Finite Group Databasebeta

← #35226235

Subgroups — #35226235

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
2#23782330C211yesmaximal, characteristic, complemented
3D4×D652C211yesmaximal, characteristic
4#23782039C211yesmaximal, characteristic, complemented
5#23781897C212yesmaximal, complemented
6#23781866C211yesmaximal, characteristic, complemented
7#23781814C212yesmaximal, complemented
8#23781805C212yesmaximal, complemented
9C4×#14579687C211yesmaximal, characteristic, complemented
10#23781730C211yesmaximal, characteristic, complemented
11#461×C163C211yesmaximal, characteristic, complemented
12C2×Q8×D163C2212yescomplemented
13C22×D652—21no—
14C2×D4×D163C2212yes—
15C2×#14579688C2212yescomplemented
16C2×#14579687C2211yescharacteristic
17C2×#14579686C2212yescomplemented
18C2×#14579686C2212yescomplemented
19SD16×D163—28no—
20C2×C8×D163C2212yescomplemented
21#19104734—21no—
22#19104732C2211yescharacteristic
23#19104713—22no—
24#19104711C2212yescomplemented
25#19104710C2212yes—
26#19104709—21no—
27#19104705—22no—
28C4×Dic326C2211yescharacteristic, complemented
29C4×D652C2211yescharacteristic
30#19104697—21no—
31#19104666C2211yescharacteristic, complemented
32#19104658C2211yescharacteristic, complemented
33#19104657C2212yescomplemented
34#19104656C2211yescharacteristic
35(C2×D163)⋊C8C2212yescomplemented
36C4×C163⋊C8C2211yescharacteristic, complemented
37C2×SD16×C163C2212yes—
38C4⋊Q8×C163C2211yescharacteristic, complemented
39C4×D4×C163C2211yescharacteristic
40C4⋊C8×C163C2211yescharacteristic, complemented
41#103×C163C2212yescomplemented
42C23×D163—22no—
43C2×Dic326—22no—
44C2×Dic326C2311yescharacteristic
45Q8×D163—24no—
46Q8×D163—42no—
47C2×D652—22no—
48C2×D652—21no—
49C2×D652—21no—
50C2×D652—44no—
51C2×D652D411yescharacteristic
52C2×D652C2312yes—
53C2×C22⋊D163—22no—
54D4×D163—24no—
55D4×D163—42no—
56C2×C4×D163—22no—
57C2×C4×D163C2312yes—
58#14579688—24no—
59#14579687D414yescomplemented
60#14579686—24no—
61#14579686—42no—
62C2×C163⋊C8C2311yescharacteristic
63C2×C163⋊C8C2311yescharacteristic
64C8×D163—24no—
65C4⋊Dic163D411yescharacteristic
66C4⋊Dic163C2311yescharacteristic
67Dic163⋊C4—22no—
68Dic163⋊C4—21no—
69(C2×D163)⋊C4—22no—
70(C2×D163)⋊C4—22no—
71(C2×D163)⋊C4D412yes—
72C4×Dic163—21no—
73C2×Q8×C163C2312yes—
74C2×D4×C163C2311yescharacteristic
75C22×C4×C163—21no—
76SD16×C163—24no—
77C2×C8×C163C2312yes—
78C4⋊C4×C163—21no—
79C4⋊C4×C163—21no—
80C4⋊C4×C163C2311yescharacteristic
81C22⋊C4×C163—21no—
82C42×C163C2311yescharacteristic
83C22×D163—22no—
84C22×D163—44no—
85C22×D163—44no—
86C22×D163—82no—
87C22×D163C2×D412yes—
88Dic326—21no—
89Dic326—44no—
90Dic326C2×D412yescomplemented
91D652—22no—
92D652—22no—
93D652—44no—
94D652—42no—
95D652—81no—
96D652SD1614yescomplemented
97C22⋊D163—44no—
98C2×Dic163—21no—
99C2×Dic163—21no—
100C2×Dic163C2×D412yes—
101C4×D163—22no—
102C4×D163—22no—
103C4×D163—44no—
104C163⋊C8—21no—
105C163⋊C8C2×D412yescomplemented
106C23×C163—21no—
107Q8×C163—22no—
108Q8×C163—22no—
109D4×C163—21no—
110D4×C163C2×D412yes—
111C2×C4×C163—21no—
112C2×C4×C163—22no—
113C2×C4×C163—42no—
114C2×C4×C163C2×D411yescharacteristic
115C2×C4×C163C2×D411yescharacteristic
116C2×C4×C163C2411yescharacteristic
117C8×C163—22no—
118C22×C163—22no—
119C22×C163—41no—
120C22×C163—41no—
121C22×C163C22×D411yescharacteristic
122C2×D163—22no—
123C2×D163—22no—
124C2×D163—42no—
125C2×D163—42no—
126C2×D163—42no—
127C2×D163—88no—
128C2×D163C2×SD1614yescomplemented
129C4×C163—21no—
130C4×C163—42no—
131C4×C163—41no—
132C4×C163C2×SD1612yescomplemented
133C4×C163C22×D411yescharacteristic
134C4×C163H(5,𝔽2)11yescharacteristic
135Dic163—22no—
136Dic163—22no—
137Dic163—41no—
138C2×C163—42no—
139C2×C163D4211yescharacteristic
140C2×C163C22×SD1611yescharacteristic
141C2×C163#57711yescharacteristic
142D163—24no—
