Finite Group Databasebeta

← #465×D163 #35224092

Subgroups — #465×D163 #35224092

Order 20864.

564 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D163C211yesmaximal, characteristic, complemented
2C22×C8×D163C211yesmaximal, characteristic, complemented
3C22⋊Q8×D163C211yesmaximal, characteristic, complemented
4#121×D163C211yesmaximal, characteristic
5#23781897C211yesmaximal, characteristic, complemented
6#23781896C211yesmaximal, characteristic, complemented
7#23781895C211yesmaximal, characteristic, complemented
8#23781894C211yesmaximal, characteristic, complemented
9#23781893C211yesmaximal, characteristic, complemented
10#23781892C211yesmaximal, characteristic, complemented
11#23781891C211yesmaximal, characteristic, complemented
12#106×D163C211yesmaximal, characteristic, complemented
13#103×D163C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×D4×D163—21no—
18C2×D4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20C2×#14579688C2211yescharacteristic, complemented
21C2×#14579687C2211yescharacteristic, complemented
22C2×#14579686C2211yescharacteristic, complemented
23SD16×D163—24no—
24C22×C163⋊C8C2211yescharacteristic, complemented
25C2×C8×D163—24no—
26C2×C8×D163C2211yescharacteristic
27C2×C8×D163C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic, complemented
29C2×C8×D163C2211yescharacteristic, complemented
30#19104715C2211yescharacteristic, complemented
31#19104714C2211yescharacteristic, complemented
32#19104713—21no—
33#19104712—21no—
34#19104711C2211yescharacteristic, complemented
35#19104710C2211yescharacteristic
36#19104709C2211yescharacteristic
37#19104708—21no—
38#19104707—21no—
39#19104706C2211yescharacteristic
40C4⋊C4×D163—21no—
41C4⋊C4×D163C2211yescharacteristic
42C4⋊C4×D163C2211yescharacteristic
43C22⋊C4×D163—21no—
44C22⋊C4×D163—21no—
45#19104667C2211yescharacteristic, complemented
46#19104666C2211yescharacteristic, complemented
47#19104666C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104658C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic, complemented
52#19104656C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic, complemented
55C22×C8×C163C2211yescharacteristic, complemented
56C22⋊Q8×C163C2211yescharacteristic, complemented
57#121×C163C2211yescharacteristic
58#106×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163D411yescharacteristic
63C2×Dic326—21no—
64C2×Dic326C2311yescharacteristic
65Q8×D163—22no—
66Q8×D163—22no—
67C2×D652—21no—
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72D4×D163—22no—
73D4×D163—22no—
74D4×D163—44no—
75C22×Dic163D411yescharacteristic
76C2×C4×D163—22no—
77C2×C4×D163—22no—
78C2×C4×D163—21no—
79C2×C4×D163—21no—
80C2×C4×D163—21no—
81C2×C4×D163D411yescharacteristic
82C2×C4×D163D411yescharacteristic
83C2×C4×D163C2311yescharacteristic
84C2×C4×D163C2311yescharacteristic
85#14579688—22no—
86#14579687—22no—
87#14579686—22no—
88C2×C163⋊C8—22no—
89C2×C163⋊C8C2311yescharacteristic
90C2×C163⋊C8C2311yescharacteristic
91C8×D163—22no—
92C8×D163—22no—
93C8×D163—22no—
94C8×D163D414yescomplemented
95C4⋊Dic163—21no—
96C4⋊Dic163C2311yescharacteristic
97C4⋊Dic163C2311yescharacteristic
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100Dic163⋊C4—21no—
101Dic163⋊C4—21no—
102C22⋊Dic163—21no—
103C22⋊Dic163—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163—21no—
110C2×D4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112SD16×C163—22no—
113C2×C8×C163—22no—
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C4⋊C4×C163—21no—
117C4⋊C4×C163C2311yescharacteristic
118C4⋊C4×C163C2311yescharacteristic
119C22⋊C4×C163—21no—
120C22⋊C4×C163—21no—
121C22×D163—22no—
122C22×D163—22no—
123C22×D163—21no—
124C22×D163—42no—
125C22×D163—42no—
