Finite Group Databasebeta

← #35228025

Subgroups — #35228025

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
1C22×#14579700C211yesmaximal, characteristic, complemented
2C2×#41×D163C211yesmaximal, characteristic, complemented
3#23782334C211yesmaximal, characteristic, complemented
4Q8×C22⋊D163C211yesmaximal, characteristic, complemented
5#23782327C211yesmaximal, characteristic, complemented
6#23782317C213yesmaximal, complemented
7#23782311C213yesmaximal, complemented
8#23782308C213yesmaximal, complemented
9#23782299C213yesmaximal, complemented
10#23782214C213yesmaximal, complemented
11#23782178C213yesmaximal, complemented
12#23782175C213yesmaximal, complemented
13C4×#14579700C211yesmaximal, characteristic, complemented
14C4×D4×D163C213yesmaximal, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×#14579700—28no—
17C2×#14579700C2211yescharacteristic
18C2×#14579700C2213yescomplemented
19C2×#14579700C2213yescomplemented
20C2×#14579700C2211yescharacteristic, complemented
21C2×#14579699C2213yescomplemented
22C2×#14579698C2213yescomplemented
23#41×D163—28no—
24C2×Q8×D163C2211yescharacteristic, complemented
25C22×D652C2213yescomplemented
26C2×D4×D163C2213yes—
27C2×D4×D163C2213yescomplemented
28C22×C4×D163C2213yescomplemented
29C22×C4×D163C2213yescomplemented
30#19104733C2213yescomplemented
31#19104731C2213yescomplemented
32#19104723C2213yescomplemented
33#19104720C2216yescomplemented
34#19104717C2216yescomplemented
35#19104715C2213yescomplemented
36#19104714C2213yescomplemented
37#19104711C2213yescomplemented
38#19104710C2213yescomplemented
39#19104709C2213yescomplemented
40#19104708C2216yescomplemented
41#19104708C2216yescomplemented
42#19104706C2213yescomplemented
43#19104705C2216yescomplemented
44#19104703C2216yescomplemented
45Q8×Dic163C2211yescharacteristic, complemented
46C4×D652C2213yescomplemented
47C4×D652C2213yescomplemented
48D4×Dic163C2213yescomplemented
49#19104697C2213yescomplemented
50#19104697C2213yescomplemented
51C4×C22⋊D163C2213yes—
52C4×C22⋊D163C2213yescomplemented
53(C22⋊D163)⋊C4C2211yescharacteristic
54(C22⋊D163)⋊C4C2216yescomplemented
55(C22⋊D163)⋊C4C2213yescomplemented
56#19104693C2213yescomplemented
57C4⋊C4×D163C2213yescomplemented
58C22⋊C4×D163C2213yescomplemented
59C22⋊C4×D163C2213yescomplemented
60C42×D163C2213yescomplemented
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×D163C2313yes—
70#14579700—212no—
71#14579700—212no—
72#14579700—24no—
73#14579700D418yescomplemented
74#14579699—212no—
75#14579698—212no—
76C2×Dic326C2313yes—
77Q8×D163—24no—
78C2×D652—212no—
79C2×D652C2313yes—
80C2×D652C2316yes—
81C2×D652C2313yes—
82C2×D652C2313yes—
83C2×C22⋊D163C2313yes—
84C2×C22⋊D163C2316yes—
85C2×C22⋊D163C2316yes—
86C2×C22⋊D163C2311yescharacteristic
87D4×D163—212no—
88D4×D163—212no—
89C22×Dic163C2311yescharacteristic
90C22×Dic163C2313yes—
91C2×C4×D163—212no—
92C2×C4×D163—212no—
93C2×C4×D163C2311yescharacteristic
