Finite Group Databasebeta

← #35227945

Subgroups — #35227945

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#23782262C212yesmaximal, complemented
3#23782257C214yesmaximal, complemented
4#23782253C212yesmaximal, complemented
5#23782248C214yesmaximal, complemented
6#23782227C211yesmaximal, characteristic, complemented
7#23782224C212yesmaximal, complemented
8#23782199C212yesmaximal, complemented
9#23782195C214yesmaximal, complemented
10#23782176C212yesmaximal, complemented
11#23782174C214yesmaximal, complemented
12C2×C4×C22⋊D163C212yesmaximal, complemented
13C2×C4×D4×C163C211yesmaximal, characteristic, complemented
14C2×#14579699C2212yescomplemented
15C22×C22⋊D163C2212yes—
16C22×C4×D163C2212yescomplemented
17#19104730C2214yescomplemented
18#19104729C2214yescomplemented
19#19104728C2214yescomplemented
20#19104727C2212yescomplemented
21#19104725C2214yescomplemented
22#19104723C2212yescomplemented
23#19104721C2214yescomplemented
24#19104720C2214yescomplemented
25#19104719C2214yescomplemented
26#19104718C2214yescomplemented
27#19104718C2214yescomplemented
28#19104717C2214yescomplemented
29#19104716C2214yescomplemented
30#19104714C2212yescomplemented
31#19104714C2212yescomplemented
32#19104714C2212yescomplemented
33#19104713C2214yescomplemented
34#19104712C2214yescomplemented
35#19104712C2214yescomplemented
36#19104709C2212yes—
37#19104709C2212yes—
38#19104709C2212yescomplemented
39#19104708C2214yescomplemented
40#19104707C2214yes—
41#19104707C2214yes—
42#19104705C2214yescomplemented
43#19104704C2214yescomplemented
44#19104703C2214yescomplemented
45C4×Dic326C2212yescomplemented
46C4×D652C2212yescomplemented
47C4×C22⋊D163C2214yescomplemented
48C4×C22⋊D163C2214yescomplemented
49C4×C22⋊D163C2214yescomplemented
50C4×C22⋊D163C2214yescomplemented
51C4×C22⋊D163C2214yescomplemented
52#19104694C2212yescomplemented
53#19104692C2214yescomplemented
54C2×Dic163⋊C4C2212yescomplemented
55C2×C22⋊Dic163C2212yes—
56C2×(C2×D163)⋊C4C2212yescomplemented
57C2×C4×Dic163C2212yescomplemented
58C22×D4×C163C2211yescharacteristic
59C23×C4×C163C2211yescharacteristic
60C23×C4×C163C2211yescharacteristic, complemented
61C4×D4×C163C2214yescomplemented
62C4×D4×C163C2214yescomplemented
63C2×C4⋊C4×C163C2211yescharacteristic, complemented
64C2×C22⋊C4×C163C2211yescharacteristic
65C2×C22⋊C4×C163C2211yescharacteristic, complemented
66C2×C42×C163C2211yescharacteristic, complemented
67C23×D163C2312yes—
68#14579699—44no—
69C2×Dic326C2312yes—
70C2×Dic326C2312yes—
71C2×D652C2312yes—
72C2×D652C2312yes—
73C2×C22⋊D163—28no—
74C2×C22⋊D163C2314yes—
75C2×C22⋊D163C2314yes—
76C2×C22⋊D163C2314yes—
77C2×C22⋊D163C2314yes—
78C22×Dic163C2312yes—
79C22×Dic163C2312yes—
80C22×Dic163C2312yes—
81C2×C4×D163—44no—
82C2×C4×D163C2314yes—
83C2×C4×D163C2314yes—
84C4⋊Dic163C2312yes—
85C4⋊Dic163C2312yes—
86C4⋊Dic163C2312yes—
87C4⋊Dic163C2312yes—
88Dic163⋊C4C2314yes—
