Finite Group Databasebeta

← #35226304

Subgroups — #35226304

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#23782473C212yesmaximal, complemented
2#23782469C212yesmaximal, complemented
3#23782055C211yesmaximal, characteristic, complemented
4#23781933C212yesmaximal, complemented
5#23781931C212yesmaximal, complemented
6#23781842C211yesmaximal, characteristic, complemented
7#23781834C212yesmaximal, complemented
8#23781784C212yesmaximal, complemented
9#454×C163C211yesmaximal, characteristic, complemented
10#19104781C2212yescomplemented
11#19104780C2211yescharacteristic
12#19104766—22no—
13#19104766C2212yescomplemented
14#19104763C2212yes—
15#19104762—22no—
16#19104760—22no—
17#19104760C2212yescomplemented
18#19104759—22no—
19#19104756—22no—
20#19104755—22no—
21#19104755C2212yes—
22C2×#14579686—22no—
23C2×#14579686C2212yescomplemented
24C2×#14579684C2212yescomplemented
25D8×D163—22no—
26#19104743C2212yescomplemented
27#19104742C2212yescomplemented
28#19104732C2211yescharacteristic, complemented
29#19104670C2212yescomplemented
30#19104663C2212yescomplemented
31#19104652C2211yescharacteristic, complemented
32#19104651C2212yescomplemented
33#19104646C2211yescharacteristic, complemented
342−1+4×C163C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C163C2212yescomplemented
36#124×C163C2211yescharacteristic, complemented
37#104×C163C2212yescomplemented
38#101×C163C2211yescharacteristic, complemented
39#14579700—22no—
40#14579699—22no—
41#14579699C2312yes—
42#14579698—22no—
43C2×D652—22no—
44C2×D652—41no—
45C2×D652C2312yes—
46C2×C22⋊D163—41no—
47D4×D163—22no—
48D4×D163—22no—
49D4×D163—42no—
50#14579689—22no—
51#14579689D412yescomplemented
52#14579688—22no—
53#14579687D412yescomplemented
54#14579686—22no—
55#14579686—22no—
56#14579686—22no—
57#14579686—22no—
58#14579686D412yescomplemented
59D1304—22no—
60#14579684—22no—
61#14579684—22no—
62#14579684—22no—
63#14579684D412yescomplemented
64#14579683—22no—
65#14579683C2312yes—
66#14579682—22no—
67C2×C163⋊C8—22no—
68C2×C163⋊C8C2312yes—
69C8×D163—22no—
70#41×C163—22no—
71#41×C163—41no—
72#41×C163C2312yes—
73C2×Q8×C163—22no—
74C2×Q8×C163C2311yescharacteristic
75C2×D4×C163C2311yescharacteristic
76SD16×C163—22no—
77D8×C163—22no—
78M16×C163C2312yes—
79C22⋊C4×C163—41no—
80C42×C163C2311yescharacteristic
81C22×D163—22no—
82C22×D163—41no—
83C22×D163—42no—
84Dic326C2×D412yes—
85D652—22no—
86D652—22no—
87D652—42no—
88D652—82no—
89D652C2×D412yes—
90C22⋊D163—22no—
91C22⋊D163—42no—
92C22⋊D163—42no—
93C2×Dic163—41no—
94C4×D163—22no—
95C4×D163—42no—
96C163⋊C8—22no—
97C163⋊C8C2×D412yes—
98C163⋊C8C2×D412yes—
99C23×C163—21no—
100Q8×C163—22no—
101Q8×C163—41no—
102Q8×C163C2×D412yes—
103D4×C163—22no—
104D4×C163—22no—
105D4×C163—41no—
106D4×C163C2×D412yes—
107C2×C4×C163—22no—
108C2×C4×C163—21no—
109C2×C4×C163—21no—
110C2×C4×C163—42no—
111C2×C4×C163C2411yescharacteristic
112C8×C163—22no—
113C22×C163—22no—
114C22×C163—41no—
115C22×C163—81no—
116C22×C163H(5,𝔽2)11yescharacteristic
117C2×D163—22no—
118C2×D163—42no—
119C2×D163—82no—
120C2×D163—161no—
121C2×D163—162no—
122C4×C163—22no—
123C4×C163—41no—
124C4×C163—41no—
125C4×C163C22×D412yes—
126Dic163—22no—
127C2×C163—21no—
128C2×C163—42no—
129C2×C163—81no—
130C2×C163D4211yescharacteristic
131D163—42no—
132D163—82no—
133C163#328811yescharacteristic, complemented, Sylow
134#3288—1631nomaximal, Sylow
135#577—1632no—
136#576—1632no—
137#496—1631no—
138#472—1632no—
139#471—1632no—
140#455—1631no—
141#454D16311yescharacteristic, complemented
