Finite Group Databasebeta

← #35227942

Subgroups — #35227942

Order 20864.

1160 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782291C211yesmaximal, characteristic, complemented
2#23782260C212yesmaximal, complemented
3#23782259C214yesmaximal, complemented
4#23782252C212yesmaximal, complemented
5#23782250C214yesmaximal, complemented
6#23782224C212yesmaximal, complemented
7#23782196C212yesmaximal, complemented
8#23782192C214yesmaximal, complemented
9#23782175C212yesmaximal, complemented
10#23782171C214yesmaximal, complemented
11C2×D4×Dic163C212yesmaximal, complemented
12C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
13#534×C163C211yesmaximal, characteristic, complemented
14C2×#14579698C2212yescomplemented
15C22×C22⋊D163C2211yescharacteristic
16C23×Dic163C2211yescharacteristic
17C23×Dic163C2212yescomplemented
18C22×C4×D163C2211yescharacteristic, complemented
19#19104730C2214yescomplemented
20#19104728C2214yescomplemented
21#19104728C2214yescomplemented
22#19104726C2212yescomplemented
23#19104725C2214yescomplemented
24#19104722C2212yescomplemented
25#19104721C2214yescomplemented
26#19104719C2214yescomplemented
27#19104718C2214yescomplemented
28#19104717C2214yescomplemented
29#19104716C2214yescomplemented
30#19104716C2214yescomplemented
31#19104715C2212yescomplemented
32#19104714C2212yescomplemented
33#19104714C2212yescomplemented
34#19104712C2214yescomplemented
35#19104712C2214yescomplemented
36#19104712C2214yescomplemented
37#19104707C2214yes—
38#19104707C2214yes—
39#19104707C2214yescomplemented
40#19104706C2212yescomplemented
41#19104704C2214yescomplemented
42#19104703C2214yescomplemented
43#19104700C2212yescomplemented
44D4×Dic163C2212yescomplemented
45D4×Dic163C2214yescomplemented
46D4×Dic163C2214yescomplemented
47D4×Dic163C2214yescomplemented
48D4×Dic163C2214yescomplemented
49(C22⋊D163)⋊C4C2214yescomplemented
50(C22⋊D163)⋊C4C2214yescomplemented
51(C22⋊D163)⋊C4C2214yescomplemented
52#19104693C2212yescomplemented
53#19104691C2214yescomplemented
54C2×C4⋊Dic163C2212yescomplemented
55C2×Dic163⋊C4C2211yescharacteristic, complemented
56C2×C22⋊Dic163C2212yes—
57C2×C22⋊Dic163C2212yescomplemented
58C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
59C2×C4×Dic163C2212yescomplemented
60C2×C4×Dic163C2211yescharacteristic, complemented
61C22×D4×C163C2212yes—
62#123×C163C2214yescomplemented
63#121×C163C2212yes—
64#121×C163C2212yes—
65#120×C163C2214yescomplemented
66C2×C22⋊C4×C163C2211yescharacteristic
67C23×D163C2311yescharacteristic
68#14579698—44no—
69C2×Dic326C2312yes—
70C2×Dic326C2312yes—
71C2×C22⋊D163—24no—
72C2×C22⋊D163C2314yes—
73C2×C22⋊D163C2314yes—
74C2×C22⋊D163C2314yes—
75C22×Dic163—42no—
76C22×Dic163—44no—
77C22×Dic163C2311yescharacteristic
78C22×Dic163C2312yes—
79C22×Dic163C2314yes—
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2312yes—
