Finite Group Databasebeta

← #35225013

Subgroups — #35225013

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×#19104765C211yesmaximal, characteristic, complemented
2C2×#19104760C211yesmaximal, characteristic, complemented
3C22×#14579688C211yesmaximal, characteristic, complemented
4#23782236C211yesmaximal, characteristic, complemented
5#23782229C211yesmaximal, characteristic, complemented
6#23782058C212yesmaximal, complemented
7#23782053C212yesmaximal, complemented
8#23782046C212yesmaximal, complemented
9#23782036C212yesmaximal, complemented
10#23781620C211yesmaximal, characteristic, complemented
11C2×#135×C163C211yesmaximal, characteristic, complemented
12C22×Dic326C2211yescharacteristic
13C22×D652C2211yescharacteristic
14#19104765—24no—
15#19104760—24no—
16C2×#14579689C2212yescomplemented
17C2×#14579688—24no—
18C2×#14579688C2212yes—
19C2×#14579688C2212yes—
20C2×#14579687C2212yescomplemented
21C2×#14579686C2212yes—
22C2×#14579684C2212yes—
23C2×#14579683C2211yescharacteristic
24#19104733C2212yescomplemented
25#19104731C2212yes—
26#19104724C2212yes—
27#19104722C2212yescomplemented
28#19104716—22no—
29#19104714—22no—
30#19104709—22no—
31#19104703—22no—
32#19104694C2211yescharacteristic
33#19104645C2212yescomplemented
34C8×Dic163C2212yescomplemented
35C2×#41×C163C2211yescharacteristic
36C2×#41×C163C2211yescharacteristic
37#135×C163—24no—
38C2×Q16×C163C2211yescharacteristic, complemented
39C2×SD16×C163C2211yescharacteristic, complemented
40C2×SD16×C163C2211yescharacteristic, complemented
41C2×D8×C163C2211yescharacteristic, complemented
42C22×C8×C163C2211yescharacteristic
43C23×D163—21no—
44C2×Dic326—21no—
45C2×Dic326—22no—
46C2×Dic326—22no—
47C2×Dic326C2312yes—
48C2×D652—22no—
49C2×D652—22no—
50C2×D652—21no—
51C2×D652C2312yes—
52C2×C22⋊D163—24no—
53C22×Dic163—21no—
54#14579689—24no—
55#14579688—24no—
56#14579688—24no—
57#14579687—24no—
58#14579686—24no—
59#14579684—24no—
60#14579683D414yescomplemented
61C2×C163⋊C8C2312yes—
62C4⋊Dic163D412yes—
63Dic163⋊C4—24no—
64C22⋊Dic163—24no—
65C22⋊Dic163D412yes—
66(C2×D163)⋊C4—24no—
67C4×Dic163C2312yes—
68#41×C163—22no—
69#41×C163—22no—
70#41×C163—22no—
71#41×C163—22no—
72C2×Q8×C163C2311yescharacteristic
73C2×Q8×C163C2311yescharacteristic
74C2×D4×C163—21no—
75C2×D4×C163—21no—
76C2×D4×C163C2311yescharacteristic
77C2×D4×C163C2311yescharacteristic
78C22×C4×C163—21no—
79C22×C4×C163—21no—
80C22×C4×C163C2311yescharacteristic
81Q16×C163—22no—
82SD16×C163—22no—
83SD16×C163—22no—
84D8×C163—22no—
85C2×C8×C163D414yescomplemented
86C2×C8×C163C2311yescharacteristic
87C2×C8×C163C2311yescharacteristic
88C22×D163—22no—
89C22×D163—42no—
90C22×D163—82no—
91Dic326—22no—
92Dic326—22no—
93Dic326—42no—
94D652—22no—
95D652—22no—
96D652—42no—
97C22⋊D163—84no—
98C2×Dic163—22no—
99C2×Dic163—42no—
100C2×Dic163C2×D414yes—
101C163⋊C8C2×D414yes—
102C23×C163—21no—
103C23×C163—21no—
104C23×C163C2×D411yescharacteristic
105Q8×C163—21no—
106Q8×C163—21no—
107Q8×C163—21no—
108Q8×C163—21no—
109D4×C163—21no—
110D4×C163—21no—
111D4×C163—21no—
112D4×C163—21no—
113D4×C163—42no—
114D4×C163—42no—
115C2×C4×C163—21no—
116C2×C4×C163—21no—
117C2×C4×C163—42no—
118C2×C4×C163—42no—
119C2×C4×C163C2×D411yescharacteristic
120C2×C4×C163C2×D412yes—
121C2×C4×C163C2×D412yes—
122C2×C4×C163C2411yescharacteristic
123C8×C163C2×D412yes—
124C8×C163C2×D412yes—
125C22×C163—22no—
126C22×C163—41no—
127C22×C163—41no—
128C22×C163—42no—
129C22×C163—42no—
130C22×C163#12712yes—
131C22×C163C22×D411yescharacteristic
132C2×D163—42no—
133C2×D163—44no—
134C2×D163—84no—
135C4×C163—41no—
136C4×C163—41no—
137C4×C163#12712yes—
138C4×C163C22×D411yescharacteristic
139C4×C163C22×D411yescharacteristic
140Dic163—24no—
141Dic163—42no—
142C2×C163—22no—
143C2×C163—81no—
144C2×C163—81no—
145C2×C163C2×#12711yescharacteristic
146C2×C163#57712yes—