94C2×C4×D163C2313yes—
95C2×C4×D163C2313yes—
96C2×C4×D163C2313yes—
97C2×C4×D163C2313yes—
98C2×C4×D163C2313yes—
99C2×C4×D163C2313yes—
100C2×C4×D163C2313yes—
101C2×C4×D163C2313yes—
102C4⋊Dic163C2313yes—
103C4⋊Dic163C2313yes—
104Dic163⋊C4C2311yescharacteristic
105Dic163⋊C4C2313yes—
106Dic163⋊C4C2316yes—
107C22⋊Dic163C2313yes—
108C22⋊Dic163C2313yes—
109(C2×D163)⋊C4C2311yescharacteristic
110(C2×D163)⋊C4C2313yes—
111(C2×D163)⋊C4C2313yes—
112(C2×D163)⋊C4C2313yes—
113(C2×D163)⋊C4C2316yes—
114(C2×D163)⋊C4C2316yes—
115C4×Dic163C2311yescharacteristic
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—26no—
133C22×D163—26no—
134C22×D163—26no—
135C22×D163—26no—
136C22×D163—43no—
137C22×D163—43no—
138C22×D163C2413yes—
139C22×D163C2413yes—
140C22×D163C2411yescharacteristic
141Dic326—26no—
142D652—212no—
143D652—26no—
144D652—26no—
145D652C2×D4112yescomplemented
146C22⋊D163—212no—
147C22⋊D163—26no—
148C22⋊D163—46no—
149C22⋊D163#4114yescomplemented
150C2×Dic163—22no—
151C2×Dic163—26no—
152C2×Dic163C2413yes—
153C2×Dic163C2413yes—
154C2×Dic163C2411yescharacteristic
155C2×Dic163C2411yescharacteristic
156C2×Dic163C2411yescharacteristic
157C4×D163—26no—
158C4×D163—26no—
159C4×D163—26no—
160C4×D163—26no—
161C4×D163—22no—
162C4×D163—26no—
163C4×D163—26no—
164C4×D163—43no—
165C4×D163C2×D4112yescomplemented
166C23×C163C2413yes—
167C23×C163C2411yescharacteristic
168Q8×C163—26no—
169Q8×C163C2×D414yescomplemented
170D4×C163—26no—
171D4×C163—26no—
172C2×C4×C163—26no—
173C2×C4×C163—26no—
174C2×C4×C163C2411yescharacteristic
175C2×C4×C163C2413yes—
176C2×C4×C163C2413yes—
177C2×C4×C163C2411yescharacteristic
178C2×C4×C163C2413yes—
179C22×C163—21no—
180C22×C163—21no—
181C22×C163—23no—
182C22×C163—23no—
183C22×C163—23no—
184C22×C163C2×#4112yescomplemented
185C22×C163C2511yescharacteristic
186C2×D163—21no—
187C2×D163—23no—
188C2×D163—23no—
189C2×D163—21no—
190C2×D163—23no—
191C2×D163—23no—
192C2×D163—23no—
193C2×D163—412no—
194C2×D163—412no—
195C2×D163C22×D416yes—
196C2×D163C2×#4112yescomplemented
197C4×C163—21no—
198C4×C163—23no—
199C4×C163—23no—
200C4×C163—21no—
201C4×C163C22×D416yes—
202Dic163—21no—
203Dic163—23no—
204Dic163—23no—
205Dic163C22×D412yes—
206Dic163C2×#4112yescomplemented
207C2×C163—22no—
208C2×C163—43no—
209C2×C163C23×D411yescharacteristic
210C2×C163C22×#4111yescharacteristic
211C2×C163#58511yescharacteristic
212D163—22no—
213D163—26no—
214D163—43no—
215C163D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—1631nomaximal, Sylow
217C22×#41—1631no—
218C22×#41—1631no—
219#550—1631no—
220D4×Q8—1631no—
221#548—1631no—
222#548D16311yescharacteristic, complemented
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—3268no—
233C2×#41—3268no—
234C2×#41—1633no—
235C2×#41—1633no—
236C2×#41—1631no—
237C2×#41—1631no—
238C2×#41—1633no—
239C2×#41—1633no—
240C2×#41C2×D16311yescharacteristic, complemented
241C22×Q8—1631no—
242C22×Q8C2×D16311yescharacteristic, complemented
