Finite Group Databasebeta

← #35227927

Subgroups — #35227927

Order 20864.

1040 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782288C212yesmaximal, complemented
2#23782287C212yesmaximal, complemented
3#23782276C211yesmaximal, characteristic, complemented
4#23782271C211yesmaximal, characteristic, complemented
5#23782270C211yesmaximal, characteristic, complemented
6#23782263C214yesmaximal, complemented
7#23782254C214yesmaximal, complemented
8#23782221C211yesmaximal, characteristic, complemented
9#23782216C211yesmaximal, characteristic, complemented
10#23782200C214yesmaximal, complemented
11#23782198C214yesmaximal, complemented
12#23782178C212yesmaximal, complemented
13C4×#14579700C212yesmaximal, complemented
14#117×D163C211yesmaximal, characteristic, complemented
15#541×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2212yes—
17C22×C4×D163C2211yescharacteristic
18#19104734C2212yescomplemented
19#19104732C2211yescharacteristic, complemented
20#19104730C2214yes—
21#19104730C2214yescomplemented
22#19104729C2214yescomplemented
23#19104727C2211yescharacteristic, complemented
24#19104725C2214yescomplemented
25#19104724C2212yescomplemented
26#19104723C2214yes—
27#19104721C2214yes—
28#19104720C2214yes—
29#19104720C2214yes—
30#19104720C2214yescomplemented
31#19104720C2214yescomplemented
32#19104717C2214yes—
33#19104715C2214yescomplemented
34#19104713C2214yes—
35#19104713C2214yescomplemented
36#19104711C2212yes—
37#19104710C2212yes—
38#19104710C2214yes—
39#19104710C2214yes—
40#19104709C2212yes—
41#19104708C2214yes—
42#19104705C2214yes—
43Q8×Dic163C2212yescomplemented
44#19104700C2214yescomplemented
45C4×D652C2214yes—
46C4×D652C2212yescomplemented
47#19104697C2214yes—
48#19104697C2214yes—
49#19104697C2214yescomplemented
50#19104697C2212yescomplemented
51(C22⋊D163)⋊C4C2214yes—
52#19104694C2211yescharacteristic
53#19104693C2212yes—
54#19104693C2212yes—
55#19104693C2214yescomplemented
56#19104692C2211yescharacteristic
57#19104692C2211yescharacteristic
58#19104691C2212yes—
59C4⋊C4×D163C2212yes—
60C22⋊C4×D163C2212yes—
61C42×D163C2211yescharacteristic
62C42×D163C2211yescharacteristic
63C42×D163C2214yescomplemented
64C4⋊Q8×C163C2211yescharacteristic, complemented
65#126×C163C2214yes—
66#125×C163C2214yescomplemented
67#125×C163C2211yescharacteristic, complemented
68C22⋊Q8×C163C2212yes—
69C4×Q8×C163C2212yescomplemented
70#117×C163C2211yescharacteristic
71C23×D163C2311yescharacteristic
72#14579700—28no—
73C2×Dic326C2314yes—
74C2×D652C2314yes—
75C2×D652C2312yes—
76C2×D652C2314yes—
77C2×D652C2314yes—
78C2×C22⋊D163C2314yes—
79C22×Dic163C2311yescharacteristic
80C2×C4×D163—24no—
81C2×C4×D163C2311yescharacteristic
82C2×C4×D163C2311yescharacteristic
83C2×C4×D163C2311yescharacteristic
84C2×C4×D163C2314yes—
85C2×C4×D163C2312yes—
86C2×C4×D163C2312yes—
87C2×C4×D163C2314yes—
88C2×C4×D163C2314yes—
89C2×C4×D163C2311yescharacteristic
90C4⋊Dic163C2312yes—
91C4⋊Dic163C2312yes—
92C4⋊Dic163C2314yes—
93Dic163⋊C4C2312yes—
94Dic163⋊C4C2312yes—
95Dic163⋊C4C2314yes—
96Dic163⋊C4C2314yes—
97Dic163⋊C4C2314yes—
98C22⋊Dic163C2312yes—
99(C2×D163)⋊C4C2312yes—
100(C2×D163)⋊C4C2312yes—
101(C2×D163)⋊C4C2314yes—
102(C2×D163)⋊C4C2314yes—
103(C2×D163)⋊C4C2314yes—
104(C2×D163)⋊C4C2314yes—
105(C2×D163)⋊C4C2314yes—
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yes—
108(C2×D163)⋊C4C2314yes—
109C4×Dic163C2312yes—
110C4×Dic163C2312yes—
111C4×Dic163C2311yescharacteristic