83C22×Dic163C2312yes—
84C22×Dic163C2314yes—
85C22×Dic163C2312yes—
86C2×C4×D163—42no—
87C2×C4×D163C2312yes—
88C2×C4×D163C2312yes—
89C4⋊Dic163C2314yes—
90C4⋊Dic163C2314yes—
91C4⋊Dic163C2312yes—
92C4⋊Dic163C2312yes—
93Dic163⋊C4C2314yes—
94Dic163⋊C4C2314yes—
95Dic163⋊C4C2314yes—
96Dic163⋊C4C2314yes—
97Dic163⋊C4C2314yes—
98C22⋊Dic163C2314yes—
99C22⋊Dic163C2314yes—
100C22⋊Dic163C2314yes—
101C22⋊Dic163C2314yes—
102C22⋊Dic163C2314yes—
103C22⋊Dic163C2314yes—
104(C2×D163)⋊C4C2314yes—
105(C2×D163)⋊C4C2314yes—
106(C2×D163)⋊C4C2314yes—
107C4×Dic163C2312yes—
108C4×Dic163C2314yes—
109C4×Dic163C2312yes—
110C4×Dic163C2314yes—
111C24×C163C2312yes—
112C24×C163C2311yescharacteristic
113C2×D4×C163—28no—
114C2×D4×C163C2314yes—
115C2×D4×C163C2314yes—
116C2×D4×C163C2312yes—
117C2×D4×C163C2312yes—
118C22×C4×C163C2312yes—
119C22×C4×C163C2311yescharacteristic
120C22×C4×C163C2311yescharacteristic
121C4⋊C4×C163C2312yes—
122C4⋊C4×C163C2312yes—
123C22⋊C4×C163C2314yes—
124C22⋊C4×C163C2314yes—
125C22⋊C4×C163C2314yes—
126C22×D163—22no—
127C22×D163—41no—
128C22×D163—41no—
129C22×D163C2412yes—
130Dic326—42no—
131Dic326—42no—
132C22⋊D163—28no—
133C22⋊D163—44no—
134C22⋊D163—44no—
135C2×Dic163—22no—
136C2×Dic163—22no—
137C2×Dic163—24no—
138C2×Dic163—28no—
139C2×Dic163—42no—
140C2×Dic163—41no—
141C2×Dic163—44no—
142C2×Dic163—42no—
143C2×Dic163—42no—
144C2×Dic163C2412yes—
145C2×Dic163C2414yes—
146C2×Dic163C2412yes—
147C2×Dic163C2411yescharacteristic
148C2×Dic163C2414yes—
149C2×Dic163C2411yescharacteristic
150C4×D163—24no—
151C4×D163—42no—
152C23×C163—24no—
153C23×C163—41no—
154C23×C163—42no—
155C23×C163—41no—
156C23×C163—42no—
157C23×C163#4114yescomplemented
158C23×C163C2414yes—
159C23×C163C2412yes—
160C23×C163C2411yescharacteristic
161D4×C163—28no—
162D4×C163—28no—
163D4×C163—42no—
164D4×C163—42no—
165C2×C4×C163—24no—
166C2×C4×C163—41no—
167C2×C4×C163—41no—
168C2×C4×C163C2412yes—
169C2×C4×C163C2412yes—
170C2×C4×C163C2414yes—
171C22×C163—21no—
172C22×C163—22no—
173C22×C163—21no—
174C22×C163—22no—
175C22×C163—24no—
176C22×C163—24no—
177C22×C163—24no—
178C22×C163—44no—
179C22×C163—48no—
180C22×C163C2×#4114yes—
181C22×C163C2×#4112yes—
182C22×C163C2511yescharacteristic
183C2×D163—22no—
184C2×D163—22no—
185C2×D163—22no—
186C2×D163—44no—
187C4×C163—24no—
188C4×C163—22no—
189C4×C163—22no—
190Dic163—22no—
191Dic163—21no—
192Dic163—22no—
193Dic163—24no—
194Dic163—24no—
195Dic163H(5,𝔽2)12yes—
196C2×C163—22no—
197C2×C163—24no—
198C2×C163—44no—
199C2×C163C22×#4111yescharacteristic
200C2×C163C2×H(5,𝔽2)11yescharacteristic
201C2×C163#58511yescharacteristic
202D163—42no—
203C163#345911yescharacteristic, complemented, Sylow
204#3459—1631nomaximal, Sylow
205#543—1631no—
206#539—1634no—
207#539—1632no—
208#538—1632no—
209#538—1634no—
210#534—1632no—
211#534D16311yescharacteristic, complemented
