Finite Group Databasebeta

← C2×#26042226 #37550171

Subgroups — C2×#26042226 #37550171

Order 23168.

1232 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 274 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
1C24×#10624382C211yesmaximal, characteristic
2C22×C4⋊#10624382C212yesmaximal, complemented
3C22×C22⋊#10624382C211yesmaximal, characteristic
4#26042226C218yesmaximal, direct product factor
5C2×(C2×#10624382)⋊C4C212yesmaximal, complemented
6C2×C22⋊C4×D181C211yesmaximal, characteristic
7C23×#10624382—28no—
8C23×#10624382—24no—
9C23×#10624382C2211yescharacteristic
10C23×#10624382C2214yes—
11C23×#10624382C2211yescharacteristic
12C23×#10624382C2212yes—
13C2×C4⋊#10624382—28no—
14C2×C4⋊#10624382C2218yescomplemented
15C2×C22⋊#10624382—24no—
16C2×C22⋊#10624382C2214yes—
17(C2×#10624382)⋊C4C2218yescomplemented
18C24×D181C2211yescharacteristic
19C22×C4×D181C2212yes—
20C2×C22⋊Dic181C411yescharacteristic, complemented
21C2×#15668426C412yescomplemented
22C22⋊C4×D181C414yescomplemented
23C22⋊C4×D181C2214yes—
24C2×C22⋊C4×C181C411yescharacteristic, complemented
25C22×#10624382—232no—
26C22×#10624382—216no—
27C22×#10624382—28no—
28C22×#10624382—24no—
29C22×#10624382—24no—
30C22×#10624382—22no—
31C22×#10624382—22no—
32C22×#10624382—24no—
33C22×#10624382—24no—
34C22×#10624382D418yes—
35C22×#10624382C2312yes—
36C22×#10624382C2312yes—
37C22×#10624382C2314yes—
38C4⋊#10624382—216no—
39C22⋊#10624382—28no—
40C23×D181—24no—
41C23×D181—24no—
42C23×D181C2×C412yes—
43C23×D181C2×C411yescharacteristic
44C23×D181D414yes—
45C23×D181Q814yes—
46C23×D181C2312yes—
47C23×D181C2311yescharacteristic
48C22×Dic181C2×C412yescomplemented
49C2×C4×D181—24no—
50C2×C4×D181—24no—
51C2×C4×D181C2×C414yescomplemented
52C2×C4×D181C2314yes—
53C22⋊Dic181C2×C414yescomplemented
54#15668426C2×C418yescomplemented
55C24×C181C2×C411yescharacteristic
56C22×C4×C181C2×C412yescomplemented
57C22⋊C4×C181C2×C414yescomplemented
58C2×#10624382—232no—
59C2×#10624382—216no—
60C2×#10624382—24no—
61C2×#10624382—24no—
62C2×#10624382—22no—
63C2×#10624382—22no—
64C2×#10624382—24no—
65C2×#10624382—24no—
66C2×#10624382—44no—
67C2×#10624382—44no—
68C2×#10624382C2×D418yes—
69C22×D181—24no—
70C22×D181—28no—
71C22×D181—28no—
72C22×D181—216no—
73C22×D181—28no—
74C22×D181—24no—
75C22×D181—24no—
76C22×D181C4⋊C418yescomplemented
77C22×D181C4⋊C414yescomplemented
78C22×D181C22×C412yes—
79C22×D181C22×C411yescharacteristic
80C22×D181C2×D414yes—
81C22×D181C2×D412yes—
82C22×D181C2×D414yes—
83C22×D181C2×Q814yes—
84C22×D181C2×Q812yes—
85C22×D181#4114yes—
86C22×D181C2411yescharacteristic
