Finite Group Databasebeta

← #548×D163 #35227998

Subgroups — #548×D163 #35227998

Order 20864.

1620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D163C211yesmaximal, characteristic, complemented
2C22×Q8×D163C211yesmaximal, characteristic, complemented
3#23782327C211yesmaximal, characteristic, complemented
4#23782326C211yesmaximal, characteristic, complemented
5#23782325C213yesmaximal, complemented
6#23782324C213yesmaximal, complemented
7#23782323C213yesmaximal, complemented
8#23782322C211yesmaximal, characteristic, complemented
9#23782321C213yesmaximal, complemented
10#124×D163C213yesmaximal, complemented
11C22⋊Q8×D163C213yesmaximal, complemented
12#121×D163C213yesmaximal, complemented
13C4×Q8×D163C211yesmaximal, characteristic, complemented
14C4×D4×D163C213yesmaximal, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic, complemented
17C2×#14579699C2213yescomplemented
18C2×#14579698C2213yescomplemented
19#41×D163—28no—
20C22×Dic326C2213yescomplemented
21C2×Q8×D163—28no—
22C2×Q8×D163C2213yes—
23C2×Q8×D163C2211yescharacteristic
24C2×Q8×D163C2211yescharacteristic, complemented
25C2×Q8×D163C2213yescomplemented
26C2×D4×D163C2213yes—
27C2×D4×D163C2213yes—
28C22×C4×D163C2213yes—
29C22×C4×D163C2213yes—
30#19104724C2213yescomplemented
31#19104723C2213yescomplemented
32#19104722C2213yescomplemented
33#19104721C2216yescomplemented
34#19104721C2216yescomplemented
35#19104715C2213yescomplemented
36#19104714C2213yescomplemented
37#19104713C2216yescomplemented
38#19104712C2216yescomplemented
39#19104711C2213yescomplemented
40#19104710C2213yescomplemented
41#19104709C2213yescomplemented
42#19104708C2216yescomplemented
43#19104707C2216yescomplemented
44#19104706C2213yescomplemented
45Q8×Dic163C2211yescharacteristic, complemented
46C4×Dic326C2213yescomplemented
47#19104700C2213yescomplemented
48C4×D652C2213yescomplemented
49D4×Dic163C2213yescomplemented
50#19104697C2213yescomplemented
51C4×C22⋊D163C2213yescomplemented
52C4×C22⋊D163C2213yescomplemented
53(C22⋊D163)⋊C4C2213yescomplemented
54(C22⋊D163)⋊C4C2213yescomplemented
55C4⋊C4×D163C2213yes—
56C4⋊C4×D163C2213yes—
57C22⋊C4×D163C2216yes—
58C22⋊C4×D163C2213yes—
59C22⋊C4×D163C2211yescharacteristic
60C42×D163C2213yes—
61C2×#41×C163C2211yescharacteristic, complemented
62C22×Q8×C163C2211yescharacteristic, complemented
63#124×C163C2213yescomplemented
64C22⋊Q8×C163C2213yescomplemented
65#121×C163C2213yescomplemented
66C4×Q8×C163C2211yescharacteristic, complemented
67C4×D4×C163C2213yescomplemented
68C23×D163C2313yes—
69C23×D163C2311yescharacteristic
70#14579700—24no—
71#14579699—212no—
72#14579698—212no—
73C2×Dic326—212no—
74C2×Dic326C2316yes—
75C2×Dic326C2313yes—
76C2×Dic326C2313yes—
77Q8×D163—212no—
78Q8×D163—212no—
79Q8×D163—24no—
80Q8×D163D418yescomplemented
81C2×D652C2313yes—
82C2×D652C2313yes—
83C2×C22⋊D163C2316yes—
84C2×C22⋊D163C2313yes—
85C2×C22⋊D163C2313yes—
86D4×D163—212no—
87D4×D163—212no—
88C22×Dic163C2313yes—
89C22×Dic163C2313yes—
90C2×C4×D163—212no—
91C2×C4×D163—212no—
92C2×C4×D163C2311yescharacteristic
93C2×C4×D163C2313yes—
