Finite Group Databasebeta

← #35224165

Subgroups — #35224165

Order 20864.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104767C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#23782178C211yesmaximal, characteristic
4C22⋊Q8×D163C211yesmaximal, characteristic, complemented
5#23781921C211yesmaximal, characteristic, complemented
6#23781913C211yesmaximal, characteristic, complemented
7#23781911C211yesmaximal, characteristic, complemented
8#23781905C211yesmaximal, characteristic, complemented
9#23781903C211yesmaximal, characteristic, complemented
10#23781897C211yesmaximal, characteristic, complemented
11#23781895C211yesmaximal, characteristic, complemented
12#23781704C211yesmaximal, characteristic, complemented
13#23781672C211yesmaximal, characteristic, complemented
14#23781665C211yesmaximal, characteristic, complemented
15#470×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19#19104767—24no—
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579688C2211yescharacteristic, complemented
22C2×#14579686C2211yescharacteristic, complemented
23C2×#14579683C2211yescharacteristic
24C2×#14579682C2211yescharacteristic
25C2×#14579682C2211yescharacteristic
26M16×D163—24no—
27C2×C8×D163C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic
29#19104720—21no—
30#19104715C2211yescharacteristic
31#19104714C2211yescharacteristic
32#19104713—21no—
33#19104712—21no—
34#19104711C2211yescharacteristic
35#19104711C2211yescharacteristic
36#19104710C2211yescharacteristic
37#19104709C2211yescharacteristic
38#19104705—21no—
39#19104697—21no—
40(C22⋊D163)⋊C4—21no—
41#19104693C2211yescharacteristic
42C4⋊C4×D163—21no—
43C4⋊C4×D163C2211yescharacteristic
44C22⋊C4×D163—21no—
45#19104669C2211yescharacteristic, complemented
46#19104668C2211yescharacteristic, complemented
47#19104666C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104659C2211yescharacteristic, complemented
51#19104658C2211yescharacteristic, complemented
52#19104657C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic, complemented
54C2×Q16×C163C2211yescharacteristic, complemented
55C2×M16×C163C2211yescharacteristic
56C22⋊Q8×C163C2211yescharacteristic
57C22⋊Q8×C163C2211yescharacteristic
58#106×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#103×C163C2211yescharacteristic, complemented
61C23×D163D411yescharacteristic
62#14579700—22no—
63#14579700—22no—
64C2×Dic326—21no—
65C2×Dic326C2311yescharacteristic
66Q8×D163—22no—
67Q8×D163—22no—
68C2×D652—21no—
69C2×D652—21no—
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C22×Dic163D411yescharacteristic
73C2×C4×D163—22no—
74C2×C4×D163—22no—
75C2×C4×D163—21no—
76C2×C4×D163—21no—
77C2×C4×D163—21no—
78C2×C4×D163D411yescharacteristic
79C2×C4×D163D411yescharacteristic
80C2×C4×D163C2311yescharacteristic
81C2×C4×D163C2311yescharacteristic
82#14579689—22no—
83#14579688—22no—
84#14579686—22no—
85#14579683—22no—
86#14579682—22no—
87#14579682D414yescomplemented
88C2×C163⋊C8C2311yescharacteristic
89C2×C163⋊C8C2311yescharacteristic
90C8×D163—22no—
91C8×D163—22no—
92C4⋊Dic163—21no—
93C4⋊Dic163C2311yescharacteristic
94C4⋊Dic163C2311yescharacteristic
95Dic163⋊C4—21no—
96Dic163⋊C4—21no—
97Dic163⋊C4—21no—
98Dic163⋊C4—21no—
99C22⋊Dic163—21no—
100(C2×D163)⋊C4—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106C4×Dic163—21no—
107C2×Q8×C163C2311yescharacteristic
108C2×Q8×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110Q16×C163—22no—
111M16×C163—22no—
