Finite Group Databasebeta

← #35228189

Subgroups — #35228189

Order 20864.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104782C211yesmaximal, characteristic, complemented
2C2×#41×D163C211yesmaximal, characteristic, complemented
3#23782335C213yesmaximal, complemented
4#23782334C211yesmaximal, characteristic, complemented
5#23782333C213yesmaximal, complemented
6#23782327C211yesmaximal, characteristic, complemented
7#23782324C213yesmaximal, complemented
8#23782322C211yesmaximal, characteristic, complemented
9#23782321C213yesmaximal, complemented
10#23782231C213yesmaximal, complemented
11#23782228C213yesmaximal, complemented
12#23782175C213yesmaximal, complemented
13#23782174C213yesmaximal, complemented
14#23782141C211yesmaximal, characteristic, complemented
15#550×C163C211yesmaximal, characteristic, complemented
16#19104782—28no—
17C2×#14579700C2211yescharacteristic
18C2×#14579699C2213yes—
19C2×#14579699C2213yes—
20C2×#14579698C2213yes—
21C2×#14579698C2213yes—
22C2×#14579698C2213yescomplemented
23#41×D163—28no—
24C22×Dic326C2213yescomplemented
25C2×Q8×D163C2211yescharacteristic
26C2×Q8×D163C2211yescharacteristic, complemented
27C2×D4×D163C2213yes—
28C22×C4×D163C2213yes—
29#19104731C2213yescomplemented
30#19104731C2216yescomplemented
31#19104723C2213yescomplemented
32#19104722C2213yescomplemented
33#19104721C2216yescomplemented
34#19104720C2211yescharacteristic
35#19104717C2213yes—
36#19104715C2213yescomplemented
37#19104714C2213yescomplemented
38#19104713C2213yescomplemented
39#19104712C2213yescomplemented
40#19104710C2213yescomplemented
41#19104709C2213yescomplemented
42#19104708C2213yescomplemented
43#19104707C2213yescomplemented
44#19104707C2216yescomplemented
45#19104707C2216yescomplemented
46#19104706C2213yescomplemented
47#19104706C2213yescomplemented
48#19104705C2213yes—
49#19104703C2213yes—
50#19104703C2216yes—
51Q8×Dic163C2211yescharacteristic, complemented
52C4×Dic326C2213yescomplemented
53#19104700C2213yescomplemented
54C4×D652C2213yescomplemented
55D4×Dic163C2216yescomplemented
56C4×C22⋊D163C2213yescomplemented
57C4×C22⋊D163C2213yescomplemented
58(C22⋊D163)⋊C4C2216yescomplemented
59#19104693C2213yes—
60#19104692C2213yes—
61C2×#41×C163C2211yescharacteristic
62C2×#41×C163C2211yescharacteristic, complemented
63#127×C163C2213yescomplemented
64#121×C163C2213yescomplemented
65#121×C163C2213yescomplemented
66C4×Q8×C163C2211yescharacteristic, complemented
67C4×D4×C163C2213yescomplemented
68C23×D163C2313yes—
69#14579700—24no—
70#14579699—212no—
71#14579699—212no—
72#14579698—212no—
73#14579698—212no—
74#14579698—212no—
75C2×Dic326—212no—
76C2×Dic326C2313yes—
77C2×Dic326C2313yes—
78C2×Dic326C2313yes—
79Q8×D163—24no—
80Q8×D163D418yescomplemented
81C2×D652C2313yes—
82C2×D652C2311yescharacteristic
83C2×D652C2313yes—
84C2×C22⋊D163C2316yes—
85C2×C22⋊D163C2313yes—
