Finite Group Databasebeta

← #41×C22⋊D163 #35228021

Subgroups — #41×C22⋊D163 #35228021

Order 20864.

1648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D163C211yesmaximal, characteristic, complemented
2#23782334C211yesmaximal, characteristic, complemented
3Q8×C22⋊D163C211yesmaximal, characteristic, complemented
4#23782326C211yesmaximal, characteristic, complemented
5#23782322C211yesmaximal, characteristic, complemented
6#23782314C213yesmaximal, complemented
7#23782309C213yesmaximal, complemented
8#23782303C213yesmaximal, complemented
9D4×C22⋊D163C213yesmaximal, complemented
10#23782211C213yesmaximal, complemented
11#23782176C213yesmaximal, complemented
12#23782172C213yesmaximal, complemented
13#41×Dic163C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊D163C213yesmaximal, complemented
15C22×#41×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic, complemented
17C2×#14579699C2213yescomplemented
18C2×#14579698C2213yescomplemented
19#41×D163—28no—
20C2×Q8×D163C2211yescharacteristic, complemented
21C22×C22⋊D163C2213yes—
22C2×D4×D163C2213yescomplemented
23C22×C4×D163C2213yescomplemented
24#19104733C2213yescomplemented
25#19104731C2213yescomplemented
26#19104724C2213yescomplemented
27#19104722C2213yescomplemented
28#19104718C2216yescomplemented
29#19104716C2216yescomplemented
30#19104714C2213yescomplemented
31#19104714C2213yescomplemented
32#19104711C2213yescomplemented
33#19104711C2213yescomplemented
34#19104709C2213yescomplemented
35#19104709C2213yescomplemented
36#19104707C2216yescomplemented
37#19104707C2216yescomplemented
38#19104706C2213yescomplemented
39#19104706C2213yescomplemented
40#19104704C2216yescomplemented
41#19104703C2216yescomplemented
42Q8×Dic163C2211yescharacteristic, complemented
43Q8×Dic163C2211yescharacteristic, complemented
44D4×Dic163C2213yescomplemented
45D4×Dic163C2213yescomplemented
46C4×C22⋊D163C2213yes—
47C4×C22⋊D163C2211yescharacteristic
48C4×C22⋊D163C2213yescomplemented
49C4×C22⋊D163C2216yescomplemented
50C4×C22⋊D163C2213yescomplemented
51C4×C22⋊D163C2213yescomplemented
52C4×C22⋊D163C2213yescomplemented
53C4×C22⋊D163C2213yescomplemented
54C4×C22⋊D163C2213yescomplemented
55#19104694C2213yescomplemented
56C2×Dic163⋊C4C2213yescomplemented
57C2×C22⋊Dic163C2213yescomplemented
58C2×(C2×D163)⋊C4C2213yescomplemented
59C2×C4×Dic163C2213yescomplemented
60C2×#41×C163—28no—
61C2×#41×C163C2211yescharacteristic
62C2×#41×C163C2213yescomplemented
63C2×#41×C163C2213yescomplemented
64C2×#41×C163C2211yescharacteristic, complemented
65C22×Q8×C163C2211yescharacteristic, complemented
66C22×D4×C163C2213yescomplemented
67C23×C4×C163C2213yescomplemented
68C23×D163C2313yes—
69#14579700—24no—
70#14579699—212no—
71#14579698—212no—
72C2×Dic326C2313yes—
73Q8×D163—24no—
74C2×D652C2313yes—
75C2×C22⋊D163—212no—
76C2×C22⋊D163C2316yes—
77C2×C22⋊D163C2313yes—
78C2×C22⋊D163C2313yes—
79C2×C22⋊D163C2313yes—
80C2×C22⋊D163C2316yes—
81C2×C22⋊D163C2311yescharacteristic
82D4×D163—212no—
83C22×Dic163C2313yes—
84C22×Dic163C2313yes—
85C22×Dic163C2313yes—
86C2×C4×D163—212no—
87C2×C4×D163C2311yescharacteristic
88C2×C4×D163C2313yes—
89C2×C4×D163C2313yes—
90C2×C4×D163C2313yes—
91C4⋊Dic163C2313yes—
92C4⋊Dic163C2313yes—
93Dic163⋊C4C2311yescharacteristic
