Finite Group Databasebeta

← #35226231

Subgroups — #35226231

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
3#23782300C211yesmaximal, characteristic
4#23781891C212yesmaximal, complemented
5#23781867C211yesmaximal, characteristic, complemented
6#23781862C211yesmaximal, characteristic, complemented
7#23781814C212yesmaximal, complemented
8#23781802C212yesmaximal, complemented
9C4×#14579688C211yesmaximal, characteristic, complemented
10C8×D652C211yesmaximal, characteristic, complemented
11#492×C163C211yesmaximal, characteristic, complemented
12C2×Q8×D163C2212yescomplemented
13C2×D4×D163—21no—
14C2×D4×D163C2212yes—
15C2×#14579688C2211yescharacteristic
16C2×#14579687C2212yescomplemented
17C2×#14579686C2212yes—
18SD16×D163—28no—
19C2×C8×D163C2212yescomplemented
20#19104734—21no—
21#19104713—22no—
22#19104711C2212yescomplemented
23#19104706—21no—
24#19104706C2212yes—
25#19104703—22no—
26C4×Dic326C2211yescharacteristic, complemented
27C4×D652C2211yescharacteristic
28C4×D652C2211yescharacteristic
29#19104697—21no—
30#19104667C2211yescharacteristic, complemented
31C4⋊(C163⋊C8)C2211yescharacteristic, complemented
32#19104660C2211yescharacteristic, complemented
33#19104659C2212yescomplemented
34#19104657C2211yescharacteristic
35#19104656C2212yescomplemented
36(C2×D163)⋊C8C2212yescomplemented
37C2×SD16×C163C2212yescomplemented
38C2×SD16×C163C2212yescomplemented
39C4⋊Q8×C163C2211yescharacteristic, complemented
40#127×C163C2211yescharacteristic
41C4×C8×C163C2211yescharacteristic, complemented
42C23×D163—22no—
43C2×Dic326—22no—
44C2×Dic326C2311yescharacteristic
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#14579688D414yescomplemented
58#14579687—24no—
59#14579686—24no—
60C2×C163⋊C8C2312yes—
61C8×D163—24no—
62C4⋊Dic163D411yescharacteristic
63C4⋊Dic163C2311yescharacteristic
64C4⋊Dic163C2311yescharacteristic
65Dic163⋊C4—22no—
66Dic163⋊C4—21no—
67C22⋊Dic163—22no—
68(C2×D163)⋊C4—22no—
69(C2×D163)⋊C4—21no—
70(C2×D163)⋊C4D412yes—
71C4×Dic163—21no—
72C2×Q8×C163C2312yes—
73C2×D4×C163—21no—
74C2×D4×C163—21no—
75C2×D4×C163C2312yes—
76SD16×C163—24no—
77SD16×C163—42no—
78C2×C8×C163C2311yescharacteristic
79C2×C8×C163C2311yescharacteristic
80C4⋊C4×C163—21no—
81C4⋊C4×C163—21no—
82C42×C163C2311yescharacteristic
83C22×D163—21no—
84C22×D163—44no—
85C22×D163—44no—
86C22×D163—82no—
87C22×D163C2×D412yes—
88Dic326—21no—
89Dic326—44no—
90Dic326C2×D412yescomplemented
91D652—21no—
92D652SD1614yescomplemented
93D652C2×D412yes—
94C22⋊D163—44no—
95C22⋊D163—44no—
96C2×Dic163—21no—
97C2×Dic163—21no—
98C2×Dic163—21no—
99C2×Dic163C2×D412yes—
100C4×D163—22no—
101C4×D163—22no—
102C4×D163—44no—
103C4×D163—42no—
104C163⋊C8—22no—
105C23×C163—22no—
106Q8×C163—22no—
107Q8×C163—22no—
108D4×C163—22no—
109D4×C163—22no—
110D4×C163—42no—
111D4×C163—81no—
112C2×C4×C163—22no—
113C2×C4×C163C2×D411yescharacteristic
114C2×C4×C163C2×D411yescharacteristic
115C2×C4×C163C2411yescharacteristic
116C8×C163—21no—
117C8×C163C2×D412yescomplemented
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—21no—
130C4×C163—42no—
131C4×C163C2×SD1612yescomplemented
132C4×C163C22×D411yescharacteristic
133C4×C163H(5,𝔽2)11yescharacteristic
134Dic163—22no—
135Dic163—22no—
136Dic163—41no—
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#492D16311yescharacteristic, complemented
150#465—1632no—
151#461—1631no—