86C2×C22⋊D163C2313yes—
87C2×C22⋊D163C2316yes—
88C2×C22⋊D163C2313yes—
89C2×C22⋊D163C2313yes—
90D4×D163—212no—
91C22×Dic163C2313yes—
92C22×Dic163C2316yes—
93C2×C4×D163—212no—
94C2×C4×D163C2311yescharacteristic
95C2×C4×D163C2313yes—
96C2×C4×D163C2313yes—
97C2×C4×D163C2313yes—
98C2×C4×D163C2313yes—
99C4⋊Dic163C2313yes—
100C4⋊Dic163C2313yes—
101Dic163⋊C4C2313yes—
102Dic163⋊C4C2313yes—
103Dic163⋊C4C2313yes—
104Dic163⋊C4C2311yescharacteristic
105C22⋊Dic163C2313yes—
106C22⋊Dic163C2313yes—
107C22⋊Dic163C2316yes—
108(C2×D163)⋊C4C2311yescharacteristic
109(C2×D163)⋊C4C2313yes—
110(C2×D163)⋊C4C2316yes—
111(C2×D163)⋊C4C2311yescharacteristic
112(C2×D163)⋊C4C2313yes—
113(C2×D163)⋊C4C2311yescharacteristic
114(C2×D163)⋊C4C2313yes—
115C4×Dic163C2313yes—
116C4×Dic163C2316yes—
117#41×C163—24no—
118#41×C163—24no—
119C2×Q8×C163C2311yescharacteristic
120C2×D4×C163C2313yes—
121C2×D4×C163C2313yes—
122C2×D4×C163C2313yes—
123C2×D4×C163C2316yes—
124C22×C4×C163C2313yes—
125C22×C4×C163C2313yes—
126C4⋊C4×C163C2311yescharacteristic
127C4⋊C4×C163C2313yes—
128C22⋊C4×C163C2313yes—
129C22⋊C4×C163C2313yes—
130C42×C163C2313yes—
131C22×D163—26no—
132C22×D163—43no—
133C22×D163—43no—
134C22×D163C2413yes—
135C22×D163C2411yescharacteristic
136C22×D163C2411yescharacteristic
137Dic326—26no—
138Dic326—26no—
139Dic326C2×D4112yes—
140D652—22no—
141D652—26no—
142D652—43no—
143C22⋊D163—26no—
144C22⋊D163—212no—
145C22⋊D163—26no—
146C22⋊D163—43no—
147C22⋊D163—46no—
148C22⋊D163—43no—
149C2×Dic163—26no—
150C2×Dic163—212no—
151C2×Dic163C2411yescharacteristic
152C2×Dic163C2413yes—
153C2×Dic163C2413yes—
154C2×Dic163C2411yescharacteristic
155C2×Dic163C2413yes—
156C4×D163—22no—
157C4×D163—26no—
158C4×D163—26no—
159C4×D163—26no—
160C4×D163C2×D4112yes—
161C23×C163C2413yes—
162C23×C163C2413yes—
163Q8×C163C2×D414yes—
164D4×C163—26no—
165D4×C163—26no—
166D4×C163—43no—
167D4×C163—46no—
168C2×C4×C163—26no—
169C2×C4×C163—26no—
170C2×C4×C163C2411yescharacteristic
171C2×C4×C163C2413yes—
172C2×C4×C163C2411yescharacteristic
173C2×C4×C163C2411yescharacteristic
174C2×C4×C163C2413yes—
175C22×C163—23no—
176C22×C163—23no—
177C22×C163—23no—
178C22×C163—23no—
179C22×C163—23no—
180C22×C163—23no—
181C22×C163C2511yescharacteristic
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—21no—
185C2×D163—23no—
186C2×D163—21no—
187C2×D163—23no—
188C2×D163—23no—
189C2×D163—21no—
190C2×D163—412no—
191C2×D163C22×D412yes—
192C4×C163—21no—
193C4×C163—21no—
194C4×C163—21no—
195C4×C163—23no—
196C4×C163C22×D416yes—
197Dic163—21no—
198Dic163—23no—
199Dic163—21no—
200Dic163—23no—
201Dic163C22×D416yes—
202C2×C163—43no—
203C2×C163—43no—
204C2×C163C23×D411yescharacteristic
205C2×C163#58511yescharacteristic
206C2×C163#58511yescharacteristic
207D163—22no—
208D163—41no—
209D163—43no—
210C163#348711yescharacteristic, complemented, Sylow
211#3487—1631nomaximal, Sylow
212C2×2−1+4—1631no—