94Dic163⋊C4C2313yes—
95Dic163⋊C4C2313yes—
96Dic163⋊C4C2313yes—
97Dic163⋊C4C2316yes—
98C22⋊Dic163C2311yescharacteristic
99C22⋊Dic163C2313yes—
100C22⋊Dic163C2313yes—
101C22⋊Dic163C2313yes—
102C22⋊Dic163C2316yes—
103C22⋊Dic163C2316yes—
104(C2×D163)⋊C4C2311yescharacteristic
105(C2×D163)⋊C4C2313yes—
106(C2×D163)⋊C4C2313yes—
107(C2×D163)⋊C4C2313yes—
108(C2×D163)⋊C4C2316yes—
109C4×Dic163C2311yescharacteristic
110C4×Dic163C2313yes—
111C4×Dic163C2313yes—
112C4×Dic163C2313yes—
113C24×C163C2313yes—
114#41×C163—24no—
115#41×C163—212no—
116#41×C163—212no—
117#41×C163D418yescomplemented
118C2×Q8×C163—24no—
119C2×Q8×C163C2313yes—
120C2×Q8×C163C2311yescharacteristic
121C2×D4×C163—212no—
122C2×D4×C163C2316yes—
123C2×D4×C163C2313yes—
124C2×D4×C163C2313yes—
125C22×C4×C163—212no—
126C22×C4×C163C2311yescharacteristic
127C22×C4×C163C2313yes—
128C22×C4×C163C2313yes—
129C22×C4×C163C2313yes—
130C22×C4×C163C2313yes—
131C22×C4×C163C2313yes—
132C22×D163—26no—
133C22×D163—26no—
134C22×D163—43no—
135C22×D163C2413yes—
136C22×D163C2411yescharacteristic
137Dic326—26no—
138D652—26no—
139C22⋊D163—212no—
140C22⋊D163—26no—
141C22⋊D163—26no—
142C22⋊D163—26no—
143C22⋊D163—46no—
144C22⋊D163#4114yesdirect product factor
145C2×Dic163—26no—
146C2×Dic163—26no—
147C2×Dic163—43no—
148C2×Dic163C2411yescharacteristic
149C2×Dic163C2413yes—
150C2×Dic163C2413yes—
151C2×Dic163C2411yescharacteristic
152C2×Dic163C2413yes—
153C2×Dic163C2411yescharacteristic
154C4×D163—22no—
155C4×D163—26no—
156C4×D163—26no—
157C4×D163—26no—
158C23×C163—26no—
159C23×C163—26no—
160C23×C163—43no—
161C23×C163C2413yes—
162C23×C163C2413yes—
163C23×C163C2411yescharacteristic
164Q8×C163—26no—
165Q8×C163C2×D414yescomplemented
166D4×C163—212no—
167D4×C163—26no—
168D4×C163C2×D4112yescomplemented
169C2×C4×C163—26no—
170C2×C4×C163—22no—
171C2×C4×C163—26no—
172C2×C4×C163—26no—
173C2×C4×C163—26no—
174C2×C4×C163C2×D4112yescomplemented
175C2×C4×C163C2411yescharacteristic
176C2×C4×C163C2413yes—
177C2×C4×C163C2413yes—
178C2×C4×C163C2411yescharacteristic
179C22×C163—21no—
180C22×C163—23no—
181C22×C163—21no—
182C22×C163—23no—
183C22×C163—23no—
184C22×C163—23no—
185C22×C163—23no—
186C22×C163—412no—
187C22×C163C22×D416yes—
188C22×C163C2×#4112yescomplemented
189C22×C163C2511yescharacteristic
190C2×D163—21no—
191C2×D163—21no—
192C2×D163—23no—
193C2×D163—23no—
194C2×D163—23no—
195C2×D163—412no—
196C2×D163C2×#4112yescomplemented
197C4×C163—21no—
198C4×C163—23no—
199C4×C163C22×D412yes—
200C4×C163C22×D416yes—
201Dic163—21no—
202Dic163—23no—
203Dic163—23no—
204Dic163—21no—
205Dic163—23no—
206Dic163C2×#4112yescomplemented
207C2×C163—26no—
208C2×C163—22no—
209C2×C163—43no—
210C2×C163C23×D411yescharacteristic
211C2×C163C22×#4111yescharacteristic
212C2×C163#58511yescharacteristic
213D163—22no—
214D163—43no—
215C163D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—1631nomaximal, Sylow
217C22×#41—1631no—
218C22×#41D16311yescharacteristic, complemented
219#550—1631no—
220D4×Q8—1631no—
221#548—1631no—
222#548—1631no—
223#547—1633no—
224#546—1633no—
225#546—1633no—
226D42—1633no—
227#532—1633no—
228#525—1633no—
229#525—1633no—
230C4×#41—1631no—
231C2×C4×D4—1633no—
232C2×#41—28no—
233C2×#41—3268no—
234C2×#41—1631no—
235C2×#41—1633no—