87C2×Dic181—24no—
88C2×Dic181C22×C414yescomplemented
89C4×D181—28no—
90C4×D181—24no—
91C4×D181—24no—
92C23×C181—24no—
93C23×C181C4⋊C414yescomplemented
94C23×C181C22×C412yes—
95C23×C181C22×C411yescharacteristic
96C2×C4×C181—24no—
97C2×C4×C181C22×C414yescomplemented
98C22×C181—24no—
99C22×C181—28no—
100C22×C181C2×C4⋊C412yes—
101C22×C181C2×C4⋊C414yescomplemented
102C22×C181C4×D414yes—
103C22×C181C23×C411yescharacteristic
104C2×D181—24no—
105C2×D181—28no—
106C2×D181—24no—
107C2×D181—28no—
108C2×D181—24no—
109C2×D181—216no—
110C2×D181C2×C4⋊C412yes—
111C2×D181C2×C4⋊C414yescomplemented
112C2×D181C4×D414yes—
113C2×D181#12014yes—
114C2×D181C22⋊Q818yes—
115C2×D181#12314yes—
116C2×D181C22×D412yes—
117C2×D181C22×D411yescharacteristic
118C2×D181C22×Q811yescharacteristic
119C2×D181C2×#4112yes—
120#10624382—28no—
121#10624382—42no—
122#10624382—42no—
123C4×C181—24no—
124Dic181—24no—
125C2×C181—24no—
126C2×C181#38514yescomplemented
127C2×C181C22×C4⋊C411yescharacteristic
128C2×C181C2×C4×D412yes—
129D181—24no—
130D181#38514yescomplemented
131D181C2×#12011yescharacteristic
132D181C2×C22⋊Q812yes—
133D181C2×#12311yescharacteristic
134C181C2×#38511yescharacteristic, complemented, Sylow
135C2×#385—1811nomaximal, Sylow
136C24×C4—1811no—
137C22×C4⋊C4—1812no—
138C22×C22⋊C4—1811no—
139C22×C22⋊C4—1811no—
140#385—1818no—
141C2×#95—1812no—
142C25—1811no—
143C23×C4—3624no—
144C23×C4—3628no—
145C23×C4—1811no—
146C23×C4—1811no—
147C23×C4—1814no—
148C23×C4—1812no—
149C23×C4—1812no—
150C2×C4⋊C4—3628no—
151C2×C4⋊C4—1818no—
152C2×C22⋊C4—3624no—
153C2×C22⋊C4—1814no—
154C2×C22⋊C4—1811no—
155C2×C22⋊C4—1812no—
156C2×C22⋊C4—1814no—
157C2×C22⋊C4—1814no—
158C2×C22⋊C4#1062438211yescharacteristic, complemented
159#95—1818no—
160C24—3624no—
161C24—3624no—
162C24—1814no—
163C24—1814no—
164C24—1812no—
165C24—1812no—
166C24—1811no—
167C24—1811no—
168C24C2×#1062438211yescharacteristic
169C22×C4—3624no—
170C22×C4—3624no—
171C22×C4—36232no—
172C22×C4—3628no—
173C22×C4—3624no—
174C22×C4—3624no—
175C22×C4—36216no—
176C22×C4—3622no—
177C22×C4—3622no—
178C22×C4—3624no—
179C22×C4—3624no—
180C22×C4—1812no—
181C22×C4—1812no—
182C22×C4—1814no—
183C22×C4—1818no—
184C22×C4—1812no—
185C22×C4—1814no—
186C22×C4—1814no—
187C22×C4C2×#1062438212yescomplemented
188C4⋊C4—36216no—
189C22⋊C4—3628no—
190C22⋊C4—1814no—
191C22⋊C4—1818no—
192C22⋊C4C2×#1062438214yescomplemented
193C23—24no—
194C23—3628no—
195C23—3624no—
196C23—3628no—
197C23—36216no—
198C23—3624no—
199C23—3628no—
200C23—3624no—
201C23—1814no—
202C23—1814no—
203C23—1818no—
204C23—1814no—
205C23—1812no—
206C23—1812no—
