Finite Group Databasebeta

← #35228204

Subgroups — #35228204

Order 20864.

1658 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×#19104781C211yesmaximal, characteristic, complemented
2C2×#41×D163C211yesmaximal, characteristic, complemented
3#23782334C211yesmaximal, characteristic, complemented
4#23782327C211yesmaximal, characteristic, complemented
5#23782322C211yesmaximal, characteristic, complemented
6#23782312C213yesmaximal, complemented
7#23782311C213yesmaximal, complemented
8#23782300C213yesmaximal, complemented
9#23782299C213yesmaximal, complemented
10#23782244C213yesmaximal, complemented
11#23782228C213yesmaximal, complemented
12#23782174C213yesmaximal, complemented
13#121×D163C213yesmaximal, complemented
14#23782147C211yesmaximal, characteristic, complemented
15#550×C163C211yesmaximal, characteristic, complemented
16#19104781—28no—
17C2×#14579700C2211yescharacteristic
18C2×#14579700C2211yescharacteristic, complemented
19C2×#14579699C2213yescomplemented
20C2×#14579699C2213yescomplemented
21C2×#14579698C2213yescomplemented
22#41×D163—28no—
23C2×Q8×D163C2211yescharacteristic, complemented
24C22×D652C2213yescomplemented
25C2×D4×D163C2213yescomplemented
26C2×D4×D163C2213yescomplemented
27C2×D4×D163C2213yescomplemented
28C2×D4×D163C2216yescomplemented
29C22×C4×D163C2213yescomplemented
30#19104731C2213yescomplemented
31#19104723C2213yescomplemented
32#19104722C2213yescomplemented
33#19104720C2213yescomplemented
34#19104717C2213yescomplemented
35#19104715C2213yescomplemented
36#19104714C2213yescomplemented
37#19104713C2211yescharacteristic, complemented
38#19104712C2213yescomplemented
39#19104710C2213yescomplemented
40#19104709C2213yescomplemented
41#19104708C2213yescomplemented
42#19104708C2216yescomplemented
43#19104707C2216yescomplemented
44#19104707C2213yescomplemented
45#19104706C2213yescomplemented
46#19104706C2216yescomplemented
47#19104706C2213yescomplemented
48#19104705C2213yescomplemented
49#19104703C2213yescomplemented
50#19104703C2216yescomplemented
51#19104703C2216yescomplemented
52Q8×Dic163C2211yescharacteristic, complemented
53C4×D652C2213yescomplemented
54C4×D652C2213yescomplemented
55#19104697C2213yescomplemented
56C4×C22⋊D163C2213yescomplemented
57C4×C22⋊D163C2213yescomplemented
58#19104692C2213yescomplemented
59C4⋊C4×D163C2213yescomplemented
60C22⋊C4×D163C2216yescomplemented
61C2×#41×C163C2211yescharacteristic, complemented
62C2×#41×C163C2211yescharacteristic, complemented
63#127×C163C2213yescomplemented
64#121×C163C2213yescomplemented
65#121×C163C2213yescomplemented
66C4×Q8×C163C2211yescharacteristic, complemented
67C4×D4×C163C2213yescomplemented
68C23×D163C2313yes—
69C23×D163C2316yes—
70#14579700—24no—
71#14579700D418yescomplemented
72#14579699—212no—
73#14579699—212no—
74#14579698—212no—
75C2×Dic326C2313yes—
76C2×Dic326C2311yescharacteristic
77Q8×D163—24no—
78C2×D652—212no—
79C2×D652C2313yes—
80C2×D652C2313yes—
81C2×D652C2313yes—
82C2×D652C2313yes—
83C2×C22⋊D163C2313yes—
84C2×C22⋊D163C2313yes—
85C2×C22⋊D163C2313yes—
86C2×C22⋊D163C2316yes—
87C2×C22⋊D163C2316yes—
88C2×C22⋊D163C2313yes—
89D4×D163—212no—
