Finite Group Databasebeta

← #35225015

Subgroups — #35225015

Order 20864.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19104755C211yesmaximal, characteristic, complemented
3C22×#14579686C211yesmaximal, characteristic, complemented
4#23782239C211yesmaximal, characteristic, complemented
5#23782229C211yesmaximal, characteristic, complemented
6#23782058C212yesmaximal, complemented
7#23782051C212yesmaximal, complemented
8#23782050C212yesmaximal, complemented
9#23782036C212yesmaximal, complemented
10#23781623C211yesmaximal, characteristic, complemented
11C2×#137×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C22×D652C2211yescharacteristic
14#19104763—24no—
15#19104755—24no—
16C2×#14579689C2211yescharacteristic, complemented
17C2×#14579688C2212yes—
18C2×#14579687C2211yescharacteristic, complemented
19C2×#14579686—24no—
20C2×#14579686C2212yes—
21C2×#14579686C2212yes—
22C2×#14579686C2211yescharacteristic, complemented
23C2×D1304C2212yes—
24C2×#14579684C2211yescharacteristic, complemented
25C22×C163⋊C8C2211yescharacteristic
26#19104733C2212yescomplemented
27#19104731C2212yes—
28#19104724C2212yes—
29#19104724C2212yescomplemented
30#19104718—22no—
31#19104711—22no—
32#19104709—22no—
33#19104703—22no—
34#19104694C2211yescharacteristic
35#19104645C2212yescomplemented
36C8×Dic163C2212yescomplemented
37C2×#41×C163C2211yescharacteristic
38C22×Q8×C163C2211yescharacteristic
39#137×C163—24no—
40C2×Q16×C163C2212yescomplemented
41C2×SD16×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—21no—
54C2×C4×D163—21no—
55#14579689—22no—
56#14579688—24no—
57#14579687—22no—
58#14579686—24no—
59#14579686—24no—
60#14579686—22no—
61D1304—24no—
62#14579684—22no—
63C2×C163⋊C8D414yescomplemented
64C2×C163⋊C8C2311yescharacteristic
65C2×C163⋊C8C2311yescharacteristic
66C4⋊Dic163D412yes—
67Dic163⋊C4—24no—
68C22⋊Dic163D412yes—
69(C2×D163)⋊C4—24no—
70(C2×D163)⋊C4—24no—
71C4×Dic163C2312yes—
72#41×C163—22no—
73#41×C163—22no—
74C2×Q8×C163—21no—
75C2×Q8×C163—22no—
76C2×Q8×C163—22no—
77C2×Q8×C163C2312yes—
78C2×Q8×C163C2311yescharacteristic
79C2×D4×C163—21no—
80C2×D4×C163C2311yescharacteristic
81C22×C4×C163—21no—
82C22×C4×C163—21no—
83C22×C4×C163C2311yescharacteristic
84Q16×C163—24no—
85SD16×C163—24no—
86M16×C163D414yescomplemented
87C2×C8×C163C2312yes—
88C22×D163—21no—
89C22×D163—22no—
90C22×D163—42no—
91C22×D163—82no—
92Dic326—21no—
93Dic326—21no—
94D652—21no—
95D652—21no—
96D652—22no—
97D652—22no—
98D652—42no—
99C22⋊D163—42no—
100C22⋊D163—84no—
101C2×Dic163—21no—
102C2×Dic163C2×D414yes—
103C4×D163—42no—
104C163⋊C8C2×D412yes—
105C163⋊C8C2×D412yes—
106C23×C163—21no—
107C23×C163C2×D411yescharacteristic
108Q8×C163—22no—
109Q8×C163—21no—
110Q8×C163—22no—
111Q8×C163—21no—
112Q8×C163—42no—
113D4×C163—21no—
114D4×C163—21no—
115D4×C163—42no—
116C2×C4×C163—21no—
117C2×C4×C163—22no—
118C2×C4×C163—42no—
119C2×C4×C163—42no—
120C2×C4×C163C2×D411yescharacteristic
121C2×C4×C163C2×D412yes—
122C2×C4×C163C2×D412yes—
123C2×C4×C163C2411yescharacteristic
124C8×C163C2×D414yes—
125C22×C163—22no—
126C22×C163—41no—
127C22×C163—42no—
128C22×C163#12712yes—
129C22×C163C22×D411yescharacteristic
130C2×D163—41no—
131C2×D163—42no—
132C2×D163—44no—
133C2×D163—42no—
134C2×D163—84no—
135C4×C163—41no—
136C4×C163—42no—
137C4×C163#12712yes—
138C4×C163C22×D411yescharacteristic
139C4×C163C22×D411yescharacteristic
140Dic163—24no—
141Dic163—41no—
142C2×C163—22no—
143C2×C163—81no—
144C2×C163C2×#12711yescharacteristic
145C2×C163#57712yes—
146D163—81no—
147D163—82no—
148C163#315411yescharacteristic, complemented, Sylow
149#3154—1631nomaximal, Sylow
150C2×#137D16311yescharacteristic, complemented
151C2×AGL(1,ℤ/8)—1631no—
152C2×#135—1631no—
153C22×SD16—1631no—
