Finite Group Databasebeta

← #35225016

Subgroups — #35225016

Order 20864.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19104759C212yesmaximal, complemented
2C2×#19104754C211yesmaximal, characteristic, complemented
3C2×D8×D163C211yesmaximal, characteristic, complemented
4#23782228C212yesmaximal, complemented
5#23782053C212yesmaximal, complemented
6#23782052C212yesmaximal, complemented
7#23782049C211yesmaximal, characteristic, complemented
8#23782048C211yesmaximal, characteristic, complemented
9#23782040C211yesmaximal, characteristic, complemented
10#23781619C211yesmaximal, characteristic, complemented
11#493×C163C211yesmaximal, characteristic, complemented
12C2×#14579698C2212yes—
13C2×D4×D163C2212yes—
14#19104759—28no—
15#19104754—24no—
16C2×Dic652C2211yescharacteristic, complemented
17C2×#14579688C2212yescomplemented
18C2×#14579687C2212yescomplemented
19C2×#14579687C2212yescomplemented
20C2×D1304C2211yescharacteristic
21C2×#14579684C2212yes—
22C2×#14579684C2212yescomplemented
23D8×D163—24no—
24C2×#14579682C2211yescharacteristic
25C2×C8×D163C2211yescharacteristic
26#19104731C2212yescomplemented
27#19104723C2212yescomplemented
28#19104722C2212yescomplemented
29#19104707—24no—
30#19104703—24no—
31#19104692C2211yescharacteristic
32#19104646C2211yescharacteristic, complemented
33#19104645C2211yescharacteristic, complemented
34C8×Dic163C2211yescharacteristic, complemented
35C2×D8×C163C2211yescharacteristic
36C2×D8×C163C2211yescharacteristic, complemented
37C2×D8×C163C2212yescomplemented
38#127×C163C2212yescomplemented
39C4×C8×C163C2211yescharacteristic, complemented
40C23×D163—22no—
41#14579698—24no—
42#14579698—24no—
43C2×Dic326C2312yes—
44C2×D652C2312yes—
45C2×C22⋊D163—22no—
46C2×C22⋊D163—22no—
47C2×C22⋊D163—24no—
48D4×D163—24no—
49D4×D163—24no—
50C22×Dic163—22no—
51C2×C4×D163C2311yescharacteristic
52Dic652—22no—
53#14579688—24no—
54#14579687—24no—
55#14579687—24no—
56D1304—22no—
57#14579684—24no—
58#14579684—24no—
59#14579682D414yescomplemented
60C2×C163⋊C8C2311yescharacteristic
61C2×C163⋊C8C2311yescharacteristic
62C8×D163D414yescomplemented
63Dic163⋊C4D412yes—
64C22⋊Dic163—24no—
65(C2×D163)⋊C4—24no—
66(C2×D163)⋊C4D412yes—
67C4×Dic163C2311yescharacteristic
68C2×D4×C163—24no—
69C2×D4×C163C2312yes—
70C2×D4×C163C2312yes—
71D8×C163—22no—
72D8×C163—22no—
73D8×C163—24no—
74C2×C8×C163C2311yescharacteristic
75C2×C8×C163C2311yescharacteristic
76C42×C163C2311yescharacteristic
77C22×D163—22no—
78C22×D163—44no—
79C22×D163—84no—
80C22×D163C2×D411yescharacteristic
81Dic326—22no—
82Dic326—22no—
83D652—22no—
84D652—22no—
85C22⋊D163—44no—
86C22⋊D163—44no—
87C22⋊D163—84no—
88C2×Dic163—22no—
89C2×Dic163—44no—
90C2×Dic163C2×D412yes—
91C2×Dic163C2×D411yescharacteristic
92C4×D163C2×D412yes—
93C4×D163C2×D412yes—
94C163⋊C8C2×D412yes—
95C163⋊C8C2×D412yes—
96C23×C163—22no—
97C23×C163—22no—
98D4×C163—22no—
99D4×C163—22no—
100D4×C163—22no—
101D4×C163—22no—
102D4×C163—84no—
103C2×C4×C163C2×D412yes—
104C2×C4×C163C2411yescharacteristic
105C8×C163C2×D412yes—
106C8×C163C2×D412yes—
107C22×C163—42no—
108C22×C163—42no—
109C22×C163—44no—
110C22×C163—44no—
111C22×C163C22×D411yescharacteristic
112C2×D163—22no—
113C2×D163—42no—
114C2×D163—44no—
115C2×D163—88no—
116C2×D163#12712yes—
117C4×C163—22no—
118C4×C163C22×D411yescharacteristic
119C4×C163C22×D411yescharacteristic
120Dic163—22no—
121Dic163—42no—
122Dic163#12712yes—
123C2×C163—82no—
124C2×C163—82no—
125C2×C163C2×#12711yescharacteristic
126C2×C163#57612yes—
127D163—22no—
128D163—82no—
129C163#315511yescharacteristic, complemented, Sylow
