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Label Name Family name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
8.4 $Q_8$ $\Dic_{2}$ $2^{3}$ $2^{2}$ 5 $C_2$ $C_2^2$ $C_2$ $C_{2}^{2}$ Nilpotent - 2
12.1 $C_3:C_4$ $\Dic_{3}$ $2^{2} \cdot 3$ $2^{2} \cdot 3$ 6 $C_2$ $S_3$ $C_3$ $C_{4}$ Solvable - 2
16.9 $Q_{16}$ $\Dic_{4}$ $2^{4}$ $2^{3}$ 7 $C_2$ $D_4$ $C_4$ $C_{2}^{2}$ Nilpotent - 3
20.1 $C_5:C_4$ $\Dic_{5}$ $2^{2} \cdot 5$ $2^{2} \cdot 5$ 8 $C_2$ $D_5$ $C_5$ $C_{4}$ Solvable - 2
24.4 $C_3:Q_8$ $\Dic_{6}$ $2^{3} \cdot 3$ $2^{2} \cdot 3$ 9 $C_2$ $D_6$ $C_6$ $C_{2}^{2}$ Solvable - 2
28.1 $C_7:C_4$ $\Dic_{7}$ $2^{2} \cdot 7$ $2^{2} \cdot 7$ 10 $C_2$ $D_7$ $C_7$ $C_{4}$ Solvable - 2
32.20 $Q_{32}$ $\Dic_{8}$ $2^{5}$ $2^{4}$ 11 $C_2$ $D_8$ $C_8$ $C_{2}^{2}$ Nilpotent - 4
36.1 $C_9:C_4$ $\Dic_{9}$ $2^{2} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 12 $C_2$ $D_9$ $C_9$ $C_{4}$ Solvable - 2
40.11 $C_5\times Q_8$ $\Dic_{10}$ $2^{3} \cdot 5$ $2^{2} \cdot 5$ 25 $C_{10}$ $C_2^2$ $C_2$ $C_{2} \times C_{10}$ Nilpotent - 2
44.1 $C_{11}:C_4$ $\Dic_{11}$ $2^{2} \cdot 11$ $2^{2} \cdot 11$ 14 $C_2$ $D_{11}$ $C_{11}$ $C_{4}$ Solvable - 2
48.8 $C_3:Q_{16}$ $\Dic_{12}$ $2^{4} \cdot 3$ $2^{3} \cdot 3$ 15 $C_2$ $D_{12}$ $C_{12}$ $C_{2}^{2}$ Solvable - 2
52.1 $C_{13}:C_4$ $\Dic_{13}$ $2^{2} \cdot 13$ $2^{2} \cdot 13$ 16 $C_2$ $D_{13}$ $C_{13}$ $C_{4}$ Solvable - 2
56.3 $C_7:Q_8$ $\Dic_{14}$ $2^{3} \cdot 7$ $2^{2} \cdot 7$ 17 $C_2$ $D_{14}$ $C_{14}$ $C_{2}^{2}$ Solvable - 2
60.3 $C_{15}:C_4$ $\Dic_{15}$ $2^{2} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 18 $C_2$ $D_{15}$ $C_{15}$ $C_{4}$ Solvable - 2
64.54 $Q_{64}$ $\Dic_{16}$ $2^{6}$ $2^{5}$ 19 $C_2$ $D_{16}$ $C_{16}$ $C_{2}^{2}$ Nilpotent - 5
68.1 $C_{17}:C_4$ $\Dic_{17}$ $2^{2} \cdot 17$ $2^{2} \cdot 17$ 20 $C_2$ $D_{17}$ $C_{17}$ $C_{4}$ Solvable - 2
72.4 $C_9:Q_8$ $\Dic_{18}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 21 $C_2$ $D_{18}$ $C_{18}$ $C_{2}^{2}$ Solvable - 2