142#453—1632no—
143#444—1632no—
1442−1+4C2×D16311yescharacteristic, complemented
145H(5,𝔽2)—1632no—
146H(5,𝔽2)—1631no—
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)C2×D16312yescomplemented
155#135—3262no—
156#135—3262no—
157#135—1632no—
158C2×SD16—3262no—
159C2×SD16—1632no—
160C2×D8—3262no—
161C2×D8—1632no—
162#131—1632no—
163#131—1632no—
164#127—1631no—
165#124C2×D16311yescharacteristic, complemented
166#108—1632no—
167#104—1632no—
168#104C2×D16312yescomplemented
169#101C2×D16311yescharacteristic, complemented
170#100—1631no—
171#100—1632no—
172#97—1631no—
173#41—22no—
174#41—41no—
175#41—3262no—
176#41—3262no—
177#41—3262no—
178#41—1632no—
179#41C22×D16312yes—
180C2×Q8—22no—
181C2×Q8C22×D16311yescharacteristic
182C2×D4—3262no—
183C2×D4—3262no—
184C2×D4—3262no—
185C2×D4—6521no—
186C2×D4—6522no—
187C2×D4—6521no—
188C2×D4—1632no—
189C2×D4C22×D16311yescharacteristic
190Q16—3262no—
191Q16—1632no—
192SD16—22no—
193SD16—3262no—
194SD16—3262no—
195SD16—3262no—
196SD16—3262no—
197SD16—3262no—
198SD16—1632no—
199SD16—1632no—
200D8—22no—
201D8—3262no—
202D8—3262no—
203D8—3262no—
204D8—3262no—
205D8—1632no—
206M16—3262no—
207M16—3262no—
208M16—1632no—
209M16C22×D16312yes—
210C2×C8—3262no—
211C2×C8—3262no—
212C2×C8—1632no—
213C22⋊C4—41no—
214C42C22×D16311yescharacteristic
215C23—21no—
216C23—3262no—
217C23—6522no—
218C23—6521no—
219Q8—22no—
220Q8—41no—
221Q8—1632no—
222Q8D4×D16312yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—13042no—
227D4—3262no—
228D4—3262no—
229D4—3262no—
230D4—6522no—
231D4—6522no—
232D4—6522no—
233D4—1632no—
234D4D4×D16312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—3262no—
240C2×C4—6522no—
241C2×C4—6521no—
242C2×C4C23×D16311yescharacteristic
243C8—22no—
244C8—3262no—
245C8—1632no—
246C8—1632no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—13042no—
251C22—26082no—
252C22—26081no—
253C22—3262no—
254C22—6522no—
255C22#1910478011yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—3262no—
260C4C2×D4×D16312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—13042no—
265C2—6522no—
266C2#2378229711yescharacteristic
subgroupG/H
centerC2#23782297
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#454×C163C2
socleC2×C163D42
hypercenter#454D163
hypocenterC163#3288
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #23782473C2semidirect
2. #23782469C2semidirect
3. #23782055C2semidirect
4. #23781933C2semidirect
5. #23781931C2semidirect
6. #23781842C2semidirect
7. #23781834C2semidirect
8. #23781784C2semidirect
9. #454×C163C2semidirect
10. #19104781C22semidirect
13. #19104766C22semidirect
17. #19104760C22semidirect
23. C2×#14579686C22semidirect
24. C2×#14579684C22semidirect
26. #19104743C22semidirect
27. #19104742C22semidirect
28. #19104732C22semidirect
29. #19104670C22semidirect
30. #19104663C22semidirect
31. #19104652C22semidirect
32. #19104651C22semidirect
33. #19104646C22semidirect
34. 2−1+4×C163C22semidirect
35. AGL(1,ℤ/8)×C163C22semidirect
36. #124×C163C22semidirect
37. #104×C163C22semidirect
38. #101×C163C22semidirect
51. #14579689D4semidirect
53. #14579687D4semidirect
58. #14579686D4semidirect
63. #14579684D4semidirect
133. C163#3288semidirect
141. #454D163semidirect
144. 2−1+4C2×D163semidirect
154. AGL(1,ℤ/8)C2×D163semidirect
165. #124C2×D163semidirect
168. #104C2×D163semidirect
169. #101C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824732
#237824692
#237820552
#237819332
#237819312
#237818422
#237818342
#237817842
#454×C1632
#3288163