Finite Group Databasebeta

← #35226302

Subgroups — #35226302

Order 20864.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#23782471C212yesmaximal, complemented
2#23782466C212yesmaximal, complemented
3#23781932C212yesmaximal, complemented
4#23781925C212yesmaximal, complemented
5#23781842C211yesmaximal, characteristic, complemented
6#23781837C211yesmaximal, characteristic, complemented
7#23781832C212yesmaximal, complemented
8#23781788C212yesmaximal, complemented
9#496×C163C211yesmaximal, characteristic, complemented
10#19104783C2211yescharacteristic, complemented
11#19104780C2211yescharacteristic
12#19104780C2212yescomplemented
13#19104766C2212yescomplemented
14#19104764—22no—
15#19104762—22no—
16#19104761C2212yes—
17#19104761C2212yescomplemented
18#19104761C2212yescomplemented
19#19104759—22no—
20#19104759—22no—
21#19104757C2212yes—
22#19104756—22no—
23#19104754—22no—
24SD16×D163—22no—
25D8×D163—22no—
26#19104741C2212yescomplemented
27#19104741C2212yescomplemented
28#19104671C2212yescomplemented
29#19104663C2212yescomplemented
30#19104663C2212yescomplemented
31#19104654C2211yescharacteristic, complemented
32#19104652C2212yescomplemented
33#19104652C2211yescharacteristic, complemented
34C2×SD16×C163C2212yescomplemented
35C2×D8×C163C2212yescomplemented
36#127×C163C2211yescharacteristic, complemented
37#124×C163C2211yescharacteristic, complemented
38#97×C163C2211yescharacteristic, complemented
39#14579700—22no—
40#14579699—41no—
41#14579699C2312yes—
42#14579699C2312yes—
43#14579698—22no—
44#14579698—22no—
45#14579698—22no—
46Q8×D163—22no—
47C2×C22⋊D163—41no—
48C2×C22⋊D163—42no—
49D4×D163—22no—
50D4×D163—22no—
51D4×D163—22no—
52Dic652D412yescomplemented
53#14579689—22no—
54#14579688D412yescomplemented
55#14579688D412yescomplemented
56#14579687—22no—
57#14579687—22no—
58#14579687—22no—
59#14579686—22no—
60D1304D412yescomplemented
61#14579684—22no—
62#14579684—22no—
63#14579684—22no—
64#14579683C2312yes—
65#14579683C2312yes—
66#14579682—22no—
67#14579682—22no—
68C8×D163—22no—
69C8×D163—22no—
70C2×Q8×C163C2311yescharacteristic
71C2×D4×C163—41no—
72C2×D4×C163C2311yescharacteristic
73C2×D4×C163C2312yes—
74SD16×C163—22no—
75SD16×C163—22no—
76D8×C163—22no—
77D8×C163—22no—
78C2×C8×C163C2312yes—
79C22⋊C4×C163—41no—
80C42×C163C2311yescharacteristic
81C22×D163—42no—
82C22×D163—41no—
83Dic326—41no—
84Dic326C2×D412yes—
85Dic326C2×D412yes—
86D652—41no—
87D652C2×D412yes—
88D652C2×D412yes—
89C22⋊D163—22no—
90C22⋊D163—22no—
91C22⋊D163—42no—
92C22⋊D163—42no—
93C22⋊D163—42no—
94C2×Dic163—42no—
95C2×Dic163—41no—
96C4×D163—22no—
97C4×D163—22no—
98C4×D163—42no—
99C163⋊C8—22no—
100C163⋊C8—22no—
101C23×C163—21no—
102C23×C163—22no—
103Q8×C163—22no—
104D4×C163—22no—
105D4×C163—22no—
106D4×C163—22no—
107D4×C163—82no—
108C2×C4×C163—21no—
109C2×C4×C163—21no—
110C2×C4×C163C2411yescharacteristic
111C8×C163C2×D412yes—
112C8×C163C2×D412yes—
113C22×C163—41no—
114C22×C163—42no—
115C22×C163—82no—
116C22×C163—81no—
117C22×C163H(5,𝔽2)11yescharacteristic
118C2×D163—22no—
119C2×D163—22no—
120C2×D163—162no—
121C2×D163—161no—
122C4×C163—41no—
123C4×C163—41no—
124C4×C163C22×D412yes—
125Dic163—22no—
126Dic163—22no—
127C2×C163—21no—
128C2×C163—82no—
129C2×C163—81no—
130C2×C163D4211yescharacteristic
131D163—42no—
132D163—42no—
133C163#328811yescharacteristic, complemented, Sylow
134#3288—1631nomaximal, Sylow
135#577—1632no—
136#576—1632no—
137#496D16311yescharacteristic, complemented
138#472—1632no—
139#471—1632no—
140#455—1631no—
141#454—1631no—
142#453—1632no—
143#444—1632no—