76.1 $C_{19}:C_4$ $\Dic_{19}$ $2^{2} \cdot 19$ $2^{2} \cdot 19$ 22 $C_2$ $D_{19}$ $C_{19}$ $C_{4}$ Solvable - 2
80.1 $C_5:C_{16}$ $\Dic_{20}$ $2^{4} \cdot 5$ $2^{4} \cdot 5$ 32 $C_8$ $D_5$ $C_5$ $C_{16}$ Solvable - 2
84.3 $C_3:C_{28}$ $\Dic_{21}$ $2^{2} \cdot 3 \cdot 7$ $2^{2} \cdot 3 \cdot 7$ 42 $C_{14}$ $S_3$ $C_3$ $C_{28}$ Solvable - 2
88.10 $Q_8\times C_{11}$ $\Dic_{22}$ $2^{3} \cdot 11$ $2^{2} \cdot 11$ 55 $C_{22}$ $C_2^2$ $C_2$ $C_{2} \times C_{22}$ Nilpotent - 2
92.1 $C_{23}:C_4$ $\Dic_{23}$ $2^{2} \cdot 23$ $2^{2} \cdot 23$ 26 $C_2$ $D_{23}$ $C_{23}$ $C_{4}$ Solvable - 2
96.1 $C_3:C_{32}$ $\Dic_{24}$ $2^{5} \cdot 3$ $2^{5} \cdot 3$ 48 $C_{16}$ $S_3$ $C_3$ $C_{32}$ Solvable - 2
100.1 $C_{25}:C_4$ $\Dic_{25}$ $2^{2} \cdot 5^{2}$ $2^{2} \cdot 5^{2}$ 28 $C_2$ $D_{25}$ $C_{25}$ $C_{4}$ Solvable - 2
104.11 $Q_8\times C_{13}$ $\Dic_{26}$ $2^{3} \cdot 13$ $2^{2} \cdot 13$ 65 $C_{26}$ $C_2^2$ $C_2$ $C_{2} \times C_{26}$ Nilpotent - 2
108.1 $C_{27}:C_4$ $\Dic_{27}$ $2^{2} \cdot 3^{3}$ $2^{2} \cdot 3^{3}$ 30 $C_2$ $D_{27}$ $C_{27}$ $C_{4}$ Solvable - 2
112.1 $C_7:C_{16}$ $\Dic_{28}$ $2^{4} \cdot 7$ $2^{4} \cdot 7$ 40 $C_8$ $D_7$ $C_7$ $C_{16}$ Solvable - 2
116.1 $C_{29}:C_4$ $\Dic_{29}$ $2^{2} \cdot 29$ $2^{2} \cdot 29$ 32 $C_2$ $D_{29}$ $C_{29}$ $C_{4}$ Solvable - 2
120.1 $C_3:C_{40}$ $\Dic_{30}$ $2^{3} \cdot 3 \cdot 5$ $2^{3} \cdot 3 \cdot 5$ 60 $C_{20}$ $S_3$ $C_3$ $C_{40}$ Solvable - 2
124.1 $C_{31}:C_4$ $\Dic_{31}$ $2^{2} \cdot 31$ $2^{2} \cdot 31$ 34 $C_2$ $D_{31}$ $C_{31}$ $C_{4}$ Solvable - 2
128.163 $Q_{128}$ $\Dic_{32}$ $2^{7}$ $2^{6}$ 35 $C_2$ $D_{32}$ $C_{32}$ $C_{2}^{2}$ Nilpotent - 6
132.3 $C_{33}:C_4$ $\Dic_{33}$ $2^{2} \cdot 3 \cdot 11$ $2^{2} \cdot 3 \cdot 11$ 36 $C_2$ $D_{33}$ $C_{33}$ $C_{4}$ Solvable - 2
136.11 $Q_8\times C_{17}$ $\Dic_{34}$ $2^{3} \cdot 17$ $2^{2} \cdot 17$ 85 $C_{34}$ $C_2^2$ $C_2$ $C_{2} \times C_{34}$ Nilpotent - 2
140.3 $C_{35}:C_4$ $\Dic_{35}$ $2^{2} \cdot 5 \cdot 7$ $2^{2} \cdot 5 \cdot 7$ 38 $C_2$ $D_{35}$ $C_{35}$ $C_{4}$ Solvable - 2