212#529—1634no—
213#529—1632no—
214#525—1632no—
215#525—1634no—
216C2×C4×D4—1632no—
217C2×C4×D4—1631no—
218C2×#41—1632no—
219C22×D4—1631no—
220C22×D4C2×D16312yes—
221C23×C4—1631no—
222C23×C4—1632no—
223C23×C4—1631no—
224#126—1634no—
225#126—1634no—
226#126—1634no—
227#125—1634no—
228#125—1632no—
229#124—1632no—
230#124—1634no—
231#123—1634no—
232#123—1634no—
233#123—1634no—
234#123—1634no—
235#123—1634no—
236#123C2×D16314yescomplemented
237C22⋊Q8—1632no—
238C22⋊Q8—1632no—
239C22⋊Q8—1632no—
240C22⋊Q8—1634no—
241C22⋊Q8—1634no—
242C22⋊Q8—1634no—
243#121—1634no—
244#121—1632no—
245#121—1634no—
246#121—1634no—
247#121C2×D16312yes—
248#121C2×D16312yes—
249#120—1634no—
250#120—1634no—
251#120C2×D16314yescomplemented
252C4×Q8—1632no—
253C4×D4—1634no—
254C4×D4—1634no—
255C4×D4—1634no—
256C4×D4—1634no—
257C4×D4—1632no—
258C4×D4—1634no—
259C4×D4—1634no—
260C4×D4—1634no—
261#117—1632no—
262#117—1634no—
263C2×C4⋊C4—1631no—
264C2×C4⋊C4—1632no—
265C2×C22⋊C4—1632no—
266C2×C22⋊C4—1632no—
267C2×C22⋊C4—1631no—
268C2×C22⋊C4C2×D16311yescharacteristic
269C2×C42—1631no—
270C2×C42—1632no—
271C24—1631no—
272C24C22×D16312yes—
273C24C22×D16311yescharacteristic
274#41—6524no—
275C2×Q8—1632no—
276C2×Q8—1632no—
277C2×D4—28no—
278C2×D4—3264no—
279C2×D4—1634no—
280C2×D4—1634no—
281C2×D4—1634no—
282C2×D4C22×D16314yes—
283C2×D4C22×D16314yes—
284C2×D4C22×D16312yes—
285C2×D4C22×D16312yes—
286C22×C4—6522no—
287C22×C4—6522no—
288C22×C4—6524no—
289C22×C4—1631no—
290C22×C4—1631no—
291C22×C4—1632no—
292C22×C4—1634no—
293C22×C4—1632no—
294C22×C4—1632no—
295C22×C4—1631no—
296C22×C4—1632no—
297C22×C4—1632no—
298C22×C4—1634no—
299C22×C4—1632no—
300C22×C4C22×D16312yes—
301C22×C4C22×D16311yescharacteristic
302C22×C4C22×D16311yescharacteristic
303C4⋊C4—1634no—
304C4⋊C4—1634no—
305C4⋊C4—1634no—
306C4⋊C4—1634no—
307C4⋊C4—1632no—
308C4⋊C4—1634no—
309C4⋊C4—1634no—
310C4⋊C4—1632no—
311C4⋊C4—1634no—
312C4⋊C4C22×D16312yes—
313C4⋊C4C22×D16312yes—
314C22⋊C4—1634no—
315C22⋊C4—1634no—
316C22⋊C4—1634no—
317C22⋊C4—1634no—
318C22⋊C4—1634no—
319C22⋊C4—1634no—
320C22⋊C4—1634no—
321C22⋊C4—1634no—
322C22⋊C4—1634no—
323C22⋊C4C22×D16314yes—
324C22⋊C4C22×D16314yes—
325C22⋊C4C22×D16314yes—
326C42—1634no—
327C42—1632no—
328C42—1634no—
329C42—1632no—
330C23—24no—
331C23—41no—
332C23—42no—
333C23—41no—
334C23—42no—
335C23—3262no—
336C23—6521no—
337C23—6521no—
338C23—1632no—
339C23#1457969814yescomplemented
340C23C23×D16314yes—
341C23C23×D16311yescharacteristic
342C23C23×D16312yes—
343Q8—6522no—
344Q8—6522no—
345D4—28no—
346D4—28no—
347D4—42no—
348D4—42no—
349D4—3268no—
350D4—6524no—
351D4—6524no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—3264no—
356C2×C4—3262no—
357C2×C4—3262no—
358C2×C4—3264no—
359C2×C4—3268no—
360C2×C4—6522no—
361C2×C4—6521no—
362C2×C4—6522no—
363C2×C4—6524no—
364C2×C4—6522no—