213C22×#41—1631no—
214#550—1631no—
215#550—1633no—
216#550—1633no—
217#550D16311yescharacteristic, complemented
218#548—1631no—
219#548—1633no—
220#548—1631no—
221#548—1633no—
222#535—1633no—
223#535—1633no—
224#525—1633no—
225#525—1633no—
226#519—1631no—
2272−1+4—3268no—
228C2×#41—3268no—
229C2×#41—1631no—
230C2×#41—1633no—
231C2×#41—1633no—
232C2×#41—1633no—
233C2×#41—1633no—
234C2×#41—1633no—
235C2×#41C2×D16311yescharacteristic
236C2×#41C2×D16311yescharacteristic, complemented
237C22×Q8—1631no—
238C22×Q8—1631no—
239C22×Q8—1633no—
240C22×D4—1633no—
241C23×C4—1633no—
242#127—1636no—
243#127—1633no—
244#127C2×D16313yescomplemented
245#124—1633no—
246#124—1636no—
247#124—1633no—
248#123—1631no—
249#123—1633no—
250C22⋊Q8—1633no—
251C22⋊Q8—1633no—
252C22⋊Q8—1633no—
253C22⋊Q8—1633no—
254#121—1633no—
255#121—1633no—
256#121—1633no—
257#121—1636no—
258#121—1636no—
259#121—1633no—
260#121—1633no—
261#121—1633no—
262#121C2×D16313yescomplemented
263#121C2×D16313yescomplemented
264#120—1636no—
265#120—1633no—
266#120—1633no—
267C4×Q8—1631no—
268C4×Q8—1633no—
269C4×Q8—1633no—
270C4×Q8C2×D16311yescharacteristic, complemented
271C4×D4—1636no—
272C4×D4—1633no—
273C4×D4—1633no—
274C4×D4—1636no—
275C4×D4—1633no—
276C4×D4C2×D16313yescomplemented
277#117—1633no—
278#117—1633no—
279C24—1633no—
280#41—24no—
281#41—24no—
282#41—32612no—
283#41—32612no—
284#41—32612no—
285#41—32612no—
286#41—3264no—
287#41—32612no—
288C2×Q8—32612no—
289C2×Q8—3264no—
290C2×Q8—1638no—
291C2×Q8—1633no—
292C2×Q8—1633no—
293C2×Q8—1633no—
294C2×Q8C22×D16311yescharacteristic
295C2×D4—32612no—
296C2×D4—1633no—
297C2×D4—1636no—
298C2×D4—1633no—
299C2×D4—1633no—
300C2×D4—1636no—
301C2×D4—1633no—
302C2×D4—1631no—
303C2×D4—1633no—
304C2×D4—1633no—
305C2×D4C22×D16313yes—
306C2×D4C22×D16313yes—
307C2×D4C22×D16316yes—
308C2×D4C22×D16313yes—
309C22×C4—32612no—
310C22×C4—1636no—
311C22×C4—1633no—
312C22×C4—1633no—
313C22×C4—1633no—
314C22×C4—1633no—
315C22×C4—1631no—
316C22×C4—1633no—
317C22×C4C22×D16313yes—
318C22×C4C22×D16313yes—
319C4⋊C4—1633no—
320C4⋊C4—1633no—
321C4⋊C4—1633no—
322C4⋊C4—1633no—
323C4⋊C4—1631no—
324C4⋊C4—1633no—
325C4⋊C4C22×D16313yes—
326C4⋊C4C22×D16311yescharacteristic
327C22⋊C4—1633no—
328C22⋊C4—1636no—
329C22⋊C4—1633no—
330C22⋊C4—1633no—
331C22⋊C4—1631no—
332C22⋊C4—1631no—
333C22⋊C4—1631no—
334C22⋊C4—1633no—
335C22⋊C4—1636no—
336C22⋊C4—1633no—
337C22⋊C4C22×D16313yes—
338C22⋊C4C22×D16313yes—
339C42—1633no—
340C42—1636no—
341C42C22×D16313yes—
342C23—3266no—
343C23—6523no—
344C23—6523no—
345C23—1633no—
346C23—1631no—
347C23—1631no—
348C23C23×D16313yes—
349C23C23×D16313yes—
350Q8—3266no—
351Q8—3266no—
352Q8—16312no—
353Q8D4×D16314yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—3262no—
359D4—3266no—
360D4—32612no—
361D4—3266no—
362D4—3266no—
363D4—6523no—
364D4—6523no—
365D4—6526no—
