Finite Group Databasebeta

← #35226224

Subgroups — #35226224

Order 20864.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#19104757C211yesmaximal, characteristic, complemented
2C2×D8×D163C211yesmaximal, characteristic, complemented
3#23782312C212yesmaximal, complemented
4#23782048C211yesmaximal, characteristic, complemented
5#23781907C211yesmaximal, characteristic, complemented
6#23781879C212yesmaximal, complemented
7#23781817C212yesmaximal, complemented
8#23781808C212yesmaximal, complemented
9#23781747C211yesmaximal, characteristic, complemented
10C8×C22⋊D163C211yesmaximal, characteristic, complemented
11#466×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2212yescomplemented
13C2×D4×D163C2212yes—
14#19104757—24no—
15C2×Dic652C2211yescharacteristic, complemented
16C2×#14579688C2212yescomplemented
17C2×#14579687C2212yescomplemented
18C2×D1304C2211yescharacteristic
19C2×#14579684C2212yes—
20D8×D163—24no—
21C2×C8×D163C2211yescharacteristic
22#19104722C2212yescomplemented
23#19104717—22no—
24#19104715C2212yes—
25#19104706—22no—
26#19104703—22no—
27#19104697C2212yescomplemented
28C4×C22⋊D163C2211yescharacteristic
29C22⋊C4×D163—22no—
30Dic163⋊C8C2211yescharacteristic, complemented
31#19104658C2212yescomplemented
32#19104655C2212yescomplemented
33C22⋊(C163⋊C8)C2211yescharacteristic, complemented
34(C2×D163)⋊C8C2211yescharacteristic
35C8×Dic163C2211yescharacteristic, complemented
36C2×D8×C163C2211yescharacteristic
37C22×C8×C163C2211yescharacteristic, complemented
38#121×C163C2212yescomplemented
39#107×C163C2211yescharacteristic, complemented
40#102×C163C2212yescomplemented
41C23×D163—22no—
42#14579699—24no—
43#14579699—42no—
44C2×Dic326C2312yes—
45C2×D652C2312yes—
46C2×C22⋊D163—22no—
47C2×C22⋊D163—22no—
48C2×C22⋊D163D411yescharacteristic
49D4×D163—24no—
50D4×D163—42no—
51C2×C4×D163—22no—
52C2×C4×D163—22no—
53C2×C4×D163C2311yescharacteristic
54Dic652—22no—
55#14579688—24no—
56#14579687—42no—
57D1304D414yescomplemented
58#14579684—24no—
59C2×C163⋊C8C2311yescharacteristic
60C2×C163⋊C8C2311yescharacteristic
61C8×D163—22no—
62C4⋊Dic163—22no—
63Dic163⋊C4D411yescharacteristic
64C22⋊Dic163—22no—
65C22⋊Dic163—22no—
66C22⋊Dic163D411yescharacteristic
67(C2×D163)⋊C4—22no—
68(C2×D163)⋊C4—22no—
69(C2×D163)⋊C4D411yescharacteristic
70C4×Dic163C2311yescharacteristic
71C2×D4×C163—22no—
72C2×D4×C163C2312yes—
73C22×C4×C163C2311yescharacteristic
74D8×C163—22no—
75C2×C8×C163—22no—
76C2×C8×C163C2311yescharacteristic
77C2×C8×C163C2311yescharacteristic
78C4⋊C4×C163C2312yes—
79C22⋊C4×C163—22no—
80C22×D163—22no—
81C22×D163—44no—
82C22×D163—44no—
83C22×D163—82no—
84C22×D163C2×D411yescharacteristic
85Dic326—22no—
86Dic326—22no—
87D652—22no—
88D652C2×D414yes—
89C22⋊D163—22no—
90C22⋊D163—44no—
91C22⋊D163—44no—
92C2×Dic163—22no—
93C2×Dic163C2×D411yescharacteristic
94C2×Dic163C2×D411yescharacteristic
95C2×Dic163C2×D411yescharacteristic
96C4×D163—21no—
97C4×D163—21no—
98C4×D163—44no—
99C4×D163—82no—
100C163⋊C8—21no—
101C163⋊C8—21no—
102C23×C163—22no—
103C23×C163C2×D411yescharacteristic
104D4×C163—22no—
105D4×C163—22no—
106D4×C163—82no—
107C2×C4×C163—21no—
108C2×C4×C163—21no—
109C2×C4×C163—22no—
110C2×C4×C163C2×D411yescharacteristic
111C2×C4×C163C2411yescharacteristic
112C8×C163—21no—
113C8×C163C2×D412yes—
114C22×C163—21no—
115C22×C163—21no—
116C22×C163—21no—
117C22×C163—42no—
118C22×C163—42no—
119C22×C163—42no—
120C22×C163C22×D411yescharacteristic
121C2×D163—21no—
122C2×D163—21no—
123C2×D163—24no—
124C2×D163—42no—
125C2×D163—42no—
126C2×D163—88no—
127C2×D163#13512yescomplemented
128C4×C163—21no—
129C4×C163—42no—
130C4×C163C22×D411yescharacteristic
131C4×C163H(5,𝔽2)11yescharacteristic
132Dic163—21no—
133Dic163—21no—
134Dic163—21no—
135Dic163—42no—
136C2×C163—41no—
137C2×C163—82no—
138C2×C163D4211yescharacteristic
139C2×C163C2×#13511yescharacteristic
140C2×C163#57611yescharacteristic
141D163—22no—
142D163—44no—
