Finite Group Databasebeta

← C32×D3×D16 #13162346

Subgroups — C32×D3×D16 #13162346

Order 1728.

700 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×D8C212yesmaximal, complemented
2C33×D16C211yesmaximal, characteristic, complemented
3C32×#826C212yesmaximal, complemented
4C32×D48C211yesmaximal, characteristic, complemented
5C32×D3×C16C211yesmaximal, characteristic, complemented
6C2×C32×D16—31nomaximal
7C3×D3×D16C314yesmaximal, direct product factor
8C32×D3×D4—22no—
9C33×D8C2212yescomplemented
10C32×#213—22no—
11C32×D24C2212yescomplemented
12C32×D3×C8C2211yescharacteristic
13C33×C16C2211yescharacteristic, complemented
14C32×C3⋊C16C2211yescharacteristic, complemented
15C2×C32×D8—32no—
16C3×D3×D8C2×C318yescomplemented
17C32×D16—24no—
18C32×D16—32no—
19C32×D16—31no—
20C32×D16D311yescharacteristic, direct product factor
21C32×D16C2×C314yescomplemented
22C2×C32×C16—31no—
23C3×#826C2×C318yescomplemented
24C3×D48C2×C314yescomplemented
25C3×D3×C16C2×C314yescomplemented
26C22×C32×D3—42no—
27C33×D4—22no—
28C32×#67—42no—
29C32×D12—22no—
30C32×C4×D3D411yescharacteristic
31C33×C8C2311yescharacteristic
32C32×C3⋊C8D411yescharacteristic
33C2×C3×D16—34no—
34D3×D16C3211yescharacteristic, direct product factor
35C2×C32×D4—62no—
36C3×D3×D4—28no—
37C32×D8—28no—
38C32×D8—32no—
39C32×D8—62no—
40C32×D8C2×D312yescomplemented
41C32×D8C22×C318yescomplemented
42C2×C32×C8—31no—
43C3×#213—28no—
44C3×D24C22×C318yescomplemented
45C3×D3×C8C22×C314yes—
46C32×C16—24no—
47C32×C16—31no—
48C32×C16C2×D311yescharacteristic, complemented
49C32×C16C22×C314yescomplemented
50C3×C3⋊C16C22×C314yescomplemented
51C22×C33—42no—
52C2×C32×D3—42no—
53C2×C32×D3—84no—
54C2×C32×D3D811yescharacteristic
55C33×C4C2×D411yescharacteristic
56C32×Dic3D811yescharacteristic
57C2×C3×D8—38no—
58D3×D8C2×C3212yescomplemented
59C3×D16—24no—
60C3×D16—38no—
61C3×D16—34no—
62C3×D16C3×D314yesdirect product factor
63C3×D16C2×C3211yescharacteristic, complemented
64C2×C3×C16—34no—
65#826C2×C3212yescomplemented
66D48C2×C3211yescharacteristic, complemented
67D3×C16C2×C3211yescharacteristic, complemented
68C23×C32—122no—
69C22×C3×D3—48no—
70C32×D4—22no—
71C32×D4—28no—
72C32×D4—48no—
73C32×D4—62no—
74C32×D4—122no—
75C2×C32×C4—31no—
76C3×#67—48no—
77C3×D12—28no—
78C3×C4×D3C3×D414yes—
79C32×C8—24no—
80C32×C8—31no—
81C32×C8C22×D311yescharacteristic
82C32×C8C23×C314yes—
83C3×C3⋊C8C3×D414yes—
84C2×D16—31noSylow
85C2×C33—82no—
86C2×C33C2×D811yescharacteristic
87C32×D3—82no—
88C32×D3D1612yesdirect product factor
89C2×C3×D4—68no—
90D3×D4—22no—
91C3×D8—28no—
92C3×D8—38no—
93C3×D8—68no—
94C3×D8C2×C3×D318yescomplemented
95C3×D8C22×C3212yescomplemented
96C2×C3×C8—34no—
97#213—22no—
98D24C22×C3212yescomplemented
99D3×C8C22×C3211yescharacteristic
100C3×C16—24no—
101C3×C16—34no—
102C3×C16C2×C3×D314yescomplemented
103C3×C16C22×C3211yescharacteristic, complemented
104C3⋊C16C22×C3211yescharacteristic, complemented
105C22×C32—31no—
106C22×C32—42no—
107C22×C32—48no—
108C22×C32—88no—
109C22×C32—122no—
110C22×C32—244no—
111C2×C3×D3—48no—
112C2×C3×D3—816no—
113C2×C3×D3C3×D814yes—
114C32×C4—24no—
115C32×C4—31no—
116C32×C4D3×D411yescharacteristic
117C32×C4C2×C3×D414yes—
118C3×Dic3C3×D814yes—
119C2×D8—32no—
120D16—32no—
121D16—31no—
122D16C32×D311yescharacteristic, direct product factor
123C2×C16—31no—
124C33C2×D1611yescharacteristic, complemented, Sylow