236C2×#41—1633no—
237C2×#41C2×D16311yescharacteristic
238C2×#41C2×D16313yescomplemented
239C2×#41C2×D16313yescomplemented
240C2×#41C2×D16311yescharacteristic, complemented
241C22×Q8—1631no—
242C22×Q8C2×D16311yescharacteristic, complemented
243C22×D4—1633no—
244C22×D4—1633no—
245C22×D4C2×D16313yescomplemented
246C23×C4—1633no—
247C23×C4C2×D16313yescomplemented
248C4⋊Q8—1633no—
249#127—1633no—
250#124—1633no—
251#124—1633no—
252#123—1636no—
253#123—1636no—
254C22⋊Q8—1633no—
255C22⋊Q8—1633no—
256C22⋊Q8—1633no—
257C22⋊Q8—1633no—
258#121—1633no—
259#121—1633no—
260#121—1636no—
261#121—1636no—
262#121—1633no—
263#121—1633no—
264#120—1636no—
265#120—1636no—
266C4×Q8—1631no—
267C4×Q8—1631no—
268C4×D4—1633no—
269C4×D4—1633no—
270C4×D4—1631no—
271C4×D4—1633no—
272C4×D4—1633no—
273C4×D4—1633no—
274C4×D4—1633no—
275C4×D4—1633no—
276C4×D4—1633no—
277C4×D4—1633no—
278C4×D4—1636no—
279#117—1633no—
280C2×C4⋊C4—1633no—
281C2×C22⋊C4—1633no—
282C2×C22⋊C4—1633no—
283C2×C42—1633no—
284C24—1633no—
285C24C22×D16313yes—
286#41—24no—
287#41—212no—
288#41—212no—
289#41—3264no—
290#41—32612no—
291#41—32612no—
292#41C22⋊D16318yesdirect product factor
293C2×Q8—24no—
294C2×Q8—3264no—
295C2×Q8—1633no—
296C2×Q8C22×D16313yes—
297C2×Q8C22×D16311yescharacteristic
298C2×D4—212no—
299C2×D4—32612no—
300C2×D4—32612no—
301C2×D4—1633no—
302C2×D4—1633no—
303C2×D4—1633no—
304C2×D4—1636no—
305C2×D4—1636no—
306C2×D4—1631no—
307C2×D4—1633no—
308C2×D4C22×D16316yes—
309C2×D4C22×D16313yes—
310C2×D4C22×D16313yes—
311C22×C4—212no—
312C22×C4—32612no—
313C22×C4—1633no—
314C22×C4—1633no—
315C22×C4—1633no—
316C22×C4—1633no—
317C22×C4—1631no—
318C22×C4—1633no—
319C22×C4—1633no—
320C22×C4C22×D16313yes—
321C22×C4C22×D16313yes—
322C22×C4C22×D16313yes—
323C22×C4C22×D16311yescharacteristic
324C22×C4C22×D16313yes—
325C22×C4C22×D16313yes—
326C4⋊C4—1631no—
327C4⋊C4—1633no—
328C4⋊C4—1633no—
329C4⋊C4—1633no—
330C4⋊C4—1633no—
331C4⋊C4—1633no—
332C4⋊C4—1636no—
333C22⋊C4—1631no—
334C22⋊C4—1633no—
335C22⋊C4—1633no—
336C22⋊C4—1633no—
337C22⋊C4—1636no—
338C22⋊C4—1636no—
339C22⋊C4—1631no—
340C22⋊C4—1633no—
341C22⋊C4—1633no—
342C22⋊C4—1633no—
343C22⋊C4—1636no—
344C42—1633no—
345C42—1633no—
346C42—1631no—
347C42—1633no—
348C23—26no—
349C23—26no—
350C23—43no—
351C23—3266no—
352C23—3266no—
353C23—6523no—
354C23—1633no—
355C23—1631no—
356C23C23×D16313yes—
357C23C23×D16313yes—
358C23C23×D16311yescharacteristic
359Q8—26no—
360Q8—3266no—
361Q8C2×C22⋊D16314yescomplemented
362D4—212no—
363D4—26no—
364D4—3266no—
365D4—32612no—
366D4—3266no—
367D4—3266no—
368D4—3266no—
369D4—6526no—
370D4—1634no—
371D4C2×C22⋊D163112yescomplemented
372C2×C4—26no—
373C2×C4—22no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—3266no—
378C2×C4—3266no—
379C2×C4—3262no—
380C2×C4—3266no—
381C2×C4—3266no—
382C2×C4—3266no—
383C2×C4—6523no—
384C2×C4—1631no—
385C2×C4—1633no—
386C2×C4—1633no—
387C2×C4—1631no—
388C2×C4—1633no—
389C2×C4—1631no—
390C2×C4C2×C22⋊D163112yescomplemented
391C2×C4C23×D16311yescharacteristic
392C2×C4C23×D16313yes—
393C2×C4C23×D16313yes—
394C2×C4C23×D16311yescharacteristic
395C22—21no—
396C22—23no—
397C22—23no—
398C22—21no—
399C22—23no—
400C22—23no—
401C22—23no—
402C22—412no—
403C22—3261no—
404C22—3261no—
405C22—3263no—