143C163#328011yescharacteristic, complemented, Sylow
144#3280—1631nomaximal, Sylow
145C2×#135—1631no—
146C22×D8—1631no—
147#546—1632no—
148#495—1631no—
149#467—1631no—
150#466D16311yescharacteristic, complemented
151#463—1632no—
152#450—1632no—
153#449—1632no—
154C4×D8—1631no—
155C8×D4—1631no—
156C2×#41—1632no—
157C22×D4—1632no—
158#135—3264no—
159C2×Q16—1631no—
160C2×SD16—1632no—
161C2×SD16—1632no—
162C2×D8—3264no—
163C2×D8—1632no—
164C2×D8—1631no—
165C2×D8C2×D16311yescharacteristic
166C22×C8—1631no—
167C22×C8C2×D16311yescharacteristic, complemented
168#124—1632no—
169#123—3262no—
170C22⋊Q8—1632no—
171#121—3262no—
172#121C2×D16312yescomplemented
173#120—3262no—
174C4×D4—1632no—
175C4×D4—1631no—
176C2×C22⋊C4—3262no—
177#107C2×D16311yescharacteristic, complemented
178C4⋊C8—1631no—
179#103—1632no—
180#102—1632no—
181#102C2×D16312yescomplemented
182C22⋊C8—1631no—
183C22⋊C8—1631no—
184C4×C8—1631no—
185C24—3262no—
186#41—3264no—
187#41—6522no—
188C2×Q8—1632no—
189C2×D4—22no—
190C2×D4—3264no—
191C2×D4—3262no—
192C2×D4—3262no—
193C2×D4—6522no—
194C2×D4—1631no—
195C2×D4—1632no—
196C2×D4C22×D16312yes—
197C22×C4—3262no—
198C22×C4—3262no—
199C22×C4—1631no—
200C22×C4C22×D16311yescharacteristic
201Q16—3262no—
202SD16—3264no—
203SD16—6522no—
204D8—22no—
205D8—3264no—
206D8—1634no—
207C2×C8—22no—
208C2×C8—3262no—
209C2×C8—1631no—
210C2×C8—1631no—
211C2×C8C22×D16311yescharacteristic
212C2×C8C22×D16311yescharacteristic
213C4⋊C4—3262no—
214C4⋊C4—1631no—
215C4⋊C4C22×D16312yes—
216C22⋊C4—22no—
217C22⋊C4—3262no—
218C22⋊C4—3262no—
219C22⋊C4—3262no—
220C22⋊C4—3262no—
221C22⋊C4—1631no—
222C22⋊C4—1631no—
223C42—1631no—
224C23—22no—
225C23—13042no—
226C23—3262no—
227C23—6524no—
228C23—6524no—
229C23—1631no—
230C23D4×D16311yescharacteristic
231Q8—3262no—
232Q8—3262no—
233D4—22no—
234D4—22no—
235D4—82no—
236D4—3262no—
237D4—3262no—
238D4—6524no—
239D4—6524no—
240D4—1634no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—13042no—
245C2×C4—3261no—
246C2×C4—3261no—
247C2×C4—3262no—
248C2×C4—6524no—
249C2×C4—1631no—
250C2×C4—1631no—
251C2×C4—1631no—
252C2×C4D4×D16311yescharacteristic
253C2×C4C23×D16311yescharacteristic
254C8—21no—
255C8—3261no—
256C8—3261no—
257C8D4×D16312yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—13048no—
265C22—3261no—
266C22—3261no—
267C22—3264no—
268C22—6522no—
269C22—6522no—
270C22—1632no—
271C22C2×D4×D16311yescharacteristic
272C4—21no—
273C4—42no—
274C4—3261no—
275C4—3261no—
276C4—3261no—
277C4—6522no—
278C4C2×D4×D16311yescharacteristic
279C4#1910478111yescharacteristic
280C2—41no—
281C2—82no—
282C2—3262no—
283C2—6524no—
284C2#2378229911yescharacteristic
285C2#135×D16311yescharacteristic
286C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#466×C163C2
socleC22×C163C22×D4
hypercenter#466D163
hypocenterC163#3280
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#19104757C2semidirect
2. C2×D8×D163C2semidirect
3. #23782312C2semidirect
4. #23782048C2semidirect
5. #23781907C2semidirect
6. #23781879C2semidirect
7. #23781817C2semidirect
8. #23781808C2semidirect
9. #23781747C2semidirect
10. C8×C22⋊D163C2semidirect
11. #466×C163C2semidirect
12. C2×#14579699C22semidirect
15. C2×Dic652C22semidirect
16. C2×#14579688C22semidirect
17. C2×#14579687C22semidirect
22. #19104722C22semidirect
27. #19104697C22semidirect
30. Dic163⋊C8C22semidirect
31. #19104658C22semidirect
32. #19104655C22semidirect
33. C22⋊(C163⋊C8)C22semidirect
35. C8×Dic163C22semidirect
37. C22×C8×C163C22semidirect
38. #121×C163C22semidirect
39. #107×C163C22semidirect
40. #102×C163C22semidirect
57. D1304D4semidirect
127. C2×D163#135semidirect
143. C163#3280semidirect
150. #466D163semidirect
167. C22×C8C2×D163semidirect
172. #121C2×D163semidirect
177. #107C2×D163semidirect
181. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047572
C2×D8×D1632
#237823122
#237820482
#237819072
#237818792
#237818172
#237818082
#237817472
C8×C22⋊D1632
#466×C1632
#3280163