144.1 $C_9:C_{16}$ $\Dic_{36}$ $2^{4} \cdot 3^{2}$ $2^{4} \cdot 3^{2}$ 48 $C_8$ $D_9$ $C_9$ $C_{16}$ Solvable - 2
148.1 $C_{37}:C_4$ $\Dic_{37}$ $2^{2} \cdot 37$ $2^{2} \cdot 37$ 40 $C_2$ $D_{37}$ $C_{37}$ $C_{4}$ Solvable - 2
152.10 $Q_8\times C_{19}$ $\Dic_{38}$ $2^{3} \cdot 19$ $2^{2} \cdot 19$ 95 $C_{38}$ $C_2^2$ $C_2$ $C_{2} \times C_{38}$ Nilpotent - 2
156.4 $C_{13}:C_{12}$ $\Dic_{39}$ $2^{2} \cdot 3 \cdot 13$ $2^{2} \cdot 3 \cdot 13$ 48 $C_6$ $D_{13}$ $C_{13}$ $C_{12}$ Solvable - 2
160.8 $C_5:Q_{32}$ $\Dic_{40}$ $2^{5} \cdot 5$ $2^{4} \cdot 5$ 43 $C_2$ $D_{40}$ $C_{40}$ $C_{2}^{2}$ Solvable - 2
164.1 $C_{41}:C_4$ $\Dic_{41}$ $2^{2} \cdot 41$ $2^{2} \cdot 41$ 44 $C_2$ $D_{41}$ $C_{41}$ $C_{4}$ Solvable - 2
168.3 $C_3:C_{56}$ $\Dic_{42}$ $2^{3} \cdot 3 \cdot 7$ $2^{3} \cdot 3 \cdot 7$ 84 $C_{28}$ $S_3$ $C_3$ $C_{56}$ Solvable - 2
172.1 $C_{43}:C_4$ $\Dic_{43}$ $2^{2} \cdot 43$ $2^{2} \cdot 43$ 46 $C_2$ $D_{43}$ $C_{43}$ $C_{4}$ Solvable - 2
176.1 $C_{11}:C_{16}$ $\Dic_{44}$ $2^{4} \cdot 11$ $2^{4} \cdot 11$ 56 $C_8$ $D_{11}$ $C_{11}$ $C_{16}$ Solvable - 2
180.3 $C_{45}:C_4$ $\Dic_{45}$ $2^{2} \cdot 3^{2} \cdot 5$ $2^{2} \cdot 3^{2} \cdot 5$ 48 $C_2$ $D_{45}$ $C_{45}$ $C_{4}$ Solvable - 2
184.1 $C_{23}:C_8$ $\Dic_{46}$ $2^{3} \cdot 23$ $2^{3} \cdot 23$ 52 $C_4$ $D_{23}$ $C_{23}$ $C_{8}$ Solvable - 2
188.1 $C_{47}:C_4$ $\Dic_{47}$ $2^{2} \cdot 47$ $2^{2} \cdot 47$ 50 $C_2$ $D_{47}$ $C_{47}$ $C_{4}$ Solvable - 2
192.179 $C_3\times Q_{64}$ $\Dic_{48}$ $2^{6} \cdot 3$ $2^{5} \cdot 3$ 57 $C_6$ $D_{16}$ $C_{16}$ $C_{2} \times C_{6}$ Nilpotent - 5
196.1 $C_{49}:C_4$ $\Dic_{49}$ $2^{2} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 52 $C_2$ $D_{49}$ $C_{49}$ $C_{4}$ Solvable - 2
200.4 $C_{25}:Q_8$ $\Dic_{50}$ $2^{3} \cdot 5^{2}$ $2^{2} \cdot 5^{2}$ 53 $C_2$ $D_{50}$ $C_{50}$ $C_{2}^{2}$ Solvable - 2
204.3 $C_{51}:C_4$ $\Dic_{51}$ $2^{2} \cdot 3 \cdot 17$ $2^{2} \cdot 3 \cdot 17$ 54 $C_2$ $D_{51}$ $C_{51}$ $C_{4}$ Solvable - 2
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