89Dic163⋊C4C2314yes—
90Dic163⋊C4C2314yes—
91Dic163⋊C4C2314yes—
92Dic163⋊C4C2314yes—
93Dic163⋊C4C2314yes—
94C22⋊Dic163C2314yes—
95C22⋊Dic163C2314yes—
96C22⋊Dic163C2314yes—
97C22⋊Dic163C2314yes—
98(C2×D163)⋊C4C2314yes—
99(C2×D163)⋊C4C2314yes—
100(C2×D163)⋊C4C2314yes—
101(C2×D163)⋊C4C2314yes—
102(C2×D163)⋊C4C2314yes—
103(C2×D163)⋊C4C2314yes—
104C4×Dic163C2314yes—
105C4×Dic163C2314yes—
106C24×C163C2311yescharacteristic
107C24×C163C2311yescharacteristic
108C2×D4×C163—24no—
109C2×D4×C163C2314yes—
110C22×C4×C163—42no—
111C22×C4×C163—42no—
112C22×C4×C163C2312yes—
113C22×C4×C163C2312yes—
114C22×C4×C163C2311yescharacteristic
115C22×C4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117C22×C4×C163C2311yescharacteristic
118C22×C4×C163C2311yescharacteristic
119C22×C4×C163C2312yes—
120C22×C4×C163C2312yes—
121C4⋊C4×C163C2314yes—
122C22⋊C4×C163C2314yes—
123C22⋊C4×C163C2314yes—
124C42×C163C2312yes—
125C42×C163C2312yes—
126C22×D163—24no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163C2414yes—
130Dic326—42no—
131Dic326—42no—
132D652—42no—
133D652—42no—
134C22⋊D163—28no—
135C22⋊D163—28no—
136C22⋊D163—44no—
137C22⋊D163—44no—
138C2×Dic163—24no—
139C2×Dic163—42no—
140C2×Dic163—42no—
141C2×Dic163C2414yes—
142C2×Dic163C2414yes—
143C2×Dic163C2414yes—
144C4×D163—28no—
145C4×D163—44no—
146C23×C163—22no—
147C23×C163—41no—
148C23×C163—41no—
149C23×C163—41no—
150C23×C163—41no—
151C23×C163#4114yescomplemented
152C23×C163C2412yes—
153C23×C163C2412yes—
154C23×C163C2411yescharacteristic
155D4×C163—28no—
156C2×C4×C163—22no—
157C2×C4×C163—22no—
158C2×C4×C163—24no—
159C2×C4×C163—24no—
160C2×C4×C163—42no—
161C2×C4×C163—42no—
162C2×C4×C163—41no—
163C2×C4×C163—41no—
164C2×C4×C163—41no—
165C2×C4×C163C2412yes—
166C2×C4×C163C2412yes—
167C2×C4×C163C2411yescharacteristic
168C2×C4×C163C2412yes—
169C2×C4×C163C2412yes—
170C2×C4×C163C2411yescharacteristic
171C22×C163—21no—
172C22×C163—22no—
173C22×C163—21no—
174C22×C163—22no—
175C22×C163—22no—
176C22×C163—22no—
177C22×C163—22no—
178C22×C163—44no—
179C22×C163—44no—
180C22×C163C2×#4112yes—
181C22×C163C2×#4114yes—
182C22×C163C2511yescharacteristic
183C2×D163—24no—
184C2×D163—24no—
185C2×D163—24no—
186C2×D163—48no—
187C4×C163—22no—
188C4×C163—21no—
189C4×C163—22no—
190C4×C163—22no—
191C4×C163—22no—
192C4×C163H(5,𝔽2)12yes—
193Dic163—24no—
194Dic163—24no—
195Dic163—24no—
196C2×C163—24no—
197C2×C163—22no—
198C2×C163—42no—
199C2×C163C22×#4111yescharacteristic
200C2×C163C2×H(5,𝔽2)11yescharacteristic
201C2×C163#58511yescharacteristic
202D163—44no—
203C163#345911yescharacteristic, complemented, Sylow
204#3459—1631nomaximal, Sylow
205#543—1631no—
206#539—1634no—
207#539—1632no—
208#538—1634no—
209#538—1632no—
210#534—1632no—
211#534—1631no—
212#529—1634no—
213#529—1632no—
214#525—1632no—
215#525—1634no—
216C2×C4×D4—1632no—
217C2×C4×D4D16311yescharacteristic, complemented
218C2×#41—1632no—
