Finite Group Databasebeta

← #35226222

Subgroups — #35226222

Order 20864.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19104754C211yesmaximal, characteristic, complemented
2C2×D8×D163C211yesmaximal, characteristic, complemented
3#23782310C212yesmaximal, complemented
4#23782049C211yesmaximal, characteristic, complemented
5#23781907C211yesmaximal, characteristic, complemented
6#23781902C211yesmaximal, characteristic, complemented
7#23781880C212yesmaximal, complemented
8#23781817C212yesmaximal, complemented
9#23781811C212yesmaximal, complemented
10C8×D652C211yesmaximal, characteristic, complemented
11C4×D8×C163C211yesmaximal, characteristic, complemented
12C2×#14579698C2212yescomplemented
13C2×D4×D163C2212yes—
14#19104754—24no—
15C2×Dic652C2211yescharacteristic, complemented
16C2×#14579688C2212yescomplemented
17C2×#14579687C2212yescomplemented
18C2×D1304C2211yescharacteristic
19C2×#14579684C2212yes—
20D8×D163—24no—
21C2×C8×D163C2211yescharacteristic
22C2×C8×D163C2211yescharacteristic, complemented
23#19104723C2212yescomplemented
24#19104719—22no—
25#19104715C2212yes—
26#19104710C2212yescomplemented
27#19104709—22no—
28#19104705—22no—
29C4×D652C2211yescharacteristic
30C2×(C2×D163)⋊C4—22no—
31C4⋊(C163⋊C8)C2211yescharacteristic, complemented
32#19104658C2212yescomplemented
33#19104655C2212yescomplemented
34(C2×D163)⋊C8C2211yescharacteristic
35(C2×D163)⋊C8C2211yescharacteristic, complemented
36C2×D8×C163C2211yescharacteristic
37C4×D4×C163C2212yescomplemented
38#107×C163C2211yescharacteristic, complemented
39#102×C163C2212yescomplemented
40C4×C8×C163C2211yescharacteristic, complemented
41C23×D163—22no—
42#14579698—24no—
43#14579698—42no—
44C2×Dic326C2312yes—
45C2×D652—22no—
46C2×D652D411yescharacteristic
47C2×D652C2312yes—
48C2×C22⋊D163—22no—
49C2×C22⋊D163—22no—
50D4×D163—24no—
51D4×D163—42no—
52C22×Dic163—22no—
53C2×C4×D163C2311yescharacteristic
54C2×C4×D163C2311yescharacteristic
55Dic652—22no—
56#14579688—42no—
57#14579687—24no—
58D1304—22no—
59#14579684—24no—
60C2×C163⋊C8C2311yescharacteristic
61C2×C163⋊C8C2311yescharacteristic
62C8×D163—22no—
63C8×D163—22no—
64C4⋊Dic163—22no—
65C4⋊Dic163D411yescharacteristic
66(C2×D163)⋊C4—22no—
67(C2×D163)⋊C4—22no—
68(C2×D163)⋊C4—22no—
69(C2×D163)⋊C4—22no—
70(C2×D163)⋊C4D411yescharacteristic
71(C2×D163)⋊C4D411yescharacteristic
72C2×D4×C163C2312yes—
73C22×C4×C163—22no—
74D8×C163D414yescomplemented
75C2×C8×C163C2311yescharacteristic
76C2×C8×C163C2311yescharacteristic
77C4⋊C4×C163C2312yes—
78C22⋊C4×C163—22no—
79C42×C163C2311yescharacteristic
80C22×D163—22no—
81C22×D163—44no—
82C22×D163—44no—
83C22×D163—82no—
84C22×D163C2×D411yescharacteristic
85C22×D163C2×D411yescharacteristic
86Dic326—22no—
87Dic326—22no—
88D652—22no—
89D652—22no—
90D652—22no—
91D652—82no—
92C22⋊D163—44no—
93C22⋊D163—44no—
94C2×Dic163—22no—
95C2×Dic163—44no—
96C2×Dic163C2×D411yescharacteristic
97C2×Dic163C2×D411yescharacteristic
98C4×D163—21no—
99C4×D163—21no—
100C4×D163—21no—
101C4×D163—21no—
102C163⋊C8—21no—
103C163⋊C8—21no—
104C23×C163—22no—
105D4×C163—22no—
106D4×C163C2×D414yes—
107C2×C4×C163—22no—
108C2×C4×C163—82no—
109C2×C4×C163C2×D411yescharacteristic
110C2×C4×C163C2×D411yescharacteristic
111C2×C4×C163C2411yescharacteristic
112C8×C163—21no—
113C8×C163C2×D412yes—
114C22×C163—24no—
115C22×C163—42no—
116C22×C163—42no—
117C22×C163C22×D411yescharacteristic
118C2×D163—21no—
119C2×D163—21no—
120C2×D163—21no—
121C2×D163—21no—
122C2×D163—21no—
123C2×D163—42no—
124C2×D163—42no—
125C2×D163—42no—
126C2×D163—88no—
127C2×D163#13512yescomplemented
128C4×C163—21no—
129C4×C163—21no—
130C4×C163—42no—
131C4×C163C22×D411yescharacteristic
132C4×C163H(5,𝔽2)11yescharacteristic