94C2×C4×D163C2313yes—
95C2×C4×D163C2313yes—
96C2×C4×D163C2313yes—
97C2×C4×D163C2313yes—
98C2×C4×D163C2313yes—
99C2×C4×D163C2311yescharacteristic
100C2×C4×D163C2313yes—
101C4⋊Dic163C2313yes—
102C4⋊Dic163C2313yes—
103Dic163⋊C4C2313yes—
104Dic163⋊C4C2313yes—
105Dic163⋊C4C2316yes—
106C22⋊Dic163C2311yescharacteristic
107C22⋊Dic163C2313yes—
108C22⋊Dic163C2316yes—
109(C2×D163)⋊C4C2311yescharacteristic
110(C2×D163)⋊C4C2313yes—
111(C2×D163)⋊C4C2311yescharacteristic
112(C2×D163)⋊C4C2313yes—
113(C2×D163)⋊C4C2316yes—
114(C2×D163)⋊C4C2316yes—
115C4×Dic163C2313yes—
116C4×Dic163C2313yes—
117C4×Dic163C2313yes—
118#41×C163—24no—
119C2×Q8×C163—24no—
120C2×Q8×C163C2313yes—
121C2×Q8×C163C2311yescharacteristic
122C2×D4×C163C2313yes—
123C2×D4×C163C2313yes—
124C22×C4×C163C2313yes—
125C22×C4×C163C2313yes—
126C4⋊C4×C163C2313yes—
127C4⋊C4×C163C2313yes—
128C22⋊C4×C163C2311yescharacteristic
129C22⋊C4×C163C2313yes—
130C22⋊C4×C163C2316yes—
131C42×C163C2313yes—
132C22×D163—22no—
133C22×D163—22no—
134C22×D163—26no—
135C22×D163—26no—
136C22×D163—26no—
137C22×D163#4114yescomplemented
138C22×D163C2413yes—
139C22×D163C2411yescharacteristic
140C22×D163C2411yescharacteristic
141Dic326—26no—
142Dic326—212no—
143Dic326C2×D4112yescomplemented
144D652—26no—
145D652—26no—
146C22⋊D163—26no—
147C22⋊D163—26no—
148C22⋊D163—212no—
149C2×Dic163—26no—
150C2×Dic163—26no—
151C2×Dic163C2411yescharacteristic
152C2×Dic163C2413yes—
153C2×Dic163C2413yes—
154C2×Dic163C2411yescharacteristic
155C2×Dic163C2413yes—
156C4×D163—22no—
157C4×D163—26no—
158C4×D163—26no—
159C4×D163—22no—
160C4×D163—26no—
161C4×D163—26no—
162C4×D163—26no—
163C4×D163—26no—
164C4×D163C2×D4112yes—
165C23×C163C2413yes—
166C23×C163C2411yescharacteristic
167Q8×C163—26no—
168Q8×C163C2×D414yescomplemented
169D4×C163—26no—
170D4×C163—26no—
171C2×C4×C163—26no—
172C2×C4×C163—26no—
173C2×C4×C163C2411yescharacteristic
174C2×C4×C163C2413yes—
175C2×C4×C163C2413yes—
176C2×C4×C163C2411yescharacteristic
177C2×C4×C163C2413yes—
178C22×C163—21no—
179C22×C163—21no—
180C22×C163—23no—
181C22×C163—23no—
182C22×C163—23no—
183C22×C163C2×#4112yescomplemented
184C22×C163C2511yescharacteristic
185C2×D163—21no—
186C2×D163—21no—
187C2×D163—28no—
188C2×D163—23no—
189C2×D163—23no—
190C2×D163—23no—
191C2×D163—412no—
192C2×D163C22×D412yes—
193C2×D163C2×#4112yes—
194C2×D163C2×#4112yescomplemented
195C2×D1632−1+412yes—
196C4×C163—21no—
197C4×C163—23no—
198C4×C163—23no—
199C4×C163—21no—
200C4×C163C22×D416yes—
201Dic163—21no—
202Dic163—23no—
203Dic163—23no—
204Dic163—21no—
205Dic163C22×D416yes—
206C2×C163—22no—
207C2×C163—43no—
208C2×C163C23×D411yescharacteristic
209C2×C163C22×#4111yescharacteristic
210C2×C163C2×2−1+411yescharacteristic
211D163—22no—
212D163—43no—
213D163#54814yesdirect product factor
214C163C2×#54811yescharacteristic, complemented, Sylow
215C2×#548—1631nomaximal, Sylow
216C22×#41—1631no—
217C23×Q8—1631no—
218#548—1631no—
219#548—1631no—
220#548—1633no—
221#548—1633no—
222#548—1633no—
223#548—1631no—
224#548—1633no—
225#548D16311yescharacteristic, direct product factor
226C2×#124—1633no—
227C2×C22⋊Q8—1633no—