143D163—82no—
144C163D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1631nomaximal, Sylow
146C22×SD16—1632no—
147D4×Q8—1631no—
148D42—1631no—
149#492—1631no—
150#465—1632no—
151#461D16311yescharacteristic, complemented
152#460—1631no—
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×D16312yes—
166C22×C8—1632no—
167C4⋊Q8—3261no—
168C4⋊Q8C2×D16311yescharacteristic, complemented
169#127—1631no—
170C22⋊Q8—3262no—
171C22⋊Q8—1632no—
172#121—3261no—
173#121—1632no—
174#120—3262no—
175C4×Q8—1631no—
176C4×D4—3261no—
177C4×D4—1631no—
178C4×D4C2×D16311yescharacteristic
179#106—1631no—
180C4⋊C8C2×D16311yescharacteristic, complemented
181#103—1631no—
182#103C2×D16312yescomplemented
183#102—1631no—
184#102—1632no—
185C22⋊C8—1632no—
186C4×C8—1631no—
187C24—3262no—
188C2×Q8—3262no—
189C2×Q8—3264no—
190C2×Q8—6522no—
191C2×Q8—1631no—
192C2×Q8C22×D16312yes—
193C2×D4—3262no—
194C2×D4—3264no—
195C2×D4—3262no—
196C2×D4—3261no—
197C2×D4—3261no—
198C2×D4—6524no—
199C2×D4—6522no—
200C2×D4—1631no—
201C2×D4—1632no—
202C2×D4C22×D16311yescharacteristic
203C22×C4—21no—
204C22×C4—3262no—
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—21no—
217C4⋊C4—3262no—
218C4⋊C4—3261no—
219C4⋊C4—1631no—
220C4⋊C4—1631no—
221C4⋊C4C22×D16311yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—3262no—
224C22⋊C4—3262no—
225C22⋊C4—1632no—
226C42—3261no—
227C42C22×D16311yescharacteristic
228C23—21no—
229C23—13042no—
230C23—3262no—
231C23—6524no—
232C23—6524no—
233C23—1632no—
234Q8—22no—
235Q8—22no—
236Q8—3261no—
237Q8—6524no—
238Q8—1632no—
239D4—21no—
240D4—13041no—
241D4—3262no—
242D4—3262no—
243D4—6524no—
244D4—6524no—
245D4—6522no—
246D4—1634no—
247D4D4×D16312yes—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—3261no—
252C2×C4—3262no—
253C2×C4—3262no—
254C2×C4—3261no—
255C2×C4—6524no—
256C2×C4—1632no—
257C2×C4D4×D16311yescharacteristic
258C2×C4D4×D16311yescharacteristic
259C2×C4C23×D16311yescharacteristic
260C8—22no—
261C8—3261no—
262C8—1632no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—13048no—
267C22—3262no—
268C22—3262no—
269C22—6522no—
270C22—6522no—
271C22—6522no—
272C22—1634no—
273C22C2×D4×D16311yescharacteristic
274C4—21no—
275C4—41no—
276C4—42no—
277C4—3262no—
278C4—3262no—
279C4—6521no—
280C4SD16×D16312yescomplemented
281C4C2×D4×D16311yescharacteristic
282C4#1910478111yescharacteristic
283C2—42no—
284C2—13042no—
285C2—3264no—
286C2#2378229911yescharacteristic
287C2C2×SD16×D16311yescharacteristic
288C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#461×C163C2
socleC22×C163C22×D4
prime radicalD4×D652C2
hypercenter#461D163
hypocenterC163D4×SD16
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. #23782330C2semidirect
4. #23782039C2semidirect
5. #23781897C2semidirect
6. #23781866C2semidirect
7. #23781814C2semidirect
8. #23781805C2semidirect
9. C4×#14579687C2semidirect
10. #23781730C2semidirect
11. #461×C163C2semidirect
12. C2×Q8×D163C22semidirect
15. C2×#14579688C22semidirect
17. C2×#14579686C22semidirect
18. C2×#14579686C22semidirect
20. C2×C8×D163C22semidirect
24. #19104711C22semidirect
28. C4×Dic326C22semidirect
31. #19104666C22semidirect
32. #19104658C22semidirect
33. #19104657C22semidirect
35. (C2×D163)⋊C8C22semidirect
36. C4×C163⋊C8C22semidirect
38. C4⋊Q8×C163C22semidirect
40. C4⋊C8×C163C22semidirect
41. #103×C163C22semidirect
59. #14579687D4semidirect
90. Dic326C2×D4semidirect
96. D652SD16semidirect
105. C163⋊C8C2×D4semidirect
128. C2×D163C2×SD16semidirect
132. C4×C163C2×SD16semidirect
144. C163D4×SD16semidirect
151. #461D163semidirect
168. C4⋊Q8C2×D163semidirect
180. C4⋊C8C2×D163semidirect
182. #103C2×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
#237823302
D4×D6522
#237820392
#237818972
#237818662
#237818142
#237818052
C4×#145796872
#237817302
#461×C1632
D4×SD16163