Finite Group Databasebeta

← #35226351

Subgroups — #35226351

Order 20864.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#23782320C211yesmaximal, characteristic, complemented
2D4×D652C211yesmaximal, characteristic, complemented
3#23782012C211yesmaximal, characteristic, complemented
4#23781980C212yesmaximal, complemented
5#23781945C211yesmaximal, characteristic, complemented
6#23781900C212yesmaximal, complemented
7#23781860C211yesmaximal, characteristic, complemented
8#23781797C212yesmaximal, complemented
9C8×D652C211yesmaximal, characteristic, complemented
10#102×D163C212yesmaximal, complemented
11C4×D8×C163C211yesmaximal, characteristic, complemented
12C2×#14579699—21no—
13C22×D652—21no—
14C2×D4×D163C2212yes—
15C2×#14579684C2212yescomplemented
16C2×C8×D163C2212yescomplemented
17#19104735C2211yescharacteristic, complemented
18#19104732C2211yescharacteristic
19#19104717—22no—
20#19104714—21no—
21#19104710C2212yes—
22#19104709—21no—
23#19104706C2212yescomplemented
24#19104705—22no—
25C4×D652C2211yescharacteristic
26C4⋊C4×D163C2212yescomplemented
27#19104668C2212yescomplemented
28C4⋊(C163⋊C8)C2211yescharacteristic, complemented
29#19104657C2212yescomplemented
30#19104657C2212yescomplemented
31#19104656C2212yes—
32#19104655C2212yescomplemented
33(C2×D163)⋊C8C2212yescomplemented
34C2×D8×C163C2211yescharacteristic
35C4×D4×C163C2211yescharacteristic
36C4×D4×C163C2211yescharacteristic, complemented
37#107×C163C2211yescharacteristic, complemented
38#102×C163C2211yescharacteristic
39#102×C163C2211yescharacteristic, complemented
40C4×C8×C163C2211yescharacteristic, complemented
41C23×D163—22no—
42#14579699—44no—
43C2×Dic326—21no—
44C2×D652—22no—
45C2×D652—21no—
46C2×D652—21no—
47C2×D652—44no—
48C2×D652D411yescharacteristic
49C2×D652C2312yes—
50C2×C22⋊D163—22no—
51C2×C22⋊D163—22no—
52D4×D163—24no—
53D4×D163—42no—
54C2×C4×D163—22no—
55C2×C4×D163C2312yes—
56#14579684—42no—
57C2×C163⋊C8C2312yes—
58C8×D163—42no—
59C4⋊Dic163—21no—
60C4⋊Dic163D411yescharacteristic
61C4⋊Dic163C2312yes—
62Dic163⋊C4—22no—
63C22⋊Dic163—22no—
64(C2×D163)⋊C4—22no—
65(C2×D163)⋊C4D412yes—
66C2×D4×C163C2311yescharacteristic
67C2×D4×C163C2311yescharacteristic
68C22×C4×C163—21no—
69C22×C4×C163—21no—
70D8×C163—22no—
71C2×C8×C163C2311yescharacteristic
72C2×C8×C163C2311yescharacteristic
73C4⋊C4×C163C2311yescharacteristic
74C4⋊C4×C163C2311yescharacteristic
75C22⋊C4×C163—21no—
76C22⋊C4×C163—21no—
77C42×C163C2311yescharacteristic
78C22×D163—22no—
79C22×D163—44no—
80C22×D163—44no—
81C22×D163—82no—
82C22×D163C2×D412yes—
83Dic326—42no—
84D652—22no—
85D652—22no—
86D652—44no—
87D652—42no—
88D652—81no—
89D652D814yescomplemented
90C22⋊D163—44no—
91C22⋊D163—44no—
92C2×Dic163—22no—
93C2×Dic163C2×D412yes—
94C4×D163—22no—
95C4×D163—22no—
96C4×D163—44no—
97C163⋊C8—22no—
98C23×C163—21no—
99C23×C163—21no—
100D4×C163—21no—
101D4×C163—21no—
102D4×C163—21no—
103D4×C163#4112yescomplemented
104C2×C4×C163—21no—
105C2×C4×C163—21no—
106C2×C4×C163—42no—
107C2×C4×C163—42no—
108C2×C4×C163C2×D411yescharacteristic
109C2×C4×C163C2×D411yescharacteristic
110C2×C4×C163C2411yescharacteristic
111C8×C163—21no—
112C8×C163—21no—
113C22×C163—22no—
114C22×C163—41no—
115C22×C163—41no—
116C22×C163—41no—
117C22×C163—41no—
118C22×C163—41no—
119C22×C163C22×D411yescharacteristic
120C2×D163—22no—
121C2×D163—22no—
122C2×D163—42no—
123C2×D163—42no—
124C2×D163—42no—
125C2×D163—88no—
126C2×D163C2×D814yes—
127C4×C163—21no—
128C4×C163—41no—
129C4×C163—41no—
130C4×C163C2×D812yes—
131C4×C163C2×#4111yescharacteristic
132C4×C163H(5,𝔽2)11yescharacteristic
133Dic163—22no—
134Dic163—42no—
135C2×C163—42no—
136C2×C163—81no—
137C2×C163#54611yescharacteristic
138C2×C163C22×D811yescharacteristic
139C2×C163#57711yescharacteristic
140D163—24no—
141D163—82no—
142C163#329411yescharacteristic, complemented, Sylow
143#3294—1631nomaximal, Sylow
144#547—1631no—
145D42—1631no—
146#486—1631no—
