Finite Group Databasebeta

← #35225003

Subgroups — #35225003

Order 20864.

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

normal subgroups only

This group has 572 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×#19104766C211yesmaximal, characteristic, complemented
2C2×#19104763C211yesmaximal, characteristic, complemented
3C2×#19104760C211yesmaximal, characteristic, complemented
4#23782211C211yesmaximal, characteristic, complemented
5C2×#19104724C211yesmaximal, characteristic, complemented
6#23782058C212yesmaximal, complemented
7#23782051C212yesmaximal, complemented
8#23782050C212yesmaximal, complemented
9#23782046C212yesmaximal, complemented
10M16×Dic163C211yesmaximal, characteristic, complemented
11C2×#137×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C22×D652C2211yescharacteristic
14#19104766—24no—
15#19104763—24no—
16#19104760—24no—
17C2×#14579689C2212yescomplemented
18C2×#14579688C2212yes—
19C2×#14579686C2212yes—
20C2×#14579686C2212yescomplemented
21C2×D1304C2212yes—
22C2×#14579684C2212yes—
23C2×#14579683C2211yescharacteristic
24#19104733C2211yescharacteristic, complemented
25#19104731C2211yescharacteristic, complemented
26#19104724—22no—
27#19104724C2212yes—
28#19104724C2212yes—
29#19104724C2211yescharacteristic, complemented
30#19104722C2211yescharacteristic, complemented
31#19104707—22no—
32C4×C22⋊D163—22no—
33C2×(C2×D163)⋊C4—22no—
34C2×C4×Dic163C2211yescharacteristic
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—22no—
53C2×C22⋊D163—21no—
54C22×Dic163D412yes—
55C2×C4×D163—21no—
56#14579689—24no—
57#14579688—24no—
58#14579686—24no—
59#14579686—24no—
60D1304—24no—
61#14579684—24no—
62#14579683D414yescomplemented
63C2×C163⋊C8C2312yes—
64Dic163⋊C4—22no—
65C22⋊Dic163—22no—
66(C2×D163)⋊C4—24no—
67(C2×D163)⋊C4—24no—
68(C2×D163)⋊C4—22no—
69C4×Dic163D412yes—
70C4×Dic163C2311yescharacteristic
71C4×Dic163C2311yescharacteristic
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—22no—
89C22×D163—21no—
90C22×D163—44no—
91C22×D163—42no—
92Dic326—21no—
93Dic326—21no—
94D652—22no—
95D652—21no—
96D652—22no—
97D652—21no—
98D652—42no—
99C22⋊D163—44no—
100C22⋊D163—42no—
101C2×Dic163—21no—
102C2×Dic163—42no—
103C2×Dic163C2×D412yes—
104C2×Dic163C2×D412yes—
105C4×D163—42no—
106C163⋊C8C2×D414yes—
107C23×C163—21no—
108C23×C163C2×D411yescharacteristic
109Q8×C163—22no—
110Q8×C163—21no—
111Q8×C163—22no—
112Q8×C163—21no—
113Q8×C163—42no—
114D4×C163—21no—
115D4×C163—21no—
116D4×C163—42no—
117C2×C4×C163—21no—
118C2×C4×C163—22no—
119C2×C4×C163—42no—
120C2×C4×C163—42no—
121C2×C4×C163C2×D411yescharacteristic
122C2×C4×C163C2×D412yes—
123C2×C4×C163C2×D412yes—
124C2×C4×C163C2411yescharacteristic
125C8×C163C2×D414yes—
126C22×C163—22no—
127C22×C163—41no—
128C22×C163—42no—
129C22×C163#12712yes—
130C22×C163C22×D411yescharacteristic
131C2×D163—41no—
132C2×D163—42no—
133C2×D163—42no—
134C2×D163—44no—
135C2×D163—84no—
136C4×C163—41no—
137C4×C163—42no—
138C4×C163#12712yes—
139C4×C163C22×D411yescharacteristic
140C4×C163C22×D411yescharacteristic
141Dic163—22no—
142Dic163—22no—
143Dic163—41no—
144C2×C163—22no—
145C2×C163—81no—
146C2×C163C2×#12711yescharacteristic
147C2×C163#57512yes—
148D163—81no—
149D163—82no—
150C163#315311yescharacteristic, complemented, Sylow
151#3153—1631nomaximal, Sylow
152C2×#137—1631no—
153C2×#137D16311yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—1631no—
155C2×AGL(1,ℤ/8)—1631no—
156#532—1631no—
157C2×#124—1631no—
158#497—1632no—
159#496—1632no—
160#495—1632no—
161#495—1632no—
162C4×M16—1631no—
163C2×#41—1631no—
164C2×#41C2×D16311yescharacteristic
