Finite Group Databasebeta

← #35226227

Subgroups — #35226227

Order 20864.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×SD16×D163C212yesmaximal, complemented
2Q8×D652C211yesmaximal, characteristic, complemented
3#23782300C211yesmaximal, characteristic
4#23782037C211yesmaximal, characteristic, complemented
5#23781893C212yesmaximal, complemented
6#23781871C211yesmaximal, characteristic, complemented
7#23781817C212yesmaximal, complemented
8#23781802C212yesmaximal, complemented
9C4×#14579686C211yesmaximal, characteristic, complemented
10#23781730C211yesmaximal, characteristic, complemented
11#460×C163C211yesmaximal, characteristic, complemented
12C2×Q8×D163C2212yescomplemented
13C2×D4×D163—21no—
14C2×D4×D163C2212yes—
15C2×#14579688C2212yes—
16C2×#14579687C2212yescomplemented
17C2×#14579687C2212yescomplemented
18C2×#14579686C2211yescharacteristic
19SD16×D163—28no—
20C2×C8×D163C2212yescomplemented
21#19104735—21no—
22#19104735C2211yescharacteristic, complemented
23#19104715—22no—
24#19104715C2212yescomplemented
25#19104706—21no—
26#19104706C2212yes—
27#19104703—22no—
28C4×D652—21no—
29C4×D652C2211yescharacteristic
30C4×D652C2211yescharacteristic
31#19104666C2211yescharacteristic, complemented
32#19104660C2212yescomplemented
33#19104659C2211yescharacteristic, complemented
34#19104655C2211yescharacteristic
35(C2×D163)⋊C8C2212yescomplemented
36C4×C163⋊C8C2211yescharacteristic, complemented
37C2×SD16×C163C2212yescomplemented
38#127×C163C2211yescharacteristic
39C4×Q8×C163C2211yescharacteristic, complemented
40C4⋊C8×C163C2211yescharacteristic, complemented
41#102×C163C2212yescomplemented
42C23×D163—22no—
43C2×Dic326—22no—
44C2×Dic326C2312yes—
45Q8×D163—24no—
46Q8×D163—42no—
47C2×D652D411yescharacteristic
48C2×D652C2311yescharacteristic
49C2×C22⋊D163—22no—
50C2×C22⋊D163—22no—
51D4×D163—24no—
52D4×D163—42no—
53D4×D163—44no—
54C2×C4×D163—22no—
55C2×C4×D163—21no—
56C2×C4×D163C2312yes—
57#14579688—24no—
58#14579687—24no—
59#14579687—42no—
60#14579686D414yescomplemented
61C2×C163⋊C8C2311yescharacteristic
62C2×C163⋊C8C2311yescharacteristic
63C8×D163—24no—
64C4⋊Dic163—22no—
65C4⋊Dic163—21no—
66C4⋊Dic163—21no—
67C4⋊Dic163D411yescharacteristic
68C4⋊Dic163C2311yescharacteristic
69C22⋊Dic163—22no—
70(C2×D163)⋊C4—22no—
71(C2×D163)⋊C4—21no—
72(C2×D163)⋊C4D412yes—
73C2×Q8×C163C2311yescharacteristic
74C2×D4×C163—21no—
75C2×D4×C163—21no—
76C2×D4×C163C2312yes—
77SD16×C163—24no—
78C2×C8×C163C2312yes—
79C4⋊C4×C163—21no—
80C4⋊C4×C163C2311yescharacteristic
81C42×C163—21no—
82C42×C163C2311yescharacteristic
83C22×D163—21no—
84C22×D163—44no—
85C22×D163—44no—
86C22×D163—82no—
87C22×D163C2×D412yes—
88Dic326—22no—
89Dic326—22no—
90Dic326—44no—
91D652—21no—
92D652SD1614yescomplemented
93D652C2×D412yes—
94C22⋊D163—44no—
95C22⋊D163—44no—
96C2×Dic163—21no—
97C2×Dic163—22no—
98C2×Dic163C2×D412yes—
99C4×D163—22no—
100C4×D163—22no—
101C4×D163—44no—
102C4×D163—42no—
103C163⋊C8—21no—
104C163⋊C8C2×D412yescomplemented
105C23×C163—22no—
106Q8×C163—21no—
107Q8×C163C2×D412yescomplemented
108D4×C163—22no—
109D4×C163—22no—
110D4×C163—42no—
111D4×C163—81no—
112C2×C4×C163—21no—
113C2×C4×C163—21no—
114C2×C4×C163C2×D411yescharacteristic
115C2×C4×C163C2×D411yescharacteristic
116C2×C4×C163C2411yescharacteristic
117C8×C163—22no—
118C22×C163—42no—
119C22×C163—42no—
120C22×C163—42no—
121C22×C163C22×D411yescharacteristic
122C2×D163—22no—
123C2×D163—22no—
124C2×D163—22no—
125C2×D163—41no—
126C2×D163—41no—
127C2×D163—88no—
128C2×D163C2×SD1614yescomplemented
129C4×C163—22no—
130C4×C163—21no—
131C4×C163—41no—
132C4×C163C2×SD1612yescomplemented
133C4×C163C22×D411yescharacteristic
134C4×C163H(5,𝔽2)11yescharacteristic
135Dic163—22no—
136Dic163—42no—
137Dic163—41no—
138C2×C163—82no—
139C2×C163D4211yescharacteristic
140C2×C163C22×SD1611yescharacteristic
141C2×C163#57711yescharacteristic
142D163—24no—
143D163—42no—
144C163D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1631nomaximal, Sylow
146C22×SD16—1632no—
147D4×Q8—1631no—
148D42—1631no—
149#492—1631no—
150#465—1632no—