228C2×#121—1633no—
229C2×C4×Q8—1631no—
230C2×C4×D4—1633no—
231C2×#41—3268no—
232C2×#41—1631no—
233C2×#41—1633no—
234C2×#41—1633no—
235C2×#41C2×D16311yescharacteristic, complemented
236C22×Q8—3268no—
237C22×Q8—1633no—
238C22×Q8—1633no—
239C22×Q8—1631no—
240C22×Q8—1631no—
241C22×Q8—1633no—
242C22×Q8C2×D16311yescharacteristic, complemented
243C22×D4—1633no—
244C22×D4—1633no—
245C23×C4—1633no—
246C23×C4—1633no—
247#124—1633no—
248#124—1633no—
249#124—1636no—
250#124—1636no—
251#124—1633no—
252#124C2×D16313yescomplemented
253C22⋊Q8—1633no—
254C22⋊Q8—1633no—
255C22⋊Q8—1636no—
256C22⋊Q8—1633no—
257C22⋊Q8—1636no—
258C22⋊Q8C2×D16313yescomplemented
259#121—1633no—
260#121—1636no—
261#121—1633no—
262#121—1633no—
263#121—1636no—
264#121C2×D16313yescomplemented
265C4×Q8—1631no—
266C4×Q8—1633no—
267C4×Q8—1633no—
268C4×Q8C2×D16311yescharacteristic, complemented
269C4×D4—1633no—
270C4×D4—1633no—
271C4×D4—1633no—
272C4×D4—1633no—
273C4×D4—1633no—
274C4×D4—1633no—
275C4×D4—1633no—
276C4×D4C2×D16313yescomplemented
277C2×C4⋊C4—1633no—
278C2×C4⋊C4—1633no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1633no—
281C2×C22⋊C4—1636no—
282C2×C42—1633no—
283C24—1633no—
284C24—1631no—
285#41—24no—
286#41—32612no—
287#41—32612no—
288#41—3264no—
289C2×Q8—24no—
290C2×Q8—32612no—
291C2×Q8—32612no—
292C2×Q8—3264no—
293C2×Q8—32612no—
294C2×Q8—1638no—
295C2×Q8—1633no—
296C2×Q8—1636no—
297C2×Q8—1633no—
298C2×Q8C22×D16313yes—
299C2×Q8C22×D16311yescharacteristic
300C2×D4—32612no—
301C2×D4—32612no—
302C2×D4—1633no—
303C2×D4—1633no—
304C2×D4—1636no—
305C2×D4—1633no—
306C2×D4—1633no—
307C2×D4C22×D16313yes—
308C2×D4C22×D16313yes—
309C22×C4—32612no—
310C22×C4—32612no—
311C22×C4—1633no—
312C22×C4—1633no—
313C22×C4—1631no—
314C22×C4—1633no—
315C22×C4—1633no—
316C22×C4—1631no—
317C22×C4—1633no—
318C22×C4—1633no—
319C22×C4—1633no—
320C22×C4—1633no—
321C22×C4—1633no—
322C22×C4C22×D16313yes—
323C22×C4C22×D16313yes—
324C4⋊C4—1633no—
325C4⋊C4—1633no—
326C4⋊C4—1633no—
327C4⋊C4—1636no—
328C4⋊C4—1633no—
329C4⋊C4C22×D16313yes—
330C4⋊C4C22×D16313yes—
331C22⋊C4—1631no—
332C22⋊C4—1633no—
333C22⋊C4—1636no—
334C22⋊C4—1631no—
335C22⋊C4—1631no—
336C22⋊C4—1633no—
337C22⋊C4—1633no—
338C22⋊C4—1636no—
339C22⋊C4—1636no—
340C22⋊C4C22×D16311yescharacteristic
341C22⋊C4C22×D16313yes—
342C22⋊C4C22×D16316yes—
343C42—1633no—
344C42—1633no—
345C42—1633no—
346C42C22×D16313yes—
347C23—3262no—
348C23—3262no—
349C23—3266no—
350C23—3266no—
351C23—3266no—
352C23—1633no—
353C23—1634no—
354C23—1631no—
355C23—1631no—
356C23C23×D16313yes—
357C23C23×D16311yescharacteristic
358Q8—26no—
359Q8—32612no—
360Q8—3266no—
361Q8—16312no—
362Q8D4×D16314yescomplemented
363D4—26no—
364D4—26no—
365D4—3266no—
366D4—3266no—
367D4—3266no—
368D4—32612no—
369D4—3266no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—3262no—
373C2×C4—3266no—
374C2×C4—3266no—
375C2×C4—3262no—
376C2×C4—3266no—
377C2×C4—3266no—
378C2×C4—3266no—
379C2×C4—3266no—
380C2×C4—3266no—
381C2×C4—3266no—
382C2×C4—1631no—
383C2×C4—1633no—
384C2×C4—1633no—
385C2×C4—1631no—
386C2×C4—1633no—
387C2×C4—16312no—