147#480—1632no—
148#474—1631no—
149#466—1632no—
150#459—1631no—
151#447—1632no—
152C4×D8D16311yescharacteristic, complemented
153C8×D4—1631no—
154C2×#102—1632no—
155C2×#41—3261no—
156C22×D4—3261no—
157C22×D4—1632no—
158C2×D8—1632no—
159C2×D8C2×D16311yescharacteristic
160C22×C8—1632no—
161C4⋊Q8—1631no—
162#127—1631no—
163#123—3262no—
164C22⋊Q8—3261no—
165#121—3261no—
166#121—1632no—
167#121—1632no—
168#120—3262no—
169C4×D4—1631no—
170C4×D4C2×D16311yescharacteristic
171C4×D4C2×D16311yescharacteristic, complemented
172C2×C4⋊C4—1632no—
173#107—1632no—
174#107C2×D16311yescharacteristic, complemented
175C4⋊C8—1631no—
176#102—1632no—
177#102—1632no—
178#102—1632no—
179#102—1632no—
180#102C2×D16311yescharacteristic
181#102C2×D16311yescharacteristic, complemented
182C22⋊C8—1632no—
183C4×C8C2×D16311yescharacteristic, complemented
184C24—3262no—
185#41—6524no—
186C2×Q8—3261no—
187C2×D4—3262no—
188C2×D4—3264no—
189C2×D4—3262no—
190C2×D4—3262no—
191C2×D4—3261no—
192C2×D4—3261no—
193C2×D4—6524no—
194C2×D4—6522no—
195C2×D4—1631no—
196C2×D4—1632no—
197C2×D4C22×D16311yescharacteristic
198C2×D4C22×D16311yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—3262no—
202C22×C4—1632no—
203D8—22no—
204D8—6522no—
205C2×C8—6522no—
206C2×C8—1632no—
207C2×C8C22×D16311yescharacteristic
208C2×C8C22×D16311yescharacteristic
209C4⋊C4—3262no—
210C4⋊C4—3261no—
211C4⋊C4—1632no—
212C4⋊C4—1631no—
213C4⋊C4C22×D16311yescharacteristic
214C4⋊C4C22×D16311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—3262no—
218C22⋊C4—3262no—
219C22⋊C4—1632no—
220C42C22×D16311yescharacteristic
221C23—21no—
222C23—21no—
223C23—13042no—
224C23—3262no—
225C23—6524no—
226C23—6524no—
227C23—1632no—
228Q8—6522no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—13041no—
233D4—3262no—
234D4—3262no—
235D4—6524no—
236D4—6524no—
237D4—6524no—
238D4—6522no—
239D4—1634no—
240D4#1457969912yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—3262no—
246C2×C4—3262no—
247C2×C4—3262no—
248C2×C4—6524no—
249C2×C4—1632no—
250C2×C4D4×D16311yescharacteristic
251C2×C4D4×D16311yescharacteristic
252C2×C4C23×D16311yescharacteristic
253C8—21no—
254C8—21no—
255C8—3262no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—13048no—
263C22—3262no—
264C22—3262no—
265C22—6522no—
266C22—6522no—
267C22—6522no—
268C22—1634no—
269C22C2×D4×D16311yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—3262no—
274C4—6522no—
275C4D8×D16312yes—
276C4C2×#1457969911yescharacteristic
277C4#1910478111yescharacteristic
278C2—42no—
279C2—81no—
280C2—13042no—
281C2—3264no—
282C2#2378231111yescharacteristic
283C2C2×D8×D16311yescharacteristic
284C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC4×D8×C163C2
socleC22×C163C22×D4
hypercenterC4×D8D163
hypocenterC163#3294
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. #23782320C2semidirect
2. D4×D652C2semidirect
3. #23782012C2semidirect
4. #23781980C2semidirect
5. #23781945C2semidirect
6. #23781900C2semidirect
7. #23781860C2semidirect
8. #23781797C2semidirect
9. C8×D652C2semidirect
10. #102×D163C2semidirect
11. C4×D8×C163C2semidirect
15. C2×#14579684C22semidirect
16. C2×C8×D163C22semidirect
17. #19104735C22semidirect
23. #19104706C22semidirect
26. C4⋊C4×D163C22semidirect
27. #19104668C22semidirect
28. C4⋊(C163⋊C8)C22semidirect
29. #19104657C22semidirect
30. #19104657C22semidirect
32. #19104655C22semidirect
33. (C2×D163)⋊C8C22semidirect
36. C4×D4×C163C22semidirect
37. #107×C163C22semidirect
39. #102×C163C22semidirect
40. C4×C8×C163C22semidirect
89. D652D8semidirect
103. D4×C163#41semidirect
142. C163#3294semidirect
152. C4×D8D163semidirect
171. C4×D4C2×D163semidirect
174. #107C2×D163semidirect
181. #102C2×D163semidirect
183. C4×C8C2×D163semidirect
240. D4#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823202
D4×D6522
#237820122
#237819802
#237819452
#237819002
#237818602
#237817972
C8×D6522
#102×D1632
C4×D8×C1632
#3294163