147D163—82no—
148C163#315411yescharacteristic, complemented, Sylow
149#3154—1631nomaximal, Sylow
150C2×#137—1631no—
151C2×AGL(1,ℤ/8)—1631no—
152C2×#135D16311yescharacteristic, complemented
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×#41C2×D16311yescharacteristic
162C2×#41C2×D16311yescharacteristic
163C22×Q8—1631no—
164C22×D4—1631no—
165#137—3264no—
166AGL(1,ℤ/8)—3264no—
167#135—24no—
168C2×Q16—1632no—
169C2×Q16C2×D16311yescharacteristic, complemented
170C2×SD16—3264no—
171C2×SD16—1632no—
172C2×SD16—1632no—
173C2×SD16—1632no—
174C2×SD16—1632no—
175C2×SD16C2×D16311yescharacteristic, complemented
176C2×SD16C2×D16311yescharacteristic, complemented
177C2×D8—1632no—
178C2×D8C2×D16311yescharacteristic, complemented
179C2×M16—1631no—
180C22×C8C2×D16311yescharacteristic
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—22no—
196#41—22no—
197C2×Q8—3261no—
198C2×Q8—3262no—
199C2×Q8—3262no—
200C2×Q8—1632no—
201C2×Q8C22×D16311yescharacteristic
202C2×Q8C22×D16311yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—3264no—
206C2×D4—3262no—
207C2×D4—3261no—
208C2×D4—3262no—
209C2×D4—1632no—
210C2×D4C22×D16311yescharacteristic
211C2×D4C22×D16311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—3261no—
215C22×C4C22×D16311yescharacteristic
216Q16—22no—
217Q16—3264no—
218SD16—22no—
219SD16—22no—
220SD16—3264no—
221SD16—3264no—
222SD16—3264no—
223SD16—3264no—
224D8—22no—
225D8—3264no—
226M16—1634no—
227C2×C8—1632no—
228C2×C8C22⋊D16314yescomplemented
229C2×C8C22×D16311yescharacteristic
230C2×C8C22×D16311yescharacteristic
231C4⋊C4—3264no—
232C4⋊C4—1632no—
233C22⋊C4—3264no—
234C22⋊C4—3264no—
235C22⋊C4—1632no—
236C42—1632no—
237C23—21no—
238C23—21no—
239C23—13042no—
240C23—3262no—
241C23—6522no—
242C23D4×D16311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—3262no—
248Q8—3262no—
249Q8—6522no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—13044no—
257D4—3262no—
258D4—3262no—
259D4—6522no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—3262no—
265C2×C4—6522no—
266C2×C4—1634no—
267C2×C4D4×D16312yes—
268C2×C4D4×D16311yescharacteristic
269C2×C4C2×C22⋊D16312yes—
270C2×C4C23×D16311yescharacteristic
271C8—1634no—
272C8C2×C22⋊D16312yes—
273C8C2×C22⋊D16312yes—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—13044no—
280C22—6522no—
281C22—6524no—
282C22#1910473112yes—
283C22C2×D4×D16311yescharacteristic
284C4—41no—
285C4—41no—
286C4—3264no—
287C4—6522no—
288C4#1910473112yes—
289C4C2×D4×D16311yescharacteristic
290C4C22×C22⋊D16311yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—13042no—
295C2C2×#1910473111yescharacteristic
296C2#2378247212yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#135×C163C2
socleC22×C163C22×D4
hypercenterC2×#135D163
hypocenterC163#3154
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#19104765C2semidirect
2. C2×#19104760C2semidirect
3. C22×#14579688C2semidirect
4. #23782236C2semidirect
5. #23782229C2semidirect
6. #23782058C2semidirect
7. #23782053C2semidirect
8. #23782046C2semidirect
9. #23782036C2semidirect
10. #23781620C2semidirect
11. C2×#135×C163C2semidirect
16. C2×#14579689C22semidirect
20. C2×#14579687C22semidirect
24. #19104733C22semidirect
27. #19104722C22semidirect
33. #19104645C22semidirect
34. C8×Dic163C22semidirect
38. C2×Q16×C163C22semidirect
39. C2×SD16×C163C22semidirect
40. C2×SD16×C163C22semidirect
41. C2×D8×C163C22semidirect
60. #14579683D4semidirect
85. C2×C8×C163D4semidirect
148. C163#3154semidirect
152. C2×#135D163semidirect
169. C2×Q16C2×D163semidirect
175. C2×SD16C2×D163semidirect
176. C2×SD16C2×D163semidirect
178. C2×D8C2×D163semidirect
228. C2×C8C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047652
C2×#191047602
C22×#145796882
#237822362
#237822292
#237820582
#237820532
#237820462
#237820362
#237816202
C2×#135×C1632
#3154163