243C22×D4—1633no—
244C22×D4—1633no—
245C22×D4—1633no—
246C23×C4—1633no—
247C23×C4—1633no—
248C4⋊Q8—1633no—
249#127—1633no—
250#124—1633no—
251#124C2×D16313yescomplemented
252#123—1636no—
253#123—1636no—
254C22⋊Q8—1633no—
255C22⋊Q8—1633no—
256C22⋊Q8—1633no—
257C22⋊Q8C2×D16313yescomplemented
258#121—1633no—
259#121—1633no—
260#121—1633no—
261#121—1636no—
262#121—1636no—
263#121C2×D16313yescomplemented
264#120—1636no—
265#120—1636no—
266C4×Q8—1631no—
267C4×Q8C2×D16311yescharacteristic, complemented
268C4×D4—1633no—
269C4×D4—1633no—
270C4×D4—1633no—
271C4×D4—1633no—
272C4×D4—1633no—
273C4×D4—1631no—
274C4×D4—1633no—
275C4×D4—1636no—
276C4×D4—1633no—
277C4×D4—1633no—
278C4×D4C2×D16313yescomplemented
279#117—1633no—
280C2×C4⋊C4—1633no—
281C2×C22⋊C4—1633no—
282C2×C22⋊C4—1633no—
283C2×C42—1633no—
284C24—1633no—
285C24—1633no—
286#41—24no—
287#41—32612no—
288#41—3264no—
289#41—32612no—
290#41—32612no—
291#41—32612no—
292#41—1638no—
293C2×Q8—24no—
294C2×Q8—3264no—
295C2×Q8—1633no—
296C2×Q8C22×D16313yes—
297C2×Q8C22×D16311yescharacteristic
298C2×D4—32612no—
299C2×D4—32612no—
300C2×D4—32612no—
301C2×D4—1633no—
302C2×D4—1636no—
303C2×D4—1636no—
304C2×D4—1631no—
305C2×D4—1633no—
306C2×D4—1636no—
307C2×D4—1633no—
308C2×D4—1633no—
309C2×D4C22×D16313yes—
310C2×D4C22×D16313yes—
311C22×C4—32612no—
312C22×C4—32612no—
313C22×C4—1631no—
314C22×C4—1633no—
315C22×C4—1633no—
316C22×C4—1633no—
317C22×C4—1633no—
318C22×C4—1631no—
319C22×C4—1633no—
320C22×C4—1633no—
321C22×C4—1633no—
322C22×C4—1633no—
323C22×C4—1633no—
324C22×C4C22×D16313yes—
325C22×C4C22×D16313yes—
326C4⋊C4—1631no—
327C4⋊C4—1633no—
328C4⋊C4—1633no—
329C4⋊C4—1633no—
330C4⋊C4—1636no—
331C4⋊C4C22×D16313yes—
332C4⋊C4C22×D16313yes—
333C22⋊C4—1633no—
334C22⋊C4—1633no—
335C22⋊C4—1631no—
336C22⋊C4—1633no—
337C22⋊C4—1633no—
338C22⋊C4—1633no—
339C22⋊C4—1636no—
340C22⋊C4—1636no—
341C22⋊C4C22×D16311yescharacteristic
342C22⋊C4C22×D16313yes—
343C22⋊C4C22×D16316yes—
344C42—1631no—
345C42—1633no—
346C42—1633no—
347C42C22×D16313yes—
348C23—3266no—
349C23—3266no—
350C23—3266no—
351C23—3266no—
352C23—6523no—
353C23—6523no—
354C23—1633no—
355C23—1631no—
356C23—1633no—
357C23C23×D16313yes—
358C23C23×D16311yescharacteristic
359Q8—26no—
360Q8—3266no—
361Q8D4×D16314yescomplemented
362D4—26no—
363D4—26no—
364D4—32612no—
365D4—3266no—
366D4—32612no—
367D4—3266no—
368D4—3266no—
369D4—6526no—
370D4—1634no—
371D4—16312no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—3266no—
375C2×C4—3266no—
376C2×C4—3262no—
377C2×C4—3266no—
378C2×C4—3266no—
379C2×C4—3262no—
380C2×C4—3266no—
381C2×C4—3266no—
382C2×C4—3266no—
383C2×C4—6523no—
384C2×C4—1633no—
385C2×C4—1631no—
386C2×C4—1633no—
387C2×C4—1631no—
388C2×C4—1631no—
389C2×C4—16312no—
390C2×C4C23×D16311yescharacteristic
391C2×C4C23×D16313yes—
392C2×C4C23×D16311yescharacteristic
393C2×C4C23×D16313yes—
394C2×C4C23×D16313yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—3261no—