406C22—3263no—
407C22—3263no—
408C22—65212no—
409C22—1632no—
410C22C22×C22⋊D16316yes—
411C22#41×D16312yescomplemented
412C22C24×D16311yescharacteristic
413C4—21no—
414C4—23no—
415C4—3261no—
416C4—3263no—
417C4—3263no—
418C4—3261no—
419C4—3263no—
420C4—1632no—
421C4C22×C22⋊D16312yes—
422C4C22×C22⋊D16316yes—
423C2—22no—
424C2—26no—
425C2—43no—
426C2—3262no—
427C2—6523no—
428C2C23×C22⋊D16311yescharacteristic
429C2C2×#41×D16311yescharacteristic
430C2#2378250111yescharacteristic
subgroupG/H
centerC2×C4C23×D163
commutator, socle, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC22×#41×C163C2
hypercenterC22×#41D163
hypocenterC163D4×#41
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×D163C2semidirect
2. #23782334C2semidirect
3. Q8×C22⋊D163C2semidirect
4. #23782326C2semidirect
5. #23782322C2semidirect
6. #23782314C2semidirect
7. #23782309C2semidirect
8. #23782303C2semidirect
9. D4×C22⋊D163C2semidirect
10. #23782211C2semidirect
11. #23782176C2semidirect
12. #23782172C2semidirect
13. #41×Dic163C2semidirect
14. C2×C4×C22⋊D163C2semidirect
15. C22×#41×C163C2semidirect
16. C2×#14579700C22semidirect
17. C2×#14579699C22semidirect
18. C2×#14579698C22semidirect
20. C2×Q8×D163C22semidirect
22. C2×D4×D163C22semidirect
23. C22×C4×D163C22semidirect
24. #19104733C22semidirect
25. #19104731C22semidirect
26. #19104724C22semidirect
27. #19104722C22semidirect
28. #19104718C22semidirect
29. #19104716C22semidirect
30. #19104714C22semidirect
31. #19104714C22semidirect
32. #19104711C22semidirect
33. #19104711C22semidirect
34. #19104709C22semidirect
35. #19104709C22semidirect
36. #19104707C22semidirect
37. #19104707C22semidirect
38. #19104706C22semidirect
39. #19104706C22semidirect
40. #19104704C22semidirect
41. #19104703C22semidirect
42. Q8×Dic163C22semidirect
43. Q8×Dic163C22semidirect
44. D4×Dic163C22semidirect
45. D4×Dic163C22semidirect
48. C4×C22⋊D163C22semidirect
49. C4×C22⋊D163C22semidirect
50. C4×C22⋊D163C22semidirect
51. C4×C22⋊D163C22semidirect
52. C4×C22⋊D163C22semidirect
53. C4×C22⋊D163C22semidirect
54. C4×C22⋊D163C22semidirect
55. #19104694C22semidirect
56. C2×Dic163⋊C4C22semidirect
57. C2×C22⋊Dic163C22semidirect
58. C2×(C2×D163)⋊C4C22semidirect
59. C2×C4×Dic163C22semidirect
62. C2×#41×C163C22semidirect
63. C2×#41×C163C22semidirect
64. C2×#41×C163C22semidirect
65. C22×Q8×C163C22semidirect
66. C22×D4×C163C22semidirect
67. C23×C4×C163C22semidirect
117. #41×C163D4semidirect
144. C22⋊D163#41direct (N × H)
165. Q8×C163C2×D4semidirect
168. D4×C163C2×D4semidirect
174. C2×C4×C163C2×D4semidirect
188. C22×C163C2×#41semidirect
196. C2×D163C2×#41semidirect
206. Dic163C2×#41semidirect
215. C163D4×#41semidirect
218. C22×#41D163semidirect
238. C2×#41C2×D163semidirect
239. C2×#41C2×D163semidirect
240. C2×#41C2×D163semidirect
242. C22×Q8C2×D163semidirect
245. C22×D4C2×D163semidirect
247. C23×C4C2×D163semidirect
292. #41C22⋊D163direct (N × H)
361. Q8C2×C22⋊D163semidirect
371. D4C2×C22⋊D163semidirect
390. C2×C4C2×C22⋊D163semidirect
411. C22#41×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1632
#237823342
Q8×C22⋊D1632
#237823262
#237823222
#237823142
#237823092
#237823032
D4×C22⋊D1632
#237822112
#237821762
#237821722
#41×Dic1632
C2×C4×C22⋊D1632
C22×#41×C1632
D4×#41163