365C2×C4—6522no—
366C2×C4—1634no—
367C2×C4—1632no—
368C2×C4—1631no—
369C2×C4—1632no—
370C2×C4—1634no—
371C2×C4—1631no—
372C2×C4C23×D16312yes—
373C2×C4C23×D16314yes—
374C2×C4C23×D16312yes—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—44no—
383C22—48no—
384C22—3262no—
385C22—3262no—
386C22—3262no—
387C22—6524no—
388C22C2×#1457969814yes—
389C22C2×#1457969812yes—
390C22C24×D16311yescharacteristic
391C4—24no—
392C4—22no—
393C4—22no—
394C4—3262no—
395C4—3261no—
396C4—3262no—
397C4—3264no—
398C4—3264no—
399C4—1632no—
400C2—24no—
401C2—22no—
402C2—44no—
403C2—6522no—
404C2C22×#1457969811yescharacteristic
405C2H(5,𝔽2)×D16311yescharacteristic
406C2#2378250011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#534×C163C2
hypercenter#534D163
hypocenterC163#3459
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #23782291C2semidirect
2. #23782260C2semidirect
3. #23782259C2semidirect
4. #23782252C2semidirect
5. #23782250C2semidirect
6. #23782224C2semidirect
7. #23782196C2semidirect
8. #23782192C2semidirect
9. #23782175C2semidirect
10. #23782171C2semidirect
11. C2×D4×Dic163C2semidirect
12. C2×(C22⋊D163)⋊C4C2semidirect
13. #534×C163C2semidirect
14. C2×#14579698C22semidirect
17. C23×Dic163C22semidirect
18. C22×C4×D163C22semidirect
19. #19104730C22semidirect
20. #19104728C22semidirect
21. #19104728C22semidirect
22. #19104726C22semidirect
23. #19104725C22semidirect
24. #19104722C22semidirect
25. #19104721C22semidirect
26. #19104719C22semidirect
27. #19104718C22semidirect
28. #19104717C22semidirect
29. #19104716C22semidirect
30. #19104716C22semidirect
31. #19104715C22semidirect
32. #19104714C22semidirect
33. #19104714C22semidirect
34. #19104712C22semidirect
35. #19104712C22semidirect
36. #19104712C22semidirect
39. #19104707C22semidirect
40. #19104706C22semidirect
41. #19104704C22semidirect
42. #19104703C22semidirect
43. #19104700C22semidirect
44. D4×Dic163C22semidirect
45. D4×Dic163C22semidirect
46. D4×Dic163C22semidirect
47. D4×Dic163C22semidirect
48. D4×Dic163C22semidirect
49. (C22⋊D163)⋊C4C22semidirect
50. (C22⋊D163)⋊C4C22semidirect
51. (C22⋊D163)⋊C4C22semidirect
52. #19104693C22semidirect
53. #19104691C22semidirect
54. C2×C4⋊Dic163C22semidirect
55. C2×Dic163⋊C4C22semidirect
57. C2×C22⋊Dic163C22semidirect
58. C2×(C2×D163)⋊C4C22semidirect
59. C2×C4×Dic163C22semidirect
60. C2×C4×Dic163C22semidirect
62. #123×C163C22semidirect
65. #120×C163C22semidirect
157. C23×C163#41semidirect
203. C163#3459semidirect
211. #534D163semidirect
236. #123C2×D163semidirect
251. #120C2×D163semidirect
339. C23#14579698semidirect

Maximal subgroups

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

subgroupindex
#237822912
#237822602
#237822592
#237822522
#237822502
#237822242
#237821962
#237821922
#237821752
#237821712
C2×D4×Dic1632
C2×(C22⋊D163)⋊C42
#534×C1632
#3459163