133Dic163—21no—
134Dic163—21no—
135Dic163—42no—
136C2×C163—44no—
137C2×C163D4211yescharacteristic
138C2×C163C2×#13511yescharacteristic
139C2×C163#57611yescharacteristic
140D163—22no—
141D163—41no—
142D163—82no—
143C163#328011yescharacteristic, complemented, Sylow
144#3280—1631nomaximal, Sylow
145C2×#135—1631no—
146C22×D8—1631no—
147#546—1632no—
148#495—1631no—
149#467—1631no—
150#466—1631no—
151#463—1632no—
152#450—1632no—
153#449—1632no—
154C4×D8D16311yescharacteristic, complemented
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×C8—1631no—
168#124—1632no—
169#123—3262no—
170C22⋊Q8—1632no—
171#121—3262no—
172#121—1632no—
173#120—3262no—
174C4×D4—1631no—
175C4×D4C2×D16312yescomplemented
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×C8C2×D16311yescharacteristic, complemented
185C24—3262no—
186#41—3264no—
187#41—6522no—
188C2×Q8—1632no—
189C2×D4—3262no—
190C2×D4—3264no—
191C2×D4—3262no—
192C2×D4—3262no—
193C2×D4—6522no—
194C2×D4—1631no—
195C2×D4—1632no—
196C2×D4C22×D16312yes—
197C22×C4—22no—
198C22×C4—3262no—
199C22×C4—1631no—
200C22×C4—1631no—
201Q16—3262no—
202SD16—3264no—
203SD16—6522no—
204D8—3264no—
205D8—3262no—
206D8D65214yescomplemented
207C2×C8—3262no—
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—
223C42C22×D16311yescharacteristic
224C23—22no—
225C23—13042no—
226C23—3262no—
227C23—6524no—
228C23—6524no—
229C23—1631no—
230C23—1631no—
231Q8—3262no—
232Q8—3262no—
233D4—22no—
234D4—13042no—
235D4—3262no—
236D4—3262no—
237D4—3262no—
238D4—6524no—
239D4—6524no—
240D4C2×D65214yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—3261no—
244C2×C4—3261no—
245C2×C4—3261no—
246C2×C4—3261no—
247C2×C4—3262no—
248C2×C4—6524no—
249C2×C4—1631no—
250C2×C4—1631no—
251C2×C4D4×D16311yescharacteristic
252C2×C4D4×D16311yescharacteristic
253C2×C4C23×D16311yescharacteristic
254C8—21no—
255C8—3261no—
256C8—3261no—
257C8C2×D65212yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—13048no—
262C22—3261no—
263C22—3261no—
264C22—3261no—
265C22—3261no—
266C22—3261no—
267C22—6522no—
268C22—6522no—
269C22—6522no—
270C22—1632no—
271C22C2×D4×D16311yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—3261no—
276C4—3261no—
277C4—6522no—
278C4C22×D65211yescharacteristic
279C4#1910478111yescharacteristic
280C2—44no—
281C2—13042no—
282C2—3262no—
283C2—6521no—
284C2D4×D65211yescharacteristic
285C2#135×D16311yescharacteristic
286C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC4×D8×C163C2
socleC22×C163C22×D4
hypercenterC4×D8D163
hypocenterC163#3280
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19104754C2semidirect
2. C2×D8×D163C2semidirect
3. #23782310C2semidirect
4. #23782049C2semidirect
5. #23781907C2semidirect
6. #23781902C2semidirect
7. #23781880C2semidirect
8. #23781817C2semidirect
9. #23781811C2semidirect
10. C8×D652C2semidirect
11. C4×D8×C163C2semidirect
12. C2×#14579698C22semidirect
15. C2×Dic652C22semidirect
16. C2×#14579688C22semidirect
17. C2×#14579687C22semidirect
22. C2×C8×D163C22semidirect
23. #19104723C22semidirect
26. #19104710C22semidirect
31. C4⋊(C163⋊C8)C22semidirect
32. #19104658C22semidirect
33. #19104655C22semidirect
35. (C2×D163)⋊C8C22semidirect
37. C4×D4×C163C22semidirect
38. #107×C163C22semidirect
39. #102×C163C22semidirect
40. C4×C8×C163C22semidirect
74. D8×C163D4semidirect
127. C2×D163#135semidirect
143. C163#3280semidirect
154. C4×D8D163semidirect
175. C4×D4C2×D163semidirect
177. #107C2×D163semidirect
181. #102C2×D163semidirect
184. C4×C8C2×D163semidirect
206. D8D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047542
C2×D8×D1632
#237823102
#237820492
#237819072
#237819022
#237818802
#237818172
#237818112
C8×D6522
C4×D8×C1632
#3280163