112C2×C8×C163C2311yescharacteristic
113C2×C8×C163C2311yescharacteristic
114C4⋊C4×C163—21no—
115C4⋊C4×C163—21no—
116C4⋊C4×C163C2311yescharacteristic
117C4⋊C4×C163C2311yescharacteristic
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22×D163—22no—
121C22×D163—22no—
122C22×D163—21no—
123C22×D163—41no—
124C22×D163C2×D411yescharacteristic
125C22×D163C2×D411yescharacteristic
126Dic326—21no—
127Dic326—21no—
128Dic326—42no—
129D652—21no—
130D652—21no—
131D652—42no—
132D652—42no—
133C22⋊D163—42no—
134C2×Dic163—22no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163C2×D411yescharacteristic
139C2×Dic163C2×D411yescharacteristic
140C4×D163—22no—
141C4×D163—21no—
142C4×D163—21no—
143C4×D163—22no—
144C4×D163—42no—
145C4×D163—42no—
146C4×D163—42no—
147C4×D163C2×D412yes—
148C4×D163#4112yes—
149C163⋊C8—21no—
150C163⋊C8C2×D412yes—
151C23×C163C2×D411yescharacteristic
152Q8×C163—21no—
153Q8×C163—21no—
154Q8×C163—21no—
155Q8×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163C2×D411yescharacteristic
163C2×C4×C163C2411yescharacteristic
164C8×C163—21no—
165C8×C163C2×D412yes—
166C22×C163—21no—
167C22×C163—21no—
168C22×C163—21no—
169C22×C163C22×D411yescharacteristic
170C2×D163—21no—
171C2×D163—21no—
172C2×D163—21no—
173C2×D163—21no—
174C2×D163—44no—
175C2×D163—41no—
176C2×D163—41no—
177C2×D163—41no—
178C2×D163#12112yes—
179C2×D163#13712yescomplemented
180C4×C163—21no—
181C4×C163—41no—
182C4×C163—41no—
183C4×C163—41no—
184C4×C163—41no—
185C4×C163C22×D411yescharacteristic
186C4×C163C2×#4111yescharacteristic
187Dic163—21no—
188Dic163—41no—
189Dic163—41no—
190Dic163—41no—
191Dic163#12112yes—
192C2×C163—41no—
193C2×C163C2×#12111yescharacteristic
194C2×C163C2×#13711yescharacteristic
195C2×C163#57511yescharacteristic
196D163—22no—
197D163—22no—
198D163—81no—
199C163#305611yescharacteristic, complemented, Sylow
200#3056—1631nomaximal, Sylow
201C2×#137—1631no—
202C22×M16—1631no—
203#525—1631no—
204C2×C22⋊Q8—1631no—
205#470—1631no—
206#470D16311yescharacteristic, complemented
207#468—1631no—
208#468—1631no—
209#467—1631no—
210#467—1631no—
211#465—1631no—
212#465—1631no—
213#428—1631no—
214#419—1631no—
215#417—1631no—
216C2×#41—1631no—
217C22×Q8—1631no—
218C23×C4—1631no—
219#137—3264no—
220C2×Q16—1631no—
221C2×Q16C2×D16311yescharacteristic, complemented
222C2×SD16—1631no—
223C2×SD16—1631no—
224C2×M16—3264no—
225C2×M16—1631no—
226C2×M16—1631no—
227C2×M16—1631no—
228C2×M16C2×D16311yescharacteristic
229C22×C8—1631no—
230C22×C8—1631no—
231#123—3261no—
232C22⋊Q8—3261no—
233C22⋊Q8—3261no—
234C22⋊Q8—1631no—
235C22⋊Q8—1631no—
236C22⋊Q8—1631no—
237C22⋊Q8—1631no—
238C22⋊Q8C2×D16311yescharacteristic
239C22⋊Q8C2×D16311yescharacteristic
240#121—1631no—
241#121—1631no—
242#120—3261no—
243C4×D4—3261no—
244C4×D4—3261no—
245#117—1631no—
246C2×C4⋊C4—3261no—
247C2×C4⋊C4—1631no—
248C2×C22⋊C4—3261no—
249#107—1631no—
250#107—1631no—
251#106—1631no—
252#106C2×D16311yescharacteristic, complemented
253#103—1631no—
254#103—1631no—
255#103—1631no—
256#103—1631no—
257#103C2×D16311yescharacteristic, complemented
258#103C2×D16311yescharacteristic, complemented
259#102—1631no—
260#102—1631no—
261C24—1631no—
262#41—3262no—
263#41—3262no—
264C2×Q8—3261no—
265C2×Q8—3262no—
266C2×Q8—3262no—
267C2×Q8—1631no—
268C2×Q8C22×D16311yescharacteristic
269C2×Q8C22×D16311yescharacteristic
270C2×D4—3261no—
271C2×D4—3261no—
272C2×D4—3261no—
273C2×D4—1631no—
274C22×C4—3262no—
275C22×C4—3262no—
276C22×C4—3261no—
277C22×C4—3261no—