388C2×C4C23×D16311yescharacteristic
389C2×C4C23×D16313yes—
390C2×C4C23×D16313yes—
391C2×C4C23×D16311yescharacteristic
392C2×C4C23×D16313yes—
393C22—21no—
394C22—21no—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—3268no—
399C22—3261no—
400C22—3261no—
401C22—3263no—
402C22—3263no—
403C22—3263no—
404C22—65212no—
405C22—1632no—
406C22—1632no—
407C22—1632no—
408C22—1632no—
409C22#41×D16312yescomplemented
410C22C24×D16311yescharacteristic
411C4—21no—
412C4—23no—
413C4—23no—
414C4—21no—
415C4—3261no—
416C4—3263no—
417C4—3263no—
418C4—3261no—
419C4—1636no—
420C4C2×D4×D16316yes—
421C2—22no—
422C2—43no—
423C2—3262no—
424C2—6523no—
425C2—1634no—
426C2C22×D4×D16311yescharacteristic
427C2C2×#41×D16311yescharacteristic
428C22−1+4×D16311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#548×C163C2
hypercenter#548D163
hypocenterC163C2×#548
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#41×D163C2semidirect
2. C22×Q8×D163C2semidirect
3. #23782327C2semidirect
4. #23782326C2semidirect
5. #23782325C2semidirect
6. #23782324C2semidirect
7. #23782323C2semidirect
8. #23782322C2semidirect
9. #23782321C2semidirect
10. #124×D163C2semidirect
11. C22⋊Q8×D163C2semidirect
12. #121×D163C2semidirect
13. C4×Q8×D163C2semidirect
14. C4×D4×D163C2semidirect
15. #548×C163C2semidirect
16. C2×#14579700C22semidirect
17. C2×#14579699C22semidirect
18. C2×#14579698C22semidirect
20. C22×Dic326C22semidirect
24. C2×Q8×D163C22semidirect
25. C2×Q8×D163C22semidirect
30. #19104724C22semidirect
31. #19104723C22semidirect
32. #19104722C22semidirect
33. #19104721C22semidirect
34. #19104721C22semidirect
35. #19104715C22semidirect
36. #19104714C22semidirect
37. #19104713C22semidirect
38. #19104712C22semidirect
39. #19104711C22semidirect
40. #19104710C22semidirect
41. #19104709C22semidirect
42. #19104708C22semidirect
43. #19104707C22semidirect
44. #19104706C22semidirect
45. Q8×Dic163C22semidirect
46. C4×Dic326C22semidirect
47. #19104700C22semidirect
48. C4×D652C22semidirect
49. D4×Dic163C22semidirect
50. #19104697C22semidirect
51. C4×C22⋊D163C22semidirect
52. C4×C22⋊D163C22semidirect
53. (C22⋊D163)⋊C4C22semidirect
54. (C22⋊D163)⋊C4C22semidirect
61. C2×#41×C163C22semidirect
62. C22×Q8×C163C22semidirect
63. #124×C163C22semidirect
64. C22⋊Q8×C163C22semidirect
65. #121×C163C22semidirect
66. C4×Q8×C163C22semidirect
67. C4×D4×C163C22semidirect
80. Q8×D163D4semidirect
137. C22×D163#41semidirect
143. Dic326C2×D4semidirect
168. Q8×C163C2×D4semidirect
183. C22×C163C2×#41semidirect
194. C2×D163C2×#41semidirect
213. D163#548direct (N × H)
214. C163C2×#548semidirect
225. #548D163direct (N × H)
235. C2×#41C2×D163semidirect
242. C22×Q8C2×D163semidirect
252. #124C2×D163semidirect
258. C22⋊Q8C2×D163semidirect
264. #121C2×D163semidirect
268. C4×Q8C2×D163semidirect
276. C4×D4C2×D163semidirect
362. Q8D4×D163semidirect
409. 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
C22×Q8×D1632
#237823272
#237823262
#237823252
#237823242
#237823232
#237823222
#237823212
#124×D1632
C22⋊Q8×D1632
#121×D1632
C4×Q8×D1632
C4×D4×D1632
#548×C1632
C2×#548163