90D4×D163—212no—
91D4×D163—212no—
92D4×D163—412no—
93C22×Dic163C2313yes—
94C2×C4×D163—212no—
95C2×C4×D163C2311yescharacteristic
96C2×C4×D163C2313yes—
97C2×C4×D163C2313yes—
98C2×C4×D163C2313yes—
99C2×C4×D163C2316yes—
100C2×C4×D163C2313yes—
101C4⋊Dic163C2313yes—
102C4⋊Dic163C2313yes—
103Dic163⋊C4C2313yes—
104Dic163⋊C4C2311yescharacteristic
105Dic163⋊C4C2311yescharacteristic
106Dic163⋊C4C2313yes—
107C22⋊Dic163C2316yes—
108C22⋊Dic163C2313yes—
109C22⋊Dic163C2313yes—
110(C2×D163)⋊C4C2313yes—
111(C2×D163)⋊C4C2316yes—
112(C2×D163)⋊C4C2313yes—
113(C2×D163)⋊C4C2313yes—
114(C2×D163)⋊C4C2311yescharacteristic
115(C2×D163)⋊C4C2311yescharacteristic
116(C2×D163)⋊C4C2313yes—
117C4×Dic163C2313yes—
118#41×C163—24no—
119#41×C163—24no—
120C2×Q8×C163C2311yescharacteristic
121C2×D4×C163C2313yes—
122C2×D4×C163C2313yes—
123C2×D4×C163C2316yes—
124C2×D4×C163C2313yes—
125C22×C4×C163C2313yes—
126C22×C4×C163C2313yes—
127C4⋊C4×C163C2311yescharacteristic
128C4⋊C4×C163C2313yes—
129C22⋊C4×C163C2313yes—
130C22⋊C4×C163C2313yes—
131C42×C163C2313yes—
132C22×D163—26no—
133C22×D163—26no—
134C22×D163—212no—
135C22×D163—43no—
136C22×D163—46no—
137C22×D163—46no—
138C22×D163C2413yes—
139C22×D163C2413yes—
140C22×D163C2411yescharacteristic
141Dic326—26no—
142Dic326—22no—
143D652—26no—
144D652—26no—
145D652—26no—
146D652C2×D4112yescomplemented
147C22⋊D163—26no—
148C22⋊D163—212no—
149C22⋊D163—26no—
150C22⋊D163—26no—
151C22⋊D163—46no—
152C22⋊D163—43no—
153C2×Dic163—26no—
154C2×Dic163C2411yescharacteristic
155C2×Dic163C2413yes—
156C2×Dic163C2413yes—
157C2×Dic163C2411yescharacteristic
158C2×Dic163C2411yescharacteristic
159C4×D163—26no—
160C4×D163—26no—
161C4×D163—26no—
162C4×D163—22no—
163C4×D163—46no—
164C4×D163C2×D4112yescomplemented
165C23×C163C2413yes—
166C23×C163C2413yes—
167Q8×C163C2×D414yescomplemented
168D4×C163—26no—
169D4×C163—26no—
170D4×C163—43no—
171D4×C163—46no—
172C2×C4×C163—26no—
173C2×C4×C163—26no—
174C2×C4×C163C2411yescharacteristic
175C2×C4×C163C2413yes—
176C2×C4×C163C2411yescharacteristic
177C2×C4×C163C2411yescharacteristic
178C2×C4×C163C2413yes—
179C22×C163—23no—
180C22×C163—23no—
181C22×C163—23no—
182C22×C163—23no—
183C22×C163—23no—
184C22×C163—23no—
185C22×C163C2511yescharacteristic
186C2×D163—21no—
187C2×D163—23no—
188C2×D163—21no—
189C2×D163—23no—
190C2×D163—23no—
191C2×D163—23no—
192C2×D163—23no—
193C2×D163—412no—
194C2×D163—812no—
195C2×D163C22×D416yes—
196C2×D163H(5,𝔽2)12yescomplemented
197C4×C163—21no—
198C4×C163—21no—
199C4×C163—23no—
200C4×C163—21no—
201C4×C163C22×D416yes—
202Dic163—21no—
203Dic163—23no—
204Dic163—23no—
205Dic163—21no—
206Dic163C22×D412yes—
207C2×C163—43no—
208C2×C163—43no—
209C2×C163C23×D411yescharacteristic
210C2×C163C2×H(5,𝔽2)11yescharacteristic
211C2×C163#58511yescharacteristic
212D163—22no—
213D163—26no—
214D163—43no—
215C163#348811yescharacteristic, complemented, Sylow
216#3488—1631nomaximal, Sylow
217C2×H(5,𝔽2)—1631no—
218C22×#41—1631no—
219#550—1631no—
220#550D16311yescharacteristic, complemented
221#548—1631no—
222#548—1631no—
223#546—1633no—
224#546—1633no—
225D42—1633no—
226D42—1633no—
227#537—1633no—
228#535—1633no—