125C23×C3—128no—
126C22×D3—42no—
127C3×D4—28no—
128C3×D4—22no—
129C3×D4—48no—
130C3×D4—68no—
131C3×D4—128no—
132C2×C3×C4—34no—
133#67—42no—
134D12—22no—
135C4×D3C32×D411yescharacteristic
136C3×C8—24no—
137C3×C8—34no—
138C3×C8C22×C3×D314yes—
139C3×C8C23×C3211yescharacteristic
140C3⋊C8C32×D411yescharacteristic
141C2×C32—24no—
142C2×C32—32no—
143C2×C32—82no—
144C2×C32—88no—
145C2×C32—168no—
146C2×C32—242no—
147C2×C32D3×D811yescharacteristic
148C2×C32C2×C3×D814yes—
149C3×D3—88no—
150C3×D3C3×D1618yesdirect product factor
151C2×D4—62no—
152D8—32no—
153D8—62no—
154D8C2×C32×D312yescomplemented
155C2×C8—31no—
156C16—31no—
157C16C2×C32×D311yescharacteristic, complemented
158C22×C3—34no—
159C22×C3—48no—
160C22×C3—42no—
161C22×C3—88no—
162C22×C3—128no—
163C22×C3—2416no—
164C2×D3—42no—
165C2×D3—84no—
166C2×D3C32×D811yescharacteristic
167C3×C4—24no—
168C3×C4—34no—
169C3×C4C3×D3×D414yes—
170C3×C4C2×C32×D411yescharacteristic
171Dic3C32×D811yescharacteristic
172C32—24no—
173C32D3×D1611yescharacteristic, direct product factor
174C32C2×C3×D1614yescomplemented
175C23—122no—
176D4—22no—
177D4—62no—
178D4—122no—
179C2×C4—31no—
180C8—31no—
181C8C22×C32×D311yescharacteristic
182C2×C3—24no—
183C2×C3—38no—
184C2×C3—88no—
185C2×C3—82no—
186C2×C3—168no—
187C2×C3—248no—
188C2×C3C3×D3×D814yes—
189C2×C3C2×C32×D811yescharacteristic
190D3—82no—
191D3C32×D1612yesdirect product factor
192C22—31no—
193C22—42no—
194C22—122no—
195C22—244no—
196C4—31no—
197C4C32×D3×D411yescharacteristic
198C3—24no—
199C3C3×D3×D1614yesdirect product factor
200C3C2×C32×D1611yescharacteristic, complemented
201C2—32no—
202C2—82no—
203C2—242no—
204C2C32×D3×D811yescharacteristic
subgroupG/H
centerC2×C32D3×D8
commutator, cosocleC3×C8C23×C32
FrattiniC8C22×C32×D3
FittingC33×D16C2
socleC2×C33C2×D8
hypercenterC32×D16D3
hypocenterC3C2×C32×D16
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×C8 ◁ C32×D16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C32×D3×D8C2semidirect
2. C33×D16C2semidirect
3. C32×#826C2semidirect
4. C32×D48C2semidirect
5. C32×D3×C16C2semidirect
7. C3×D3×D16C3direct (N × H)
9. C33×D8C22semidirect
11. C32×D24C22semidirect
13. C33×C16C22semidirect
14. C32×C3⋊C16C22semidirect
16. C3×D3×D8C2×C3semidirect
20. C32×D16D3direct (N × H)
21. C32×D16C2×C3semidirect
23. C3×#826C2×C3semidirect
24. C3×D48C2×C3semidirect
25. C3×D3×C16C2×C3semidirect
34. D3×D16C32direct (N × H)
40. C32×D8C2×D3semidirect
41. C32×D8C22×C3semidirect
44. C3×D24C22×C3semidirect
48. C32×C16C2×D3semidirect
49. C32×C16C22×C3semidirect
50. C3×C3⋊C16C22×C3semidirect
58. D3×D8C2×C32semidirect
62. C3×D16C3×D3direct (N × H)
63. C3×D16C2×C32semidirect
65. #826C2×C32semidirect
66. D48C2×C32semidirect
67. D3×C16C2×C32semidirect
88. C32×D3D16direct (N × H)
94. C3×D8C2×C3×D3semidirect
95. C3×D8C22×C32semidirect
98. D24C22×C32semidirect
102. C3×C16C2×C3×D3semidirect
103. C3×C16C22×C32semidirect
104. C3⋊C16C22×C32semidirect
122. D16C32×D3direct (N × H)
124. C33C2×D16semidirect
150. C3×D3C3×D16direct (N × H)
154. D8C2×C32×D3semidirect
157. C16C2×C32×D3semidirect
173. C32D3×D16direct (N × H)
174. C32C2×C3×D16semidirect
191. D3C32×D16direct (N × H)
199. C3C3×D3×D16direct (N × H)
200. C3C2×C32×D16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×D3×D82
C33×D162
C32×#8262
C32×D482
C32×D3×C162
C2×C32×D163
C3×D3×D163