207C23—1812no—
208C23—1814no—
209C23—1814no—
210C23—1811no—
211C23—1811no—
212C23C4⋊#1062438214yescomplemented
213C23C22×#1062438212yes—
214C23C22×#1062438211yescharacteristic
215C2×C4—24no—
216C2×C4—3624no—
217C2×C4—3624no—
218C2×C4—3628no—
219C2×C4—36232no—
220C2×C4—36216no—
221C2×C4—3624no—
222C2×C4—3624no—
223C2×C4—3624no—
224C2×C4—3622no—
225C2×C4—3622no—
226C2×C4—3624no—
227C2×C4—3624no—
228C2×C4—1818no—
229C2×C4—1814no—
230C2×C4—7244no—
231C2×C4—7244no—
232C2×C4C22×#1062438214yescomplemented
233C22—28no—
234C22—24no—
235C22—3624no—
236C22—3624no—
237C22—3628no—
238C22—36216no—
239C22—3628no—
240C22—3624no—
241C22—1814no—
242C22—1814no—
243C22—1814no—
244C22—1818no—
245C22—1811no—
246C22—1812no—
247C22—1811no—
248C22—1812no—
249C22—1812no—
250C22—1814no—
251C22C2×C4⋊#1062438212yes—
252C22C2×C4⋊#1062438214yescomplemented
253C22D4×#1062438214yes—
254C22C23×#1062438211yescharacteristic
255C4—24no—
256C4—3628no—
257C4—3624no—
258C4—7242no—
259C4—7242no—
260C2—24no—
261C2—3624no—
262C2—1811no—
263C2—1811no—
264C2—1812no—
265C2—1814no—
266C2#2604222614yesdirect product factor
267C2C22×C4⋊#1062438211yescharacteristic
268C2C2×D4×#1062438212yes—
subgroupG/H
centerC23C22×#10624382
commutatorC22×C181C23×C4
FrattiniC22C23×#10624382
FittingC2×C22⋊C4×C181C4
socleC23×C181C22×C4
prime radicalC24×D181C22
cosocleC22×D181C24
hypercenterC2×C22⋊C4#10624382
hypocenterC181C2×#385
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊#10624382C2semidirect
4. #26042226C2direct (N × H)
5. C2×(C2×#10624382)⋊C4C2semidirect
14. C2×C4⋊#10624382C22semidirect
17. (C2×#10624382)⋊C4C22semidirect
20. C2×C22⋊Dic181C4semidirect
21. C2×#15668426C4semidirect
22. C22⋊C4×D181C4semidirect
24. C2×C22⋊C4×C181C4semidirect
48. C22×Dic181C2×C4semidirect
51. C2×C4×D181C2×C4semidirect
53. C22⋊Dic181C2×C4semidirect
54. #15668426C2×C4semidirect
56. C22×C4×C181C2×C4semidirect
57. C22⋊C4×C181C2×C4semidirect
76. C22×D181C4⋊C4semidirect
77. C22×D181C4⋊C4semidirect
88. C2×Dic181C22×C4semidirect
93. C23×C181C4⋊C4semidirect
97. C2×C4×C181C22×C4semidirect
101. C22×C181C2×C4⋊C4semidirect
111. C2×D181C2×C4⋊C4semidirect
126. C2×C181#385semidirect
130. D181#385semidirect
134. C181C2×#385semidirect
158. C2×C22⋊C4#10624382semidirect
187. C22×C4C2×#10624382semidirect
192. C22⋊C4C2×#10624382semidirect
212. C23C4⋊#10624382semidirect
232. C2×C4C22×#10624382semidirect
252. C22C2×C4⋊#10624382semidirect
266. C2#26042226direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#106243822
C22×C4⋊#106243822
C22×C22⋊#106243822
#260422262
C2×(C2×#10624382)⋊C42
C2×C22⋊C4×D1812
C2×#385181