229#525—1633no—
230C2×#121—1633no—
231#520—1631no—
232H(5,𝔽2)—3268no—
233C2×#41—3268no—
234C2×#41—1631no—
235C2×#41—1631no—
236C2×#41—1633no—
237C2×#41—1633no—
238C2×#41—1633no—
239C2×#41C2×D16311yescharacteristic, complemented
240C2×#41C2×D16311yescharacteristic, complemented
241C22×Q8—1631no—
242C22×D4—1633no—
243C22×D4—1633no—
244C22×D4—1633no—
245C22×D4—1636no—
246C22×D4—1633no—
247C23×C4—1633no—
248#127—1633no—
249#127C2×D16313yescomplemented
250#124—1633no—
251#124—1633no—
252#123—1633no—
253#123—1633no—
254C22⋊Q8—1633no—
255C22⋊Q8—1631no—
256C22⋊Q8—1633no—
257C22⋊Q8—1633no—
258#121—1636no—
259#121—1633no—
260#121—1633no—
261#121—1636no—
262#121—1633no—
263#121—1633no—
264#121—1633no—
265#121—1633no—
266#121—1636no—
267#121C2×D16313yescomplemented
268#121C2×D16313yescomplemented
269#120—1633no—
270#120—1636no—
271#120—1636no—
272#120—1633no—
273C4×Q8—1631no—
274C4×Q8C2×D16311yescharacteristic, complemented
275C4×D4—1633no—
276C4×D4—1633no—
277C4×D4—1633no—
278C4×D4—1633no—
279C4×D4—1633no—
280C4×D4C2×D16313yescomplemented
281#117—1633no—
282C2×C4⋊C4—1633no—
283C2×C22⋊C4—1636no—
284C24—1633no—
285C24—1636no—
286#41—24no—
287#41—24no—
288#41—32612no—
289#41—32612no—
290#41—3264no—
291#41—32612no—
292#41—1638no—
293C2×Q8—3264no—
294C2×Q8—1631no—
295C2×Q8—1633no—
296C2×Q8C22×D16311yescharacteristic
297C2×D4—32612no—
298C2×D4—32612no—
299C2×D4—32612no—
300C2×D4—32612no—
301C2×D4—65212no—
302C2×D4—1633no—
303C2×D4—1636no—
304C2×D4—1633no—
305C2×D4—1633no—
306C2×D4—1633no—
307C2×D4—1636no—
308C2×D4—1633no—
309C2×D4—1633no—
310C2×D4—1633no—
311C2×D4—1633no—
312C2×D4C22×D16313yes—
313C2×D4C22×D16313yes—
314C2×D4C22×D16313yes—
315C2×D4C22×D16316yes—
316C22×C4—32612no—
317C22×C4—1633no—
318C22×C4—1633no—
319C22×C4—1633no—
320C22×C4—1633no—
321C22×C4—1631no—
322C22×C4—1636no—
323C22×C4—1633no—
324C22×C4C22×D16313yes—
325C22×C4C22×D16313yes—
326C4⋊C4—1633no—
327C4⋊C4—1633no—
328C4⋊C4—1631no—
329C4⋊C4—1631no—
330C4⋊C4—1633no—
331C4⋊C4—1633no—
332C4⋊C4C22×D16313yes—
333C4⋊C4C22×D16311yescharacteristic
334C22⋊C4—1636no—
335C22⋊C4—1633no—
336C22⋊C4—1633no—
337C22⋊C4—1633no—
338C22⋊C4—1636no—
339C22⋊C4—1633no—
340C22⋊C4—1633no—
341C22⋊C4—1631no—
342C22⋊C4—1631no—
343C22⋊C4—1633no—
344C22⋊C4C22×D16313yes—
345C22⋊C4C22×D16313yes—
346C42—1633no—
347C42C22×D16313yes—
348C23—32612no—
349C23—3266no—
350C23—3266no—
351C23—6526no—
352C23—6526no—
353C23—6523no—
354C23—1633no—
355C23—1631no—
356C23—1633no—
357C23C23×D16313yes—
358C23C23×D16313yes—
359Q8—3262no—
360Q8—3266no—
361Q8D4×D16314yescomplemented
362D4—26no—
363D4—26no—
364D4—46no—
365D4—43no—
366D4—3266no—
367D4—3266no—
368D4—3266no—
369D4—32612no—
370D4—3266no—
371D4—3266no—
372D4—3266no—
373D4—6526no—
374D4—6523no—
375D4—16312no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—3266no—
379C2×C4—3262no—
380C2×C4—3266no—
381C2×C4—3266no—
382C2×C4—3266no—
383C2×C4—6526no—
384C2×C4—1631no—
385C2×C4—1633no—
386C2×C4—1633no—
387C2×C4—1631no—
388C2×C4—1631no—
389C2×C4—16312no—
390C2×C4C23×D16311yescharacteristic
391C2×C4C23×D16313yes—
392C2×C4C23×D16311yescharacteristic
393C2×C4C23×D16311yescharacteristic
