Finite Group Databasebeta

← #35224984

Subgroups — #35224984

Order 20864.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#19104763C211yesmaximal, characteristic, complemented
2C2×#19104761C211yesmaximal, characteristic, complemented
3C2×#19104760C211yesmaximal, characteristic, complemented
4#23782211C211yesmaximal, characteristic, complemented
5C2×#19104731C211yesmaximal, characteristic, complemented
6#23782053C212yesmaximal, complemented
7#23782051C212yesmaximal, complemented
8#23782040C212yesmaximal, complemented
9#23782036C212yesmaximal, complemented
10M16×Dic163C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C22×D652C2211yescharacteristic
14C22×C22⋊D163—22no—
15#19104763—24no—
16#19104761—24no—
17#19104760—24no—
18C2×#14579688C2212yescomplemented
19C2×#14579687C2212yescomplemented
20C2×#14579686C2212yescomplemented
21C2×D1304C2212yescomplemented
22C2×#14579684C2212yescomplemented
23C2×#14579684C2212yescomplemented
24C2×#14579683C2211yescharacteristic
25#19104733C2211yescharacteristic, complemented
26#19104731—22no—
27#19104731C2212yes—
28#19104731C2212yes—
29#19104731C2211yescharacteristic, complemented
30#19104724C2211yescharacteristic, complemented
31#19104722C2211yescharacteristic, complemented
32#19104707—22no—
33C4×C22⋊D163—22no—
34C2×C4×Dic163C2211yescharacteristic
35#19104645C2212yescomplemented
36C8×Dic163C2212yescomplemented
37C2×#41×C163C2211yescharacteristic
38C22×D4×C163C2211yescharacteristic
39AGL(1,ℤ/8)×C163—24no—
40C2×SD16×C163C2212yescomplemented
41C2×D8×C163C2212yescomplemented
42C2×M16×C163C2211yescharacteristic
43C23×D163—21no—
44#14579699—22no—
45#14579699—22no—
46C2×Dic326C2311yescharacteristic
47C2×D652—22no—
48C2×D652—22no—
49C2×D652—21no—
50C2×D652C2312yes—
51C2×D652C2311yescharacteristic
52C2×C22⋊D163—24no—
53C2×C22⋊D163—24no—
54C2×C22⋊D163—22no—
55C2×C22⋊D163—21no—
56C2×C22⋊D163—48no—
57C22×Dic163D412yes—
58C2×C4×D163—21no—
59#14579688—24no—
60#14579687—24no—
61#14579686—24no—
62D1304—24no—
63#14579684—24no—
64#14579684—24no—
65#14579683D414yescomplemented
66C2×C163⋊C8C2312yes—
67Dic163⋊C4—22no—
68C22⋊Dic163—22no—
69(C2×D163)⋊C4—22no—
70C4×Dic163D412yes—
71C4×Dic163C2311yescharacteristic
72C4×Dic163C2311yescharacteristic
73C24×C163—21no—
74#41×C163—22no—
75#41×C163—22no—
76C2×Q8×C163C2311yescharacteristic
77C2×D4×C163—21no—
78C2×D4×C163—22no—
79C2×D4×C163—22no—
80C2×D4×C163—21no—
81C2×D4×C163C2311yescharacteristic
82C2×D4×C163C2312yes—
83C22×C4×C163—21no—
84C22×C4×C163C2311yescharacteristic
85SD16×C163—24no—
86D8×C163—24no—
87M16×C163D414yescomplemented
88C2×C8×C163C2312yes—
89C22×D163—21no—
90C22×D163—22no—
91C22×D163—44no—
92C22×D163—42no—
93Dic326—21no—
94Dic326—21no—
95D652—21no—
96D652—22no—
97D652—21no—
98D652—22no—
99D652—42no—
100C22⋊D163—48no—
101C22⋊D163—42no—
102C22⋊D163—44no—
103C22⋊D163—48no—
104C2×Dic163—24no—
105C2×Dic163—21no—
106C2×Dic163C2×D412yes—
107C2×Dic163C2×D412yes—
108C4×D163—42no—
109C163⋊C8C2×D414yescomplemented
110C23×C163—21no—
111C23×C163—22no—
112C23×C163—44no—
113C23×C163—42no—
114C23×C163C2×D411yescharacteristic
115Q8×C163—21no—
116Q8×C163—21no—
117D4×C163—21no—
118D4×C163—22no—
119D4×C163—22no—
120D4×C163—21no—
121D4×C163—42no—
122D4×C163—42no—
123C2×C4×C163—21no—
124C2×C4×C163—42no—
125C2×C4×C163C2×D411yescharacteristic
126C2×C4×C163C2×D412yes—
127C2×C4×C163C2×D412yes—
128C2×C4×C163C2411yescharacteristic
129C8×C163C2×D414yescomplemented
130C22×C163—22no—
131C22×C163—41no—
132C22×C163—42no—
133C22×C163—44no—
134C22×C163—42no—
135C22×C163—84no—
136C22×C163#12712yes—
137C22×C163C22×D411yescharacteristic
138C2×D163—41no—
139C2×D163—42no—
140C2×D163—44no—
141C2×D163—42no—
142C2×D163—84no—
143C4×C163—41no—
144C4×C163#12712yes—
145C4×C163C22×D411yescharacteristic
146C4×C163C22×D411yescharacteristic
147Dic163—22no—
148Dic163—41no—
149Dic163AGL(1,ℤ/8)14yescomplemented
150C2×C163—22no—
151C2×C163—81no—
152C2×C163—82no—
153C2×C163C2×#12711yescharacteristic
154C2×C163C2×AGL(1,ℤ/8)12yes—
155D163—81no—
156D163—82no—
157C163#315111yescharacteristic, complemented, Sylow
158#3151—1631nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1631no—
160C2×AGL(1,ℤ/8)—1631no—