165C22×Q8C2×D16311yescharacteristic
166C22×D4—1631no—
167#137—24no—
168#137—3264no—
169AGL(1,ℤ/8)—3264no—
170AGL(1,ℤ/8)—3264no—
171C2×Q16—1632no—
172C2×Q16C2×D16312yescomplemented
173C2×SD16—1632no—
174C2×SD16—1632no—
175C2×SD16—1632no—
176C2×SD16C2×D16312yescomplemented
177C2×D8—1632no—
178C2×D8—1632no—
179C2×M16—1631no—
180C2×M16C2×D16311yescharacteristic
181C4⋊Q8—1631no—
182#127—1631no—
183#124—3262no—
184#124—1632no—
185#124—1632no—
186#124—1631no—
187#124—1631no—
188#121—3262no—
189C4×D4—3262no—
190C2×C22⋊C4—3262no—
191C2×C42—1631no—
192#97—1632no—
193C4×C8—1632no—
194C24—3261no—
195#41—22no—
196#41—22no—
197#41—3262no—
198#41—3262no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—1631no—
203C2×Q8C22×D16312yes—
204C2×Q8C22×D16311yescharacteristic
205C2×D4—21no—
206C2×D4—3261no—
207C2×D4—3262no—
208C2×D4—3262no—
209C2×D4—3261no—
210C2×D4—3262no—
211C2×D4—1632no—
212C2×D4—1631no—
213C2×D4C22×D16311yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—3261no—
217C22×C4—1632no—
218C22×C4C22×D16311yescharacteristic
219Q16—24no—
220Q16—3264no—
221SD16—24no—
222SD16—3264no—
223SD16—3264no—
224SD16—3264no—
225D8—3264no—
226D8—3264no—
227M16—1634no—
228M16C22⋊D16314yescomplemented
229C2×C8—1632no—
230C2×C8C22×D16312yes—
231C4⋊C4—3262no—
232C22⋊C4—3264no—
233C22⋊C4—3264no—
234C22⋊C4—3262no—
235C22⋊C4—3262no—
236C42—1632no—
237C42—1631no—
238C42—1631no—
239C23—21no—
240C23—3262no—
241C23—3261no—
242C23—6524no—
243C23—6522no—
244C23D4×D16311yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—3261no—
251Q8—3261no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—3262no—
256D4—3261no—
257D4—3261no—
258D4—3262no—
259D4—6524no—
260D4—6522no—
261D4—6522no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—3261no—
267C2×C4—6522no—
268C2×C4—6522no—
269C2×C4—1632no—
270C2×C4—1632no—
271C2×C4D4×D16312yes—
272C2×C4D4×D16311yescharacteristic
273C2×C4C2×C22⋊D16312yes—
274C2×C4C23×D16311yescharacteristic
275C8—1634no—
276C8C2×C22⋊D16314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—13044no—
281C22—6521no—
282C22—6522no—
283C22—6522no—
284C22—6524no—
285C22#1910473112yes—
286C22C2×D4×D16311yescharacteristic
287C4—41no—
288C4—42no—
289C4—3262no—
290C4—3262no—
291C4—6521no—
292C4#1910473112yes—
293C4C2×D4×D16311yescharacteristic
294C4C22×C22⋊D16311yescharacteristic
295C2—22no—
296C2—81no—
297C2—13042no—
298C2—13041no—
299C2C2×#1910473111yescharacteristic
300C2#2378246412yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#137×C163C2
socleC22×C163C22×D4
hypercenterC2×#137D163
hypocenterC163#3153
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×#19104766C2semidirect
2. C2×#19104763C2semidirect
3. C2×#19104760C2semidirect
4. #23782211C2semidirect
5. C2×#19104724C2semidirect
6. #23782058C2semidirect
7. #23782051C2semidirect
8. #23782050C2semidirect
9. #23782046C2semidirect
10. M16×Dic163C2semidirect
11. C2×#137×C163C2semidirect
17. C2×#14579689C22semidirect
20. C2×#14579686C22semidirect
24. #19104733C22semidirect
25. #19104731C22semidirect
29. #19104724C22semidirect
30. #19104722C22semidirect
35. #19104645C22semidirect
36. C8×Dic163C22semidirect
40. C2×Q16×C163C22semidirect
41. C2×SD16×C163C22semidirect
62. #14579683D4semidirect
86. M16×C163D4semidirect
150. C163#3153semidirect
153. C2×#137D163semidirect
172. C2×Q16C2×D163semidirect
176. C2×SD16C2×D163semidirect
228. M16C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047662
C2×#191047632
C2×#191047602
#237822112
C2×#191047242
#237820582
#237820512
#237820502
#237820462
M16×Dic1632
C2×#137×C1632
#3153163