366D4—6523no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—3262no—
370C2×C4—3266no—
371C2×C4—3266no—
372C2×C4—32612no—
373C2×C4—3266no—
374C2×C4—3266no—
375C2×C4—1631no—
376C2×C4—1633no—
377C2×C4—1633no—
378C2×C4—1631no—
379C2×C4—1633no—
380C2×C4—16312no—
381C2×C4C23×D16313yes—
382C2×C4C23×D16311yescharacteristic
383C2×C4C23×D16311yescharacteristic
384C2×C4C23×D16311yescharacteristic
385C2×C4C23×D16313yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—3261no—
393C22—3261no—
394C22—3261no—
395C22—3263no—
396C22—3261no—
397C22—3263no—
398C22—3261no—
399C22—3263no—
400C22—65212no—
401C22—1632no—
402C22C24×D16311yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—3261no—
408C4—3263no—
409C4—3261no—
410C4—3263no—
411C4—1636no—
412C4C2×D4×D16316yes—
413C2—43no—
414C2—43no—
415C2—3262no—
416C2—6523no—
417C2—6521no—
418C2C22×D4×D16311yescharacteristic
419C2#2378250011yescharacteristic
420C2#2378250111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#550×C163C2
hypercenter#550D163
hypocenterC163#3487
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#19104782C2semidirect
2. C2×#41×D163C2semidirect
3. #23782335C2semidirect
4. #23782334C2semidirect
5. #23782333C2semidirect
6. #23782327C2semidirect
7. #23782324C2semidirect
8. #23782322C2semidirect
9. #23782321C2semidirect
10. #23782231C2semidirect
11. #23782228C2semidirect
12. #23782175C2semidirect
13. #23782174C2semidirect
14. #23782141C2semidirect
15. #550×C163C2semidirect
22. C2×#14579698C22semidirect
24. C22×Dic326C22semidirect
26. C2×Q8×D163C22semidirect
29. #19104731C22semidirect
30. #19104731C22semidirect
31. #19104723C22semidirect
32. #19104722C22semidirect
33. #19104721C22semidirect
36. #19104715C22semidirect
37. #19104714C22semidirect
38. #19104713C22semidirect
39. #19104712C22semidirect
40. #19104710C22semidirect
41. #19104709C22semidirect
42. #19104708C22semidirect
43. #19104707C22semidirect
44. #19104707C22semidirect
45. #19104707C22semidirect
46. #19104706C22semidirect
47. #19104706C22semidirect
51. Q8×Dic163C22semidirect
52. C4×Dic326C22semidirect
53. #19104700C22semidirect
54. C4×D652C22semidirect
55. D4×Dic163C22semidirect
56. C4×C22⋊D163C22semidirect
57. C4×C22⋊D163C22semidirect
58. (C22⋊D163)⋊C4C22semidirect
62. C2×#41×C163C22semidirect
63. #127×C163C22semidirect
64. #121×C163C22semidirect
65. #121×C163C22semidirect
66. C4×Q8×C163C22semidirect
67. C4×D4×C163C22semidirect
80. Q8×D163D4semidirect
210. C163#3487semidirect
217. #550D163semidirect
236. C2×#41C2×D163semidirect
244. #127C2×D163semidirect
262. #121C2×D163semidirect
263. #121C2×D163semidirect
270. C4×Q8C2×D163semidirect
276. C4×D4C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047822
C2×#41×D1632
#237823352
#237823342
#237823332
#237823272
#237823242
#237823222
#237823212
#237822312
#237822282
#237821752
#237821742
#237821412
#550×C1632
#3487163