161C2×AGL(1,ℤ/8)—1631no—
162C2×AGL(1,ℤ/8)D16311yescharacteristic, complemented
163#532—1631no—
164C2×#127—1631no—
165#496—1632no—
166#496—1632no—
167#493—1632no—
168#492—1632no—
169C4×M16—1631no—
170C2×#41—1631no—
171C2×#41C2×D16311yescharacteristic
172C22×D4—3262no—
173C22×D4—1631no—
174C22×D4C2×D16311yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3264no—
177AGL(1,ℤ/8)—3264no—
178AGL(1,ℤ/8)—3264no—
179C2×SD16—1632no—
180C2×SD16—1632no—
181C2×SD16—1632no—
182C2×SD16C2×D16312yescomplemented
183C2×D8—1632no—
184C2×D8—1632no—
185C2×D8—1632no—
186C2×D8C2×D16312yescomplemented
187C2×M16—1631no—
188C2×M16C2×D16311yescharacteristic
189C4⋊Q8—1631no—
190#127—3262no—
191#127—1632no—
192#127—1632no—
193#127—1631no—
194#124—1631no—
195#124—1631no—
196#121—3262no—
197C4×D4—3262no—
198C2×C42—1631no—
199#97—1632no—
200C4×C8—1632no—
201C24—21no—
202C24—3261no—
203#41—22no—
204#41—22no—
205#41—3262no—
206#41—3262no—
207C2×Q8—1631no—
208C2×Q8C22×D16311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—3261no—
214C2×D4—3262no—
215C2×D4—3264no—
216C2×D4—3264no—
217C2×D4—3262no—
218C2×D4—3261no—
219C2×D4—3262no—
220C2×D4—6528no—
221C2×D4—1631no—
222C2×D4—1632no—
223C2×D4C22×D16311yescharacteristic
224C2×D4C22×D16312yes—
225C22×C4—21no—
226C22×C4—3261no—
227C22×C4—1632no—
228C22×C4C22×D16311yescharacteristic
229SD16—24no—
230SD16—3264no—
231SD16—3264no—
232SD16—3264no—
233D8—24no—
234D8—3264no—
235D8—3264no—
236D8—3264no—
237M16—1634no—
238M16C22⋊D16314yescomplemented
239C2×C8—1632no—
240C2×C8C22×D16312yes—
241C4⋊C4—3262no—
242C22⋊C4—3262no—
243C22⋊C4—3262no—
244C42—1632no—
245C42—1631no—
246C42—1631no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3261no—
252C23—3262no—
253C23—6524no—
254C23—6522no—
255C23D4×D16311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3261no—
259Q8—3261no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3261no—
267D4—3262no—
268D4—3261no—
269D4—3262no—
270D4—6528no—
271D4—6522no—
272D4—6528no—
273D4—6524no—
274D4—6522no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—3264no—
278C2×C4—3261no—
279C2×C4—6522no—
280C2×C4—1632no—
281C2×C4—1632no—
282C2×C4D4×D16312yes—
283C2×C4D4×D16311yescharacteristic
284C2×C4C2×C22⋊D16312yes—
285C2×C4C23×D16311yescharacteristic
286C8—1634no—
287C8C2×C22⋊D16314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—13044no—
295C22—6524no—
296C22—6522no—
297C22—6521no—
298C22—6522no—
299C22#1910473112yes—
300C22C2×D4×D16311yescharacteristic
301C4—41no—
302C4—3262no—
303C4—6521no—
304C4—1634no—
305C4#1910473112yes—
306C4C2×D4×D16311yescharacteristic
307C4C22×C22⋊D16311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—13042no—
312C2—13041no—
313C2C2×#1910473111yescharacteristic
314C2AGL(1,ℤ/8)×D16312yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×AGL(1,ℤ/8)×C163C2
socleC22×C163C22×D4
hypercenterC2×AGL(1,ℤ/8)D163
hypocenterC163#3151
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19104763C2semidirect
2. C2×#19104761C2semidirect
3. C2×#19104760C2semidirect
4. #23782211C2semidirect
5. C2×#19104731C2semidirect
6. #23782053C2semidirect
7. #23782051C2semidirect
8. #23782040C2semidirect
9. #23782036C2semidirect
10. M16×Dic163C2semidirect
11. C2×AGL(1,ℤ/8)×C163C2semidirect
18. C2×#14579688C22semidirect
19. C2×#14579687C22semidirect
20. C2×#14579686C22semidirect
21. C2×D1304C22semidirect
22. C2×#14579684C22semidirect
23. C2×#14579684C22semidirect
25. #19104733C22semidirect
29. #19104731C22semidirect
30. #19104724C22semidirect
31. #19104722C22semidirect
35. #19104645C22semidirect
36. C8×Dic163C22semidirect
40. C2×SD16×C163C22semidirect
41. C2×D8×C163C22semidirect
65. #14579683D4semidirect
87. M16×C163D4semidirect
109. C163⋊C8C2×D4semidirect
129. C8×C163C2×D4semidirect
149. Dic163AGL(1,ℤ/8)semidirect
157. C163#3151semidirect
162. C2×AGL(1,ℤ/8)D163semidirect
182. C2×SD16C2×D163semidirect
186. C2×D8C2×D163semidirect
238. M16C22⋊D163semidirect
287. C8C2×C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047632
C2×#191047612
C2×#191047602
#237822112
C2×#191047312
#237820532
#237820512
#237820402
#237820362
M16×Dic1632
C2×AGL(1,ℤ/8)×C1632
#3151163