394C2×C4C23×D16313yes—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—23no—
401C22—130412no—
402C22—3261no—
403C22—3263no—
404C22—3263no—
405C22—3261no—
406C22—3263no—
407C22—3263no—
408C22—3263no—
409C22—65212no—
410C22—1636no—
411C22—1632no—
412C22C24×D16311yescharacteristic
413C4—21no—
414C4—21no—
415C4—23no—
416C4—21no—
417C4—3261no—
418C4—3263no—
419C4—3263no—
420C4—3261no—
421C4—1632no—
422C4C2×D4×D16316yes—
423C2—43no—
424C2—43no—
425C2—3266no—
426C2—3262no—
427C2—6523no—
428C2C22×D4×D16311yescharacteristic
429C2H(5,𝔽2)×D16311yescharacteristic
430C2#2378250111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#550×C163C2
hypercenter#550D163
hypocenterC163#3488
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#19104781C2semidirect
2. C2×#41×D163C2semidirect
3. #23782334C2semidirect
4. #23782327C2semidirect
5. #23782322C2semidirect
6. #23782312C2semidirect
7. #23782311C2semidirect
8. #23782300C2semidirect
9. #23782299C2semidirect
10. #23782244C2semidirect
11. #23782228C2semidirect
12. #23782174C2semidirect
13. #121×D163C2semidirect
14. #23782147C2semidirect
15. #550×C163C2semidirect
18. C2×#14579700C22semidirect
19. C2×#14579699C22semidirect
20. C2×#14579699C22semidirect
21. C2×#14579698C22semidirect
23. C2×Q8×D163C22semidirect
24. C22×D652C22semidirect
25. C2×D4×D163C22semidirect
26. C2×D4×D163C22semidirect
27. C2×D4×D163C22semidirect
28. C2×D4×D163C22semidirect
29. C22×C4×D163C22semidirect
30. #19104731C22semidirect
31. #19104723C22semidirect
32. #19104722C22semidirect
33. #19104720C22semidirect
34. #19104717C22semidirect
35. #19104715C22semidirect
36. #19104714C22semidirect
37. #19104713C22semidirect
38. #19104712C22semidirect
39. #19104710C22semidirect
40. #19104709C22semidirect
41. #19104708C22semidirect
42. #19104708C22semidirect
43. #19104707C22semidirect
44. #19104707C22semidirect
45. #19104706C22semidirect
46. #19104706C22semidirect
47. #19104706C22semidirect
48. #19104705C22semidirect
49. #19104703C22semidirect
50. #19104703C22semidirect
51. #19104703C22semidirect
52. Q8×Dic163C22semidirect
53. C4×D652C22semidirect
54. C4×D652C22semidirect
55. #19104697C22semidirect
56. C4×C22⋊D163C22semidirect
57. C4×C22⋊D163C22semidirect
58. #19104692C22semidirect
59. C4⋊C4×D163C22semidirect
60. C22⋊C4×D163C22semidirect
61. C2×#41×C163C22semidirect
62. C2×#41×C163C22semidirect
63. #127×C163C22semidirect
64. #121×C163C22semidirect
65. #121×C163C22semidirect
66. C4×Q8×C163C22semidirect
67. C4×D4×C163C22semidirect
71. #14579700D4semidirect
146. D652C2×D4semidirect
164. C4×D163C2×D4semidirect
167. Q8×C163C2×D4semidirect
196. C2×D163H(5,𝔽2)semidirect
215. C163#3488semidirect
220. #550D163semidirect
239. C2×#41C2×D163semidirect
240. C2×#41C2×D163semidirect
249. #127C2×D163semidirect
267. #121C2×D163semidirect
268. #121C2×D163semidirect
274. C4×Q8C2×D163semidirect
280. C4×D4C2×D163semidirect
361. Q8D4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047812
C2×#41×D1632
#237823342
#237823272
#237823222
#237823122
#237823112
#237823002
#237822992
#237822442
#237822282
#237821742
#121×D1632
#237821472
#550×C1632
#3488163