219C22×D4—1632no—
220C22×D4C2×D16311yescharacteristic
221C23×C4—1632no—
222C23×C4C2×D16311yescharacteristic
223C23×C4C2×D16311yescharacteristic, complemented
224#126—1634no—
225#126—1634no—
226#126—1634no—
227#125—1634no—
228#125—1632no—
229#124—1634no—
230#124—1632no—
231#123—1634no—
232#123—1634no—
233#123—1634no—
234#123—1634no—
235#123—1634no—
236#123—1634no—
237C22⋊Q8—1632no—
238C22⋊Q8—1632no—
239C22⋊Q8—1632no—
240C22⋊Q8—1634no—
241C22⋊Q8—1634no—
242C22⋊Q8—1634no—
243#121—1634no—
244#121—1634no—
245#121—1632no—
246#121—1632no—
247#121—1632no—
248#121—1634no—
249#120—1634no—
250#120—1634no—
251#120—1634no—
252C4×Q8—1632no—
253C4×D4—1634no—
254C4×D4—1634no—
255C4×D4—1634no—
256C4×D4—1634no—
257C4×D4—1634no—
258C4×D4—1632no—
259C4×D4C2×D16314yescomplemented
260C4×D4C2×D16314yescomplemented
261#117—1632no—
262#117—1634no—
263C2×C4⋊C4—1632no—
264C2×C4⋊C4C2×D16311yescharacteristic, complemented
265C2×C22⋊C4—1632no—
266C2×C22⋊C4—1632no—
267C2×C22⋊C4C2×D16311yescharacteristic
268C2×C22⋊C4C2×D16311yescharacteristic, complemented
269C2×C42—1632no—
270C2×C42C2×D16311yescharacteristic, complemented
271C24—1632no—
272C24C22×D16311yescharacteristic
273C24C22×D16311yescharacteristic
274#41—6524no—
275C2×Q8—1632no—
276C2×Q8—1632no—
277C2×D4—24no—
278C2×D4—3268no—
279C2×D4—1634no—
280C2×D4—1634no—
281C2×D4—1634no—
282C2×D4—1634no—
283C2×D4—1632no—
284C2×D4—1632no—
285C2×D4C22×D16314yes—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—6524no—
289C22×C4—1632no—
290C22×C4—1634no—
291C22×C4—1632no—
292C22×C4—1632no—
293C22×C4—1634no—
294C22×C4C22×D16311yescharacteristic
295C22×C4C22×D16312yes—
296C22×C4C22×D16311yescharacteristic
297C22×C4C22×D16312yes—
298C22×C4C22×D16311yescharacteristic
299C22×C4C22×D16312yes—
300C22×C4C22×D16311yescharacteristic
301C22×C4C22×D16311yescharacteristic
302C22×C4C22×D16312yes—
303C4⋊C4—1634no—
304C4⋊C4—1634no—
305C4⋊C4—1634no—
306C4⋊C4—1632no—
307C4⋊C4—1634no—
308C4⋊C4—1632no—
309C4⋊C4—1634no—
310C4⋊C4—1632no—
311C4⋊C4—1634no—
312C4⋊C4—1632no—
313C4⋊C4C22×D16314yes—
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⋊C4—1634no—
324C22⋊C4C22×D16314yes—
325C22⋊C4C22×D16314yes—
326C42—1634no—
327C42—1634no—
328C42C22×D16312yes—
329C42C22×D16312yes—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—3264no—
336C23—6522no—
337C23—6522no—
338C23—1634no—
339C23#1457969914yescomplemented
340C23C23×D16312yes—
341C23C23×D16311yescharacteristic
342C23C23×D16312yes—
343Q8—6522no—
344Q8—6522no—
345D4—28no—
346D4—3268no—
347D4—3268no—
348D4—6522no—
349D4—6522no—
350D4—6524no—
351D4—6524no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—3264no—
362C2×C4—3268no—
363C2×C4—6524no—
364C2×C4—6522no—
365C2×C4—6522no—
366C2×C4—1634no—
367C2×C4—1634no—
368C2×C4—1634no—
369C2×C4C23×D16312yes—
370C2×C4C23×D16312yes—
371C2×C4C23×D16311yescharacteristic
372C2×C4C23×D16312yes—
373C2×C4C23×D16312yes—