154#536—1631no—
155#535—1631no—
156#497—1632no—
157#496—1632no—
158#495—1632no—
159#492—1632no—
160#405—1631no—
161C2×#41—1631no—
162C2×#41C2×D16311yescharacteristic
163C22×Q8C2×D16311yescharacteristic
164C22×D4—1631no—
165#137—24no—
166AGL(1,ℤ/8)—3264no—
167#135—3264no—
168C2×Q16—1631no—
169C2×Q16C2×D16312yescomplemented
170C2×SD16—3264no—
171C2×SD16—1632no—
172C2×SD16—1632no—
173C2×SD16—1631no—
174C2×SD16—1631no—
175C2×SD16—1632no—
176C2×SD16C2×D16312yescomplemented
177C2×D8—1631no—
178C2×D8—1632no—
179C2×M16C2×D16311yescharacteristic
180C22×C8—1631no—
181C4⋊Q8—1632no—
182#127—1632no—
183#124—1632no—
184#124—1632no—
185#123—3262no—
186C22⋊Q8—3262no—
187#121—3262no—
188#120—3262no—
189#117—1631no—
190#97—1632no—
191C4×C8—1632no—
192C24—3261no—
193#41—22no—
194#41—22no—
195#41—3262no—
196#41—3262no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1631no—
201C2×Q8C22×D16311yescharacteristic
202C2×Q8C22×D16312yes—
203C2×D4—21no—
204C2×D4—3261no—
205C2×D4—3264no—
206C2×D4—3262no—
207C2×D4—3261no—
208C2×D4—3262no—
209C2×D4—1631no—
210C2×D4—1632no—
211C2×D4C22×D16311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—3261no—
215C22×C4C22×D16311yescharacteristic
216Q16—24no—
217Q16—3262no—
218SD16—24no—
219SD16—3262no—
220SD16—3264no—
221SD16—3264no—
222SD16—3262no—
223SD16—3264no—
224D8—3262no—
225D8—3264no—
226M16C22⋊D16314yescomplemented
227C2×C8—1634no—
228C2×C8—1631no—
229C2×C8—1631no—
230C2×C8C22×D16312yes—
231C4⋊C4—3264no—
232C4⋊C4—1632no—
233C22⋊C4—3264no—
234C22⋊C4—3264no—
235C22⋊C4—1632no—
236C42—1632no—
237C23—21no—
238C23—13042no—
239C23—3261no—
240C23—3262no—
241C23—6522no—
242C23D4×D16311yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—3261no—
249Q8—3261no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—13044no—
254D4—3261no—
255D4—3262no—
256D4—3261no—
257D4—3262no—
258D4—6522no—
259D4—6522no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—3261no—
265C2×C4—6522no—
266C2×C4—1634no—
267C2×C4D4×D16312yes—
268C2×C4D4×D16311yescharacteristic
269C2×C4C2×C22⋊D16312yes—
270C2×C4C23×D16311yescharacteristic
271C8—1632no—
272C8—1632no—
273C8C2×C22⋊D16314yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—13044no—
278C22—6521no—
279C22—6522no—
280C22—6524no—
281C22—6522no—
282C22#1910473112yes—
283C22C2×D4×D16311yescharacteristic
284C4—41no—
285C4—42no—
286C4—3264no—
287C4—6521no—
288C4#1910473112yes—
289C4C2×D4×D16311yescharacteristic
290C4C22×C22⋊D16311yescharacteristic
291C2—22no—
292C2—81no—
293C2—13042no—
294C2—13041no—
295C2C2×#1910473111yescharacteristic
296C2#2378247412yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#137×C163C2
socleC22×C163C22×D4
hypercenterC2×#137D163
hypocenterC163#3154
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#19104763C2semidirect
2. C2×#19104755C2semidirect
3. C22×#14579686C2semidirect
4. #23782239C2semidirect
5. #23782229C2semidirect
6. #23782058C2semidirect
7. #23782051C2semidirect
8. #23782050C2semidirect
9. #23782036C2semidirect
10. #23781623C2semidirect
11. C2×#137×C163C2semidirect
16. C2×#14579689C22semidirect
18. C2×#14579687C22semidirect
22. C2×#14579686C22semidirect
24. C2×#14579684C22semidirect
26. #19104733C22semidirect
29. #19104724C22semidirect
35. #19104645C22semidirect
36. C8×Dic163C22semidirect
40. C2×Q16×C163C22semidirect
41. C2×SD16×C163C22semidirect
63. C2×C163⋊C8D4semidirect
86. M16×C163D4semidirect
148. C163#3154semidirect
150. C2×#137D163semidirect
169. C2×Q16C2×D163semidirect
176. C2×SD16C2×D163semidirect
226. M16C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047632
C2×#191047552
C22×#145796862
#237822392
#237822292
#237820582
#237820512
#237820502
#237820362
#237816232
C2×#137×C1632
#3154163