130#3155—1631nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1632no—
132C2×#135—1631no—
133C22×D8—1631no—
134#535—1632no—
135#496—1632no—
136#496—1632no—
137#495—1631no—
138#495—1631no—
139#493—1631no—
140#493D16311yescharacteristic, complemented
141#405—1631no—
142C2×#41—1632no—
143C22×D4—1632no—
144AGL(1,ℤ/8)—3268no—
145#135—3264no—
146C2×Q16—1631no—
147C2×SD16—1632no—
148C2×SD16—1632no—
149C2×SD16—1632no—
150C2×D8—3264no—
151C2×D8—1632no—
152C2×D8—1632no—
153C2×D8—1631no—
154C2×D8C2×D16311yescharacteristic
155C2×D8C2×D16311yescharacteristic, complemented
156C2×D8C2×D16312yescomplemented
157C2×M16—1631no—
158C22×C8—1631no—
159#127—1632no—
160#127C2×D16312yescomplemented
161#124—1632no—
162#124—1632no—
163#121—3264no—
164#120—3264no—
165#117—1631no—
166#97—1631no—
167#97—1631no—
168C4×C8—1631no—
169C4×C8C2×D16311yescharacteristic, complemented
170C24—3262no—
171#41—3264no—
172#41—3264no—
173C2×Q8—1632no—
174C2×D4—24no—
175C2×D4—3262no—
176C2×D4—3264no—
177C2×D4—3264no—
178C2×D4—3264no—
179C2×D4—3262no—
180C2×D4—1632no—
181C2×D4C22×D16312yes—
182C2×D4C22×D16312yes—
183C22×C4—3262no—
184C22×C4—1631no—
185Q16—3262no—
186SD16—3264no—
187SD16—3264no—
188SD16—3264no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—3264no—
193D8—3264no—
194D8—3262no—
195M16—1634no—
196C2×C8—1631no—
197C2×C8—1631no—
198C2×C8—1634no—
199C2×C8C22×D16311yescharacteristic
200C2×C8C22×D16311yescharacteristic
201C4⋊C4—1632no—
202C22⋊C4—3264no—
203C22⋊C4—3264no—
204C22⋊C4—1632no—
205C42—1631no—
206C42C22×D16311yescharacteristic
207C23—22no—
208C23—22no—
209C23—13044no—
210C23—3262no—
211C23—6524no—
212C23—1631no—
213Q8—3262no—
214Q8—3262no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—13044no—
221D4—3262no—
222D4—3262no—
223D4—6524no—
224D4—6524no—
225C2×C4—3262no—
226C2×C4—6524no—
227C2×C4—1632no—
228C2×C4—1631no—
229C2×C4—1632no—
230C2×C4—1632no—
231C2×C4D4×D16312yes—
232C2×C4C23×D16311yescharacteristic
233C8—1632no—
234C8—1632no—
235C8D4×D16312yes—
236C8D4×D16312yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—13048no—
242C22—3262no—
243C22—6522no—
244C22—6524no—
245C22—1632no—
246C22C2×D4×D16311yescharacteristic
247C4—22no—
248C4—3262no—
249C4—6522no—
250C4—1632no—
251C4C2×D4×D16311yescharacteristic
252C4C2×D4×D16311yescharacteristic
253C2—82no—
254C2—82no—
255C2—13042no—
256C2—3262no—
257C2#127×D16311yescharacteristic
258C2#2378246612yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#493×C163C2
socleC22×C163C22×D4
hypercenter#493D163
hypocenterC163#3155
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. C2×#19104754C2semidirect
3. C2×D8×D163C2semidirect
4. #23782228C2semidirect
5. #23782053C2semidirect
6. #23782052C2semidirect
7. #23782049C2semidirect
8. #23782048C2semidirect
9. #23782040C2semidirect
10. #23781619C2semidirect
11. #493×C163C2semidirect
16. C2×Dic652C22semidirect
17. C2×#14579688C22semidirect
18. C2×#14579687C22semidirect
19. C2×#14579687C22semidirect
22. C2×#14579684C22semidirect
26. #19104731C22semidirect
27. #19104723C22semidirect
28. #19104722C22semidirect
32. #19104646C22semidirect
33. #19104645C22semidirect
34. C8×Dic163C22semidirect
36. C2×D8×C163C22semidirect
37. C2×D8×C163C22semidirect
38. #127×C163C22semidirect
39. C4×C8×C163C22semidirect
59. #14579682D4semidirect
62. C8×D163D4semidirect
129. C163#3155semidirect
140. #493D163semidirect
155. C2×D8C2×D163semidirect
156. C2×D8C2×D163semidirect
160. #127C2×D163semidirect
169. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
C2×#191047542
C2×D8×D1632
#237822282
#237820532
#237820522
#237820492
#237820482
#237820402
#237816192
#493×C1632
#3155163