278C22×C4—3261no—
279C22×C4—1631no—
280C22×C4—1631no—
281C22×C4—1631no—
282C22×C4—1631no—
283C22×C4—1631no—
284C22×C4C22×D16311yescharacteristic
285Q16—22no—
286Q16—3262no—
287SD16—3262no—
288SD16—3262no—
289M16—22no—
290M16—3262no—
291M16—3262no—
292M16—1634no—
293C2×C8—3262no—
294C2×C8—3262no—
295C2×C8—1631no—
296C2×C8—1631no—
297C2×C8C22×D16311yescharacteristic
298C2×C8C22×D16311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—3261no—
302C4⋊C4—3261no—
303C4⋊C4—3261no—
304C4⋊C4—3261no—
305C4⋊C4—3261no—
306C4⋊C4—1631no—
307C4⋊C4—1631no—
308C4⋊C4C22×D16311yescharacteristic
309C4⋊C4C22×D16311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3261no—
313C22⋊C4—3261no—
314C22⋊C4—3261no—
315C22⋊C4—3261no—
316C22⋊C4—3261no—
317C22⋊C4—3261no—
318C22⋊C4—3261no—
319C42—3261no—
320C23—3262no—
321C23—3262no—
322C23—3261no—
323C23—6521no—
324C23—1631no—
325C23—1631no—
326C23D4×D16311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—3261no—
332Q8—3261no—
333Q8—6522no—
334D4—3261no—
335D4—3261no—
336D4—6522no—
337D4—6522no—
338D4—6522no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—3262no—
346C2×C4—3262no—
347C2×C4—3262no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—3261no—
351C2×C4—3261no—
352C2×C4—3261no—
353C2×C4—6522no—
354C2×C4—6522no—
355C2×C4—6522no—
356C2×C4—1631no—
357C2×C4—1631no—
358C2×C4—1632no—
359C2×C4—1632no—
360C2×C4D4×D16311yescharacteristic
361C2×C4C23×D16311yescharacteristic
362C8—21no—
363C8—3261no—
364C8—1632no—
365C8D4×D16312yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—3261no—
370C22—3261no—
371C22—3261no—
372C22—3261no—
373C22—6524no—
374C22—6521no—
375C22—6521no—
376C22—6521no—
377C22—1632no—
378C22—1632no—
379C22C2×D4×D16311yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3261no—
386C4—6521no—
387C4—6521no—
388C4—6521no—
389C4—1632no—
390C4C2×D4×D16311yescharacteristic
391C4#41×D16311yescharacteristic
392C2—41no—
393C2—13041no—
394C2—3262no—
395C2—3262no—
396C2#121×D16311yescharacteristic
397C2#137×D16311yescharacteristic
398C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#470×C163C2
socleC22×C163C22×D4
prime radical#23782178C2
hypercenter#470D163
hypocenterC163#3056
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#19104767C2semidirect
2. C2×M16×D163C2semidirect
4. C22⋊Q8×D163C2semidirect
5. #23781921C2semidirect
6. #23781913C2semidirect
7. #23781911C2semidirect
8. #23781905C2semidirect
9. #23781903C2semidirect
10. #23781897C2semidirect
11. #23781895C2semidirect
12. #23781704C2semidirect
13. #23781672C2semidirect
14. #23781665C2semidirect
15. #470×C163C2semidirect
20. C2×#14579689C22semidirect
21. C2×#14579688C22semidirect
22. C2×#14579686C22semidirect
45. #19104669C22semidirect
46. #19104668C22semidirect
47. #19104666C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104659C22semidirect
51. #19104658C22semidirect
52. #19104657C22semidirect
53. #19104655C22semidirect
54. C2×Q16×C163C22semidirect
58. #106×C163C22semidirect
59. #103×C163C22semidirect
60. #103×C163C22semidirect
87. #14579682D4semidirect
179. C2×D163#137semidirect
199. C163#3056semidirect
206. #470D163semidirect
221. C2×Q16C2×D163semidirect
252. #106C2×D163semidirect
257. #103C2×D163semidirect
258. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047672
C2×M16×D1632
#237821782
C22⋊Q8×D1632
#237819212
#237819132
#237819112
#237819052
#237819032
#237818972
#237818952
#237817042
#237816722
#237816652
#470×C1632
#3056163