurs because in some cells (such as those found in certain molds) mitosis occurs repeatedly without cytokineses taking place. In this case, each repeated replication of genetic material with no division of cytoplasm (or final separation into new daughter cells) results in cells with two nuclei."