112C4×Dic163C2311yescharacteristic
113C4×Dic163C2314yes—
114C4×Dic163C2314yes—
115C2×Q8×C163C2312yes—
116C22×C4×C163C2311yescharacteristic
117C4⋊C4×C163C2312yes—
118C4⋊C4×C163C2312yes—
119C4⋊C4×C163C2314yes—
120C4⋊C4×C163C2314yes—
121C4⋊C4×C163C2314yes—
122C4⋊C4×C163C2314yes—
123C22⋊C4×C163C2312yes—
124C22⋊C4×C163C2314yes—
125C42×C163C2311yescharacteristic
126C42×C163C2311yescharacteristic
127C42×C163C2314yes—
128C22×D163—22no—
129C22×D163—42no—
130C22×D163C2411yescharacteristic
131C22×D163C2414yes—
132C22×D163C2411yescharacteristic
133Dic326—44no—
134D652—28no—
135D652—24no—
136D652—44no—
137D652—44no—
138C22⋊D163—44no—
139C2×Dic163—22no—
140C2×Dic163C2412yes—
141C2×Dic163C2412yes—
142C2×Dic163C2414yes—
143C2×Dic163C2411yescharacteristic
144C2×Dic163C2411yescharacteristic
145C4×D163—22no—
146C4×D163—22no—
147C4×D163—22no—
148C4×D163—28no—
149C4×D163—42no—
150C4×D163—44no—
151C4×D163—44no—
152C4×D163—42no—
153C4×D163#4114yes—
154C23×C163C2411yescharacteristic
155Q8×C163—24no—
156C2×C4×C163—22no—
157C2×C4×C163C2412yes—
158C2×C4×C163C2414yes—
159C2×C4×C163C2412yes—
160C2×C4×C163C2414yes—
161C2×C4×C163C2411yescharacteristic
162C2×C4×C163C2411yescharacteristic
163C22×C163—21no—
164C22×C163—22no—
165C22×C163C2511yescharacteristic
166C2×D163—22no—
167C2×D163—21no—
168C2×D163—24no—
169C2×D163—24no—
170C2×D163—24no—
171C2×D163—22no—
172C2×D163—44no—
173C2×D163C2×#4112yes—
174C4×C163—21no—
175C4×C163—24no—
176C4×C163—22no—
177C4×C163—22no—
178C4×C163—24no—
179C4×C163C2×#4112yes—
180Dic163—21no—
181Dic163—22no—
182Dic163—24no—
183Dic163—22no—
184Dic163C2×#4112yes—
185C2×C163—41no—
186C2×C163C22×#4111yescharacteristic
187C2×C163#58512yes—
188D163—22no—
189D163—44no—
190D163—41no—
191C163#345511yescharacteristic, complemented, Sylow
192#3455—1631nomaximal, Sylow
193#542—1632no—
194#542—1632no—
195#541—1631no—
196#541D16311yescharacteristic, complemented
197#540—1631no—
198#540—1631no—
199#539—1634no—
200#538—1634no—
201#533—1631no—
202#532—1631no—
203#529—1634no—
204#529—1634no—
205#525—1632no—
206C4×#41—1632no—
207C2×#117—1631no—
208C2×#41—1632no—
209C23×C4—1631no—
210C4⋊Q8—1632no—
211C4⋊Q8C2×D16311yescharacteristic, complemented
212#127—1631no—
213#126—1634no—
214#126—1634no—
215#126—1634no—
216#126C2×D16314yes—
217#125—1634no—
218#125—1631no—
219#125C2×D16314yescomplemented
220#125C2×D16311yescharacteristic, complemented
221#124—1632no—
222#124—1634no—
223#124—1634no—
224#123—1634no—
225#123—1634no—
226#123—1634no—
227#123—1634no—
228#123—1634no—
229C22⋊Q8—1632no—
230C22⋊Q8—1634no—
231C22⋊Q8—1634no—
232C22⋊Q8—1634no—
233C22⋊Q8C2×D16312yes—
234#121—1632no—
235#121—1632no—
236#121—1634no—
237#121—1634no—
238#121—1634no—
239#120—1634no—
240C4×Q8—1632no—
241C4×Q8—1634no—
242C4×Q8C2×D16312yescomplemented
243C4×D4—1634no—
244C4×D4—1632no—
245C4×D4—1634no—
246C4×D4—1634no—
247C4×D4—1634no—
248C4×D4—1634no—
249C4×D4—1632no—
250#117—1631no—
251#117—1632no—
252#117—1632no—
253#117—1634no—
254#117—1632no—
255#117—1631no—
256#117—1631no—
257#117C2×D16311yescharacteristic
258C2×C4⋊C4—1632no—
259C2×C22⋊C4—1632no—
260C2×C42—1631no—
261C2×C42—1631no—
262C2×C42—1634no—
263C24—1631no—
264#41—3268no—
265C2×Q8—1634no—
266C2×Q8C22×D16312yes—
267C2×D4—1634no—
268C2×D4—1634no—
269C2×D4—1632no—
270C2×D4—1634no—
271C2×D4—1634no—
272C22×C4—3264no—
273C22×C4—1631no—
274C22×C4—1634no—
275C22×C4—1631no—
276C22×C4—1631no—
277C22×C4—1631no—
278C22×C4—1632no—