374C2×C4C23×D16311yescharacteristic
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—44no—
383C22—44no—
384C22—3264no—
385C22—3264no—
386C22—3264no—
387C22—6528no—
388C22C2×#1457969912yes—
389C22C2×#1457969914yes—
390C22C24×D16311yescharacteristic
391C4—22no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—3264no—
397C4—3264no—
398C4—3264no—
399C4#1910478012yes—
400C2—22no—
401C2—24no—
402C2—42no—
403C2—6524no—
404C2C22×#1457969911yescharacteristic
405C2C2×#1910478011yescharacteristic
406C2#2378250111yescharacteristic
subgroupG/H
centerC2×C4C23×D163
commutator, socle, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×C4×D4×C163C2
hypercenterC2×C4×D4D163
hypocenterC163#3459
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #23782291C2semidirect
2. #23782262C2semidirect
3. #23782257C2semidirect
4. #23782253C2semidirect
5. #23782248C2semidirect
6. #23782227C2semidirect
7. #23782224C2semidirect
8. #23782199C2semidirect
9. #23782195C2semidirect
10. #23782176C2semidirect
11. #23782174C2semidirect
12. C2×C4×C22⋊D163C2semidirect
13. C2×C4×D4×C163C2semidirect
14. C2×#14579699C22semidirect
16. C22×C4×D163C22semidirect
17. #19104730C22semidirect
18. #19104729C22semidirect
19. #19104728C22semidirect
20. #19104727C22semidirect
21. #19104725C22semidirect
22. #19104723C22semidirect
23. #19104721C22semidirect
24. #19104720C22semidirect
25. #19104719C22semidirect
26. #19104718C22semidirect
27. #19104718C22semidirect
28. #19104717C22semidirect
29. #19104716C22semidirect
30. #19104714C22semidirect
31. #19104714C22semidirect
32. #19104714C22semidirect
33. #19104713C22semidirect
34. #19104712C22semidirect
35. #19104712C22semidirect
38. #19104709C22semidirect
39. #19104708C22semidirect
42. #19104705C22semidirect
43. #19104704C22semidirect
44. #19104703C22semidirect
45. C4×Dic326C22semidirect
46. C4×D652C22semidirect
47. C4×C22⋊D163C22semidirect
48. C4×C22⋊D163C22semidirect
49. C4×C22⋊D163C22semidirect
50. C4×C22⋊D163C22semidirect
51. C4×C22⋊D163C22semidirect
52. #19104694C22semidirect
53. #19104692C22semidirect
54. C2×Dic163⋊C4C22semidirect
56. C2×(C2×D163)⋊C4C22semidirect
57. C2×C4×Dic163C22semidirect
60. C23×C4×C163C22semidirect
61. C4×D4×C163C22semidirect
62. C4×D4×C163C22semidirect
63. C2×C4⋊C4×C163C22semidirect
65. C2×C22⋊C4×C163C22semidirect
66. C2×C42×C163C22semidirect
151. C23×C163#41semidirect
203. C163#3459semidirect
217. C2×C4×D4D163semidirect
223. C23×C4C2×D163semidirect
259. C4×D4C2×D163semidirect
260. C4×D4C2×D163semidirect
264. C2×C4⋊C4C2×D163semidirect
268. C2×C22⋊C4C2×D163semidirect
270. C2×C42C2×D163semidirect
339. C23#14579699semidirect

Maximal subgroups

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

subgroupindex
#237822912
#237822622
#237822572
#237822532
#237822482
#237822272
#237822242
#237821992
#237821952
#237821762
#237821742
C2×C4×C22⋊D1632
C2×C4×D4×C1632
#3459163