151#461—1631no—
152#460D16311yescharacteristic, complemented
153#450—1632no—
154#448—1632no—
155C4×SD16—1631no—
156C8×D4—1631no—
157C22×Q8—1632no—
158C22×D4—3261no—
159C22×D4—1632no—
160C2×SD16—3268no—
161C2×SD16—1632no—
162C2×SD16—1632no—
163C2×SD16—1631no—
164C2×SD16—1632no—
165C2×SD16C2×D16312yescomplemented
166C22×C8—1632no—
167C4⋊Q8—3261no—
168C4⋊Q8—1631no—
169#127C2×D16311yescharacteristic
170C22⋊Q8—3262no—
171C22⋊Q8—1632no—
172#121—3261no—
173#121—1632no—
174#120—3262no—
175C4×Q8C2×D16311yescharacteristic, complemented
176C4×D4—3261no—
177C4×D4—1631no—
178C4×D4—1631no—
179#106—1631no—
180C4⋊C8C2×D16311yescharacteristic, complemented
181#103—1631no—
182#103—1632no—
183#102—1631no—
184#102C2×D16312yescomplemented
185C22⋊C8—1632no—
186C4×C8—1631no—
187C24—3262no—
188C2×Q8—3262no—
189C2×Q8—3264no—
190C2×Q8—6522no—
191C2×Q8—1632no—
192C2×Q8C22×D16311yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—3264no—
196C2×D4—3262no—
197C2×D4—3262no—
198C2×D4—6522no—
199C2×D4—6524no—
200C2×D4—1631no—
201C2×D4—1631no—
202C2×D4C22×D16312yes—
203C22×C4—3262no—
204C22×C4—3261no—
205C22×C4—1632no—
206SD16—24no—
207SD16—3264no—
208SD16—3264no—
209SD16—6522no—
210SD16—1634no—
211C2×C8—3264no—
212C2×C8—1631no—
213C2×C8—1631no—
214C2×C8C22×D16312yes—
215C4⋊C4—21no—
216C4⋊C4—3262no—
217C4⋊C4—3261no—
218C4⋊C4—3261no—
219C4⋊C4—1631no—
220C4⋊C4—1631no—
221C4⋊C4C22×D16311yescharacteristic
222C22⋊C4—3262no—
223C22⋊C4—3262no—
224C22⋊C4—3261no—
225C22⋊C4—1632no—
226C42—21no—
227C42C22×D16311yescharacteristic
228C23—22no—
229C23—13042no—
230C23—3261no—
231C23—6524no—
232C23—6524no—
233C23—1632no—
234Q8—21no—
235Q8—3262no—
236Q8—3262no—
237Q8—6524no—
238Q8D4×D16312yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—3261no—
244D4—6524no—
245D4—6524no—
246D4—1632no—
247D4—1634no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—3262no—
251C2×C4—3262no—
252C2×C4—3261no—
253C2×C4—3262no—
254C2×C4—6524no—
255C2×C4—6522no—
256C2×C4—1632no—
257C2×C4D4×D16311yescharacteristic
258C2×C4D4×D16311yescharacteristic
259C2×C4C23×D16311yescharacteristic
260C8—22no—
261C8—3261no—
262C8—1632no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—13048no—
267C22—3262no—
268C22—3262no—
269C22—3262no—
270C22—6521no—
271C22—6521no—
272C22—1634no—
273C22C2×D4×D16311yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—3262no—
278C4—6521no—
279C4—6522no—
280C4SD16×D16312yescomplemented
281C4C2×D4×D16311yescharacteristic
282C4#1910478111yescharacteristic
283C2—82no—
284C2—3264no—
285C2—6522no—
286C2#2378229911yescharacteristic
287C2C2×SD16×D16311yescharacteristic
288C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#460×C163C2
socleC22×C163C22×D4
prime radical#23782300C2
hypercenter#460D163
hypocenterC163D4×SD16
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. Q8×D652C2semidirect
4. #23782037C2semidirect
5. #23781893C2semidirect
6. #23781871C2semidirect
7. #23781817C2semidirect
8. #23781802C2semidirect
9. C4×#14579686C2semidirect
10. #23781730C2semidirect
11. #460×C163C2semidirect
12. C2×Q8×D163C22semidirect
16. C2×#14579687C22semidirect
17. C2×#14579687C22semidirect
20. C2×C8×D163C22semidirect
22. #19104735C22semidirect
24. #19104715C22semidirect
31. #19104666C22semidirect
32. #19104660C22semidirect
33. #19104659C22semidirect
35. (C2×D163)⋊C8C22semidirect
36. C4×C163⋊C8C22semidirect
37. C2×SD16×C163C22semidirect
39. C4×Q8×C163C22semidirect
40. C4⋊C8×C163C22semidirect
41. #102×C163C22semidirect
60. #14579686D4semidirect
92. D652SD16semidirect
104. C163⋊C8C2×D4semidirect
107. Q8×C163C2×D4semidirect
128. C2×D163C2×SD16semidirect
132. C4×C163C2×SD16semidirect
144. C163D4×SD16semidirect
152. #460D163semidirect
165. C2×SD16C2×D163semidirect
175. C4×Q8C2×D163semidirect
180. C4⋊C8C2×D163semidirect
184. #102C2×D163semidirect
238. Q8D4×D163semidirect
280. C4SD16×D163semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1632
Q8×D6522
#237823002
#237820372
#237818932
#237818712
#237818172
#237818022
C4×#145796862
#237817302
#460×C1632
D4×SD16163