279C22×C4—1632no—
280C22×C4—1634no—
281C22×C4—1634no—
282C22×C4—1631no—
283C22×C4C22×D16311yescharacteristic
284C4⋊C4—1632no—
285C4⋊C4—1632no—
286C4⋊C4—1632no—
287C4⋊C4—1632no—
288C4⋊C4—1634no—
289C4⋊C4—1634no—
290C4⋊C4—1634no—
291C4⋊C4—1634no—
292C4⋊C4C22×D16312yes—
293C4⋊C4C22×D16312yes—
294C4⋊C4C22×D16314yes—
295C4⋊C4C22×D16314yes—
296C4⋊C4C22×D16314yes—
297C4⋊C4C22×D16314yes—
298C22⋊C4—1632no—
299C22⋊C4—1632no—
300C22⋊C4—1632no—
301C22⋊C4—1634no—
302C22⋊C4—1634no—
303C22⋊C4—1634no—
304C22⋊C4—1634no—
305C22⋊C4—1634no—
306C22⋊C4—1634no—
307C22⋊C4—1634no—
308C22⋊C4—1634no—
309C22⋊C4C22×D16312yes—
310C22⋊C4C22×D16314yes—
311C42—1632no—
312C42—1631no—
313C42—1632no—
314C42—1634no—
315C42—1631no—
316C42—1634no—
317C42C22×D16311yescharacteristic
318C42C22×D16311yescharacteristic
319C42C22×D16314yes—
320C23—3262no—
321C23—6522no—
322C23—1631no—
323C23—1634no—
324C23—1631no—
325C23C23×D16311yescharacteristic
326Q8—24no—
327Q8—6524no—
328D4—3268no—
329D4—3264no—
330D4—6524no—
331D4—6524no—
332D4—6524no—
333C2×C4—22no—
334C2×C4—3262no—
335C2×C4—3262no—
336C2×C4—3268no—
337C2×C4—3262no—
338C2×C4—3262no—
339C2×C4—6522no—
340C2×C4—6524no—
341C2×C4—6522no—
342C2×C4—6524no—
343C2×C4—1632no—
344C2×C4—1632no—
345C2×C4—1634no—
346C2×C4—1631no—
347C2×C4—1631no—
348C2×C4—1634no—
349C2×C4C23×D16312yes—
350C2×C4C23×D16314yes—
351C2×C4C23×D16314yes—
352C2×C4C23×D16312yes—
353C2×C4C23×D16311yescharacteristic
354C2×C4C23×D16311yescharacteristic
355C22—21no—
356C22—22no—
357C22—3262no—
358C22—3261no—
359C22—3264no—
360C22—3264no—
361C22—3264no—
362C22—3262no—
363C22—6524no—
364C22—1632no—
365C22C24×D16311yescharacteristic
366C4—21no—
367C4—24no—
368C4—22no—
369C4—22no—
370C4—24no—
371C4—3261no—
372C4—3262no—
373C4—3264no—
374C4—3262no—
375C4—1632no—
376C4#41×D16312yes—
377C2—41no—
378C2—3262no—
379C2—6521no—
380C2—6524no—
381C2C2×#41×D16311yescharacteristic
382C2#2378250212yes—
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#541×C163C2
hypercenter#541D163
hypocenterC163#3455
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #23782288C2semidirect
2. #23782287C2semidirect
3. #23782276C2semidirect
4. #23782271C2semidirect
5. #23782270C2semidirect
6. #23782263C2semidirect
7. #23782254C2semidirect
8. #23782221C2semidirect
9. #23782216C2semidirect
10. #23782200C2semidirect
11. #23782198C2semidirect
12. #23782178C2semidirect
13. C4×#14579700C2semidirect
14. #117×D163C2semidirect
15. #541×C163C2semidirect
18. #19104734C22semidirect
19. #19104732C22semidirect
21. #19104730C22semidirect
22. #19104729C22semidirect
23. #19104727C22semidirect
24. #19104725C22semidirect
25. #19104724C22semidirect
30. #19104720C22semidirect
31. #19104720C22semidirect
33. #19104715C22semidirect
35. #19104713C22semidirect
43. Q8×Dic163C22semidirect
44. #19104700C22semidirect
46. C4×D652C22semidirect
49. #19104697C22semidirect
50. #19104697C22semidirect
55. #19104693C22semidirect
63. C42×D163C22semidirect
64. C4⋊Q8×C163C22semidirect
66. #125×C163C22semidirect
67. #125×C163C22semidirect
69. C4×Q8×C163C22semidirect
191. C163#3455semidirect
196. #541D163semidirect
211. C4⋊Q8C2×D163semidirect
219. #125C2×D163semidirect
220. #125C2×D163semidirect
242. C4×Q8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822882
#237822872
#237822762
#237822712
#237822702
#237822632
#237822542
#237822212
#237822162
#237822002
#237821982
#237821782
C4×#145797002
#117×D1632
#541×C1632
#3455163