126C22×D163—42no—
127C22×D163SD1614yescomplemented
128C22×D163C2×D411yescharacteristic
129C22×D163C2×D411yescharacteristic
130Dic326—21no—
131Dic326—21no—
132Dic326—42no—
133D652—21no—
134D652—21no—
135D652—42no—
136C22⋊D163—42no—
137C22⋊D163—42no—
138C22⋊D163—42no—
139C2×Dic163—22no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163C2×D411yescharacteristic
144C2×Dic163C2×D411yescharacteristic
145C4×D163—22no—
146C4×D163—21no—
147C4×D163—21no—
148C4×D163—22no—
149C4×D163—42no—
150C4×D163—42no—
151C4×D163—42no—
152C4×D163C2×D412yes—
153C4×D163#4112yes—
154C163⋊C8—21no—
155C163⋊C8C2×D412yescomplemented
156C23×C163—21no—
157C23×C163C2×D411yescharacteristic
158Q8×C163—21no—
159Q8×C163—21no—
160D4×C163—21no—
161D4×C163—21no—
162D4×C163—42no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2411yescharacteristic
170C8×C163—21no—
171C8×C163C2×D412yescomplemented
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163C2×SD1612yescomplemented
178C22×C163C22×D411yescharacteristic
179C2×D163—21no—
180C2×D163—21no—
181C2×D163—28no—
182C2×D163—41no—
183C2×D163—41no—
184C2×D163—41no—
185C2×D163—84no—
186C2×D163#12112yes—
187C2×D163C2×SD1612yes—
188C2×D163C2×SD1612yescomplemented
189C2×D163#13512yes—
190C4×C163—21no—
191C4×C163—41no—
192C4×C163—41no—
193C4×C163—41no—
194C4×C163C22×D411yescharacteristic
195C4×C163C2×#4111yescharacteristic
196Dic163—21no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200Dic163#12112yes—
201C2×C163—22no—
202C2×C163—81no—
203C2×C163C2×#12111yescharacteristic
204C2×C163C22×SD1611yescharacteristic
205C2×C163C2×#13511yescharacteristic
206D163—22no—
207D163—81no—
208D163#46514yesdirect product factor
209C163C2×#46511yescharacteristic, complemented, Sylow
210C2×#465—1631nomaximal, Sylow
211C22×SD16—1631no—
212C23×C8—1631no—
213C2×C22⋊Q8—1631no—
214C2×#121—1631no—
215#465—1631no—
216#465—1631no—
217#465—1631no—
218#465—1631no—
219#465—1631no—
220#465—1631no—
221#465—1631no—
222#465D16311yescharacteristic, direct product factor
223C2×#106—1631no—
224C2×#103—1631no—
225C2×#102—1631no—
226C22×Q8—1631no—
227C22×D4—3261no—
228C22×D4—1631no—
229C23×C4—1631no—
230C2×SD16—3264no—
231C2×SD16—1631no—
232C2×SD16—1631no—
233C2×SD16—1631no—
234C2×SD16C2×D16311yescharacteristic, complemented
235C22×C8—3264no—
236C22×C8—1631no—
237C22×C8—1631no—
238C22×C8—1631no—
239C22×C8—1631no—
240C22×C8—1631no—
241C22×C8C2×D16311yescharacteristic, complemented
242C22⋊Q8—3261no—
243C22⋊Q8—3261no—
244C22⋊Q8—1631no—
245C22⋊Q8—1631no—
246C22⋊Q8—1631no—
247C22⋊Q8C2×D16311yescharacteristic, complemented
248#121—3261no—
249#121—3261no—
250#121—1631no—
251#121—1631no—
252#121—1631no—
253#121C2×D16311yescharacteristic
254C2×C4⋊C4—3261no—
255C2×C4⋊C4—1631no—
256C2×C4⋊C4—1631no—
257C2×C22⋊C4—3261no—
258C2×C22⋊C4—3261no—
259#106—1631no—
260#106—1631no—
261#106—1631no—
262#106C2×D16311yescharacteristic, complemented
263#103—1631no—
264#103—1631no—
265#103—1631no—
266#103C2×D16311yescharacteristic, complemented
267#102—1631no—
268#102—1631no—
269#102—1631no—
270#102C2×D16311yescharacteristic, complemented
271C24—3261no—
272C24—1631no—
273C2×Q8—3261no—
274C2×Q8—3262no—
275C2×Q8—3262no—
276C2×Q8—1631no—
277C2×Q8C22×D16311yescharacteristic
278C2×D4—21no—
279C2×D4—3261no—
280C2×D4—3261no—
281C2×D4—3261no—
282C2×D4—3262no—
283C2×D4—3262no—
284C2×D4—3261no—
285C2×D4—6524no—
286C2×D4—1631no—
287C2×D4C22×D16311yescharacteristic
288C22×C4—3262no—
289C22×C4—3262no—
290C22×C4—3261no—
291C22×C4—3261no—
292C22×C4—3261no—
293C22×C4—1631no—
294C22×C4—1631no—
295C22×C4—1631no—
296C22×C4—1631no—
297C22×C4—1631no—
298C22×C4C22×D16311yescharacteristic
299SD16—22no—
300SD16—3262no—
301SD16—3262no—
302SD16—3262no—
303C2×C8—22no—
304C2×C8—3262no—