1442−1+4—1631no—
145H(5,𝔽2)—1631no—
146H(5,𝔽2)—1632no—
147#137—3262no—
148#137—1632no—
149AGL(1,ℤ/8)—3262no—
150AGL(1,ℤ/8)—3262no—
151AGL(1,ℤ/8)—3262no—
152AGL(1,ℤ/8)—1632no—
153AGL(1,ℤ/8)—1632no—
154AGL(1,ℤ/8)—1632no—
155#135—3262no—
156#135—3262no—
157#135—1632no—
158C2×SD16—3262no—
159C2×SD16C2×D16312yescomplemented
160C2×D8—3262no—
161C2×D8C2×D16312yescomplemented
162#131—1632no—
163#131—1632no—
164#127C2×D16311yescharacteristic, complemented
165#124C2×D16311yescharacteristic, complemented
166#108—1632no—
167#104—1632no—
168#104—1632no—
169#101—1631no—
170#100—1632no—
171#100—1631no—
172#97C2×D16311yescharacteristic, complemented
173#41—3262no—
174#41—3262no—
175#41—3262no—
176#41—3262no—
177#41—6521no—
178#41—1632no—
179#41—1632no—
180C2×Q8—3262no—
181C2×Q8C22×D16311yescharacteristic
182C2×D4—41no—
183C2×D4—3262no—
184C2×D4—3262no—
185C2×D4—3262no—
186C2×D4—6521no—
187C2×D4—6522no—
188C2×D4C22×D16311yescharacteristic
189C2×D4C22×D16312yes—
190Q16—3262no—
191Q16—1632no—
192SD16—22no—
193SD16—22no—
194SD16—3262no—
195SD16—3262no—
196SD16—3262no—
197SD16—3262no—
198SD16—1632no—
199SD16—1632no—
200D8—22no—
201D8—22no—
202D8—3262no—
203D8—3262no—
204D8—3262no—
205D8—1632no—
206M16—3262no—
207M16—3262no—
208M16—1632no—
209M16—1632no—
210C2×C8—3262no—
211C2×C8—3262no—
212C2×C8C22×D16312yes—
213C22⋊C4—41no—
214C42C22×D16311yescharacteristic
215C23—21no—
216C23—22no—
217C23—6522no—
218C23—6521no—
219Q8—22no—
220Q8—6521no—
221Q8—1632no—
222Q8—1632no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—3262no—
228D4—3262no—
229D4—6521no—
230D4—6522no—
231D4—6522no—
232D4—6522no—
233D4—1632no—
234D4—1632no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—3262no—
238C2×C4—3262no—
239C2×C4—6522no—
240C2×C4—6521no—
241C2×C4—6522no—
242C2×C4C23×D16311yescharacteristic
243C8—3262no—
244C8—3262no—
245C8D4×D16312yes—
246C8D4×D16312yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—26081no—
252C22—26082no—
253C22—3262no—
254C22—3262no—
255C22#1910478011yescharacteristic
256C4—41no—
257C4—41no—
258C4—3262no—
259C4—3262no—
260C4C2×D4×D16312yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—6522no—
265C2—6522no—
266C2#2378230011yescharacteristic
subgroupG/H
centerC2#23782300
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#496×C163C2
socleC2×C163D42
hypercenter#496D163
hypocenterC163#3288
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #23782471C2semidirect
2. #23782466C2semidirect
3. #23781932C2semidirect
4. #23781925C2semidirect
5. #23781842C2semidirect
6. #23781837C2semidirect
7. #23781832C2semidirect
8. #23781788C2semidirect
9. #496×C163C2semidirect
10. #19104783C22semidirect
12. #19104780C22semidirect
13. #19104766C22semidirect
17. #19104761C22semidirect
18. #19104761C22semidirect
26. #19104741C22semidirect
27. #19104741C22semidirect
28. #19104671C22semidirect
29. #19104663C22semidirect
30. #19104663C22semidirect
31. #19104654C22semidirect
32. #19104652C22semidirect
33. #19104652C22semidirect
34. C2×SD16×C163C22semidirect
35. C2×D8×C163C22semidirect
36. #127×C163C22semidirect
37. #124×C163C22semidirect
38. #97×C163C22semidirect
52. Dic652D4semidirect
54. #14579688D4semidirect
55. #14579688D4semidirect
60. D1304D4semidirect
133. C163#3288semidirect
137. #496D163semidirect
159. C2×SD16C2×D163semidirect
161. C2×D8C2×D163semidirect
164. #127C2×D163semidirect
165. #124C2×D163semidirect
172. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824712
#237824662
#237819322
#237819252
#237818422
#237818372
#237818322
#237817882
#496×C1632
#3288163