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×SD16C2×D16312yescomplemented
165C2×SD16C2×D16312yescomplemented
166C22×C8—1632no—
167C4⋊Q8—3261no—
168C4⋊Q8C2×D16311yescharacteristic, complemented
169#127C2×D16311yescharacteristic
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×D4—1631no—
179#106—1631no—
180C4⋊C8—1631no—
181#103—1632no—
182#103—1631no—
183#102—1632no—
184#102—1631no—
185C22⋊C8—1632no—
186C4×C8C2×D16311yescharacteristic, complemented
187C24—3262no—
188C2×Q8—3262no—
189C2×Q8—3264no—
190C2×Q8—6522no—
191C2×Q8—1631no—
192C2×Q8C22×D16312yes—
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—42no—
208SD16—3264no—
209SD16—3264no—
210SD16—1634no—
211C2×C8—3264no—
212C2×C8—1632no—
213C2×C8C22×D16311yescharacteristic
214C2×C8C22×D16311yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—3262no—
218C4⋊C4—3261no—
219C4⋊C4—1631no—
220C4⋊C4—1631no—
221C4⋊C4—1631no—
222C22⋊C4—3262no—
223C22⋊C4—3262no—
224C22⋊C4—3261no—
225C22⋊C4—1632no—
226C42—3261no—
227C42C22×D16311yescharacteristic
228C23—22no—
229C23—13042no—
230C23—3261no—
231C23—6524no—
232C23—6524no—
233C23—1632no—
234Q8—22no—
235Q8—22no—
236Q8—3261no—
237Q8—6524no—
238Q8—1632no—
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—3261no—
244D4—6524no—
245D4—6524no—
246D4—1632no—
247D4—1634no—
248C2×C4—22no—
249C2×C4—3261no—
250C2×C4—3262no—
251C2×C4—3262no—
252C2×C4—3261no—
253C2×C4—3261no—
254C2×C4—6524no—
255C2×C4—6522no—
256C2×C4—1632no—
257C2×C4D4×D16311yescharacteristic
258C2×C4D4×D16311yescharacteristic
259C2×C4C23×D16311yescharacteristic
260C8—21no—
261C8—3262no—
262C8D4×D16312yescomplemented
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—42no—
276C4—3262no—
277C4—3262no—
278C4—6521no—
279C4—6521no—
280C4SD16×D16312yescomplemented
281C4C2×D4×D16311yescharacteristic
282C4#1910478011yescharacteristic
283C2—82no—
284C2—3264no—
285C2—6522no—
286C2#2378230011yescharacteristic
287C2C2×SD16×D16311yescharacteristic
288C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#492×C163C2
socleC22×C163C22×D4
prime radical#23782300C2
hypercenter#492D163
hypocenterC163D4×SD16
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. #23782330C2semidirect
4. #23781891C2semidirect
5. #23781867C2semidirect
6. #23781862C2semidirect
7. #23781814C2semidirect
8. #23781802C2semidirect
9. C4×#14579688C2semidirect
10. C8×D652C2semidirect
11. #492×C163C2semidirect
12. C2×Q8×D163C22semidirect
16. C2×#14579687C22semidirect
19. C2×C8×D163C22semidirect
22. #19104711C22semidirect
26. C4×Dic326C22semidirect
30. #19104667C22semidirect
31. C4⋊(C163⋊C8)C22semidirect
32. #19104660C22semidirect
33. #19104659C22semidirect
35. #19104656C22semidirect
36. (C2×D163)⋊C8C22semidirect
37. C2×SD16×C163C22semidirect
38. C2×SD16×C163C22semidirect
39. C4⋊Q8×C163C22semidirect
41. C4×C8×C163C22semidirect
57. #14579688D4semidirect
90. Dic326C2×D4semidirect
92. D652SD16semidirect
117. C8×C163C2×D4semidirect
128. C2×D163C2×SD16semidirect
131. C4×C163C2×SD16semidirect
144. C163D4×SD16semidirect
149. #492D163semidirect
164. C2×SD16C2×D163semidirect
165. C2×SD16C2×D163semidirect
168. C4⋊Q8C2×D163semidirect
186. C4×C8C2×D163semidirect
262. C8D4×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
#237823002
#237818912
#237818672
#237818622
#237818142
#237818022
C4×#145796882
C8×D6522
#492×C1632
D4×SD16163