401C22—3263no—
402C22—3263no—
403C22—3261no—
404C22—3263no—
405C22—3263no—
406C22—3263no—
407C22—65212no—
408C22—65212no—
409C22—1636no—
410C22—1632no—
411C22#41×D16312yescomplemented
412C22C24×D16311yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—3261no—
418C4—3263no—
419C4—3263no—
420C4—1632no—
421C4—1632no—
422C4C2×D4×D16316yes—
423C2—22no—
424C2—43no—
425C2—3262no—
426C2—3266no—
427C2—6523no—
428C2C22×D4×D16311yescharacteristic
429C2C2×#41×D16311yescharacteristic
430C2#2378250211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#548×C163C2
hypercenter#548D163
hypocenterC163D4×#41
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#14579700C2semidirect
2. C2×#41×D163C2semidirect
3. #23782334C2semidirect
4. Q8×C22⋊D163C2semidirect
5. #23782327C2semidirect
6. #23782317C2semidirect
7. #23782311C2semidirect
8. #23782308C2semidirect
9. #23782299C2semidirect
10. #23782214C2semidirect
11. #23782178C2semidirect
12. #23782175C2semidirect
13. C4×#14579700C2semidirect
14. C4×D4×D163C2semidirect
15. #548×C163C2semidirect
18. C2×#14579700C22semidirect
19. C2×#14579700C22semidirect
20. C2×#14579700C22semidirect
21. C2×#14579699C22semidirect
22. C2×#14579698C22semidirect
24. C2×Q8×D163C22semidirect
25. C22×D652C22semidirect
27. C2×D4×D163C22semidirect
28. C22×C4×D163C22semidirect
29. C22×C4×D163C22semidirect
30. #19104733C22semidirect
31. #19104731C22semidirect
32. #19104723C22semidirect
33. #19104720C22semidirect
34. #19104717C22semidirect
35. #19104715C22semidirect
36. #19104714C22semidirect
37. #19104711C22semidirect
38. #19104710C22semidirect
39. #19104709C22semidirect
40. #19104708C22semidirect
41. #19104708C22semidirect
42. #19104706C22semidirect
43. #19104705C22semidirect
44. #19104703C22semidirect
45. Q8×Dic163C22semidirect
46. C4×D652C22semidirect
47. C4×D652C22semidirect
48. D4×Dic163C22semidirect
49. #19104697C22semidirect
50. #19104697C22semidirect
52. C4×C22⋊D163C22semidirect
54. (C22⋊D163)⋊C4C22semidirect
55. (C22⋊D163)⋊C4C22semidirect
56. #19104693C22semidirect
57. C4⋊C4×D163C22semidirect
58. C22⋊C4×D163C22semidirect
59. C22⋊C4×D163C22semidirect
60. C42×D163C22semidirect
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
73. #14579700D4semidirect
145. D652C2×D4semidirect
149. C22⋊D163#41semidirect
165. C4×D163C2×D4semidirect
169. Q8×C163C2×D4semidirect
184. C22×C163C2×#41semidirect
196. C2×D163C2×#41semidirect
206. Dic163C2×#41semidirect
215. C163D4×#41semidirect
222. #548D163semidirect
240. C2×#41C2×D163semidirect
242. C22×Q8C2×D163semidirect
251. #124C2×D163semidirect
257. C22⋊Q8C2×D163semidirect
263. #121C2×D163semidirect
267. C4×Q8C2×D163semidirect
278. C4×D4C2×D163semidirect
361. Q8D4×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
C22×#145797002
C2×#41×D1632
#237823342
Q8×C22⋊D1632
#237823272
#237823172
#237823112
#237823082
#237822992
#237822142
#237821782
#237821752
C4×#145797002
C4×D4×D1632
#548×C1632
D4×#41163