305C2×C8—3262no—
306C2×C8—3262no—
307C2×C8—3262no—
308C2×C8—1631no—
309C2×C8—1631no—
310C2×C8—1634no—
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3261no—
315C4⋊C4—3261no—
316C4⋊C4—3261no—
317C4⋊C4—3261no—
318C4⋊C4—3261no—
319C4⋊C4—1631no—
320C4⋊C4—1631no—
321C4⋊C4C22×D16311yescharacteristic
322C4⋊C4C22×D16311yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C22⋊C4—3261no—
330C22⋊C4—3261no—
331C23—21no—
332C23—3262no—
333C23—3262no—
334C23—3261no—
335C23—6522no—
336C23—6522no—
337C23—6522no—
338C23—1631no—
339C23—1634no—
340C23—1631no—
341C23D4×D16311yescharacteristic
342Q8—21no—
343Q8—21no—
344Q8—3261no—
345Q8—3261no—
346Q8—6522no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—3261no—
351D4—3261no—
352D4—6522no—
353D4—6522no—
354D4—6522no—
355D4—6522no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—3262no—
362C2×C4—3262no—
363C2×C4—3262no—
364C2×C4—3261no—
365C2×C4—3261no—
366C2×C4—3261no—
367C2×C4—3261no—
368C2×C4—3261no—
369C2×C4—6522no—
370C2×C4—6522no—
371C2×C4—6522no—
372C2×C4—1631no—
373C2×C4—1631no—
374C2×C4—1632no—
375C2×C4—1632no—
376C2×C4D4×D16311yescharacteristic
377C2×C4C23×D16311yescharacteristic
378C8—21no—
379C8—3261no—
380C8—1632no—
381C8D4×D16312yescomplemented
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—13044no—
388C22—3268no—
389C22—3261no—
390C22—3261no—
391C22—6521no—
392C22—6521no—
393C22—6521no—
394C22—1632no—
395C22—1632no—
396C22—1632no—
397C22—1632no—
398C22SD16×D16312yescomplemented
399C22C2×D4×D16311yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—3261no—
405C4—6521no—
406C4—6521no—
407C4—6521no—
408C4—1632no—
409C4C2×D4×D16311yescharacteristic
410C4#41×D16311yescharacteristic
411C2—22no—
412C2—81no—
413C2—13041no—
414C2—3262no—
415C2—1634no—
416C2#121×D16311yescharacteristic
417C2C2×SD16×D16311yescharacteristic
418C2#135×D16311yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
prime radical#121×D163C2
hypercenter#465D163
hypocenterC163C2×#465
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. C22×C8×D163C2semidirect
3. C22⋊Q8×D163C2semidirect
5. #23781897C2semidirect
6. #23781896C2semidirect
7. #23781895C2semidirect
8. #23781894C2semidirect
9. #23781893C2semidirect
10. #23781892C2semidirect
11. #23781891C2semidirect
12. #106×D163C2semidirect
13. #103×D163C2semidirect
14. #102×D163C2semidirect
15. #465×C163C2semidirect
20. C2×#14579688C22semidirect
21. C2×#14579687C22semidirect
22. C2×#14579686C22semidirect
24. C22×C163⋊C8C22semidirect
28. C2×C8×D163C22semidirect
29. C2×C8×D163C22semidirect
30. #19104715C22semidirect
31. #19104714C22semidirect
34. #19104711C22semidirect
45. #19104667C22semidirect
46. #19104666C22semidirect
47. #19104666C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104658C22semidirect
51. #19104657C22semidirect
52. #19104656C22semidirect
53. #19104655C22semidirect
54. C2×SD16×C163C22semidirect
55. C22×C8×C163C22semidirect
56. C22⋊Q8×C163C22semidirect
58. #106×C163C22semidirect
59. #103×C163C22semidirect
60. #102×C163C22semidirect
94. C8×D163D4semidirect
127. C22×D163SD16semidirect
155. C163⋊C8C2×D4semidirect
171. C8×C163C2×D4semidirect
177. C22×C163C2×SD16semidirect
188. C2×D163C2×SD16semidirect
208. D163#465direct (N × H)
209. C163C2×#465semidirect
222. #465D163direct (N × H)
234. C2×SD16C2×D163semidirect
241. C22×C8C2×D163semidirect
247. C22⋊Q8C2×D163semidirect
262. #106C2×D163semidirect
266. #103C2×D163semidirect
270. #102C2×D163semidirect
381. C8D4×D163semidirect
398. C22SD16×D163semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1632
C22×C8×D1632
C22⋊Q8×D1632
#121×D1632
#237818972
#237818962
#237818952
#237818942
#237818932
#237818922
#237818912
#106×D1632
#103×D1632
#102×D1632
#465×C1632
C2×#465163