Finite Group Databasebeta

← #35225102

Subgroups — #35225102

Order 20864.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782202C211yesmaximal, characteristic, complemented
2#23782197C211yesmaximal, characteristic, complemented
3#23782065C211yesmaximal, characteristic, complemented
4#23782046C211yesmaximal, characteristic, complemented
5#23782021C211yesmaximal, characteristic, complemented
6#23782018C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781900C211yesmaximal, characteristic
9#23781895C211yesmaximal, characteristic, complemented
10#23781891C211yesmaximal, characteristic, complemented
11C4×#14579689C211yesmaximal, characteristic, complemented
12C4×#14579688C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14C4×C8×D163C211yesmaximal, characteristic, complemented
15C4×SD16×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579688C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104730—21no—
22#19104728—21no—
23#19104726C2211yescharacteristic
24#19104726C2211yescharacteristic
25#19104724C2211yescharacteristic
26#19104722C2211yescharacteristic
27#19104720—21no—
28#19104717—21no—
29#19104715C2211yescharacteristic
30#19104711C2211yescharacteristic
31#19104710C2211yescharacteristic
32#19104706C2211yescharacteristic
33C4×Dic326C2211yescharacteristic
34C4×D652—21no—
35C4×D652C2211yescharacteristic
36C4×C22⋊D163—21no—
37#19104694—21no—
38#19104692—21no—
39C42×D163C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104668C2211yescharacteristic
42#19104667C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49C4×C163⋊C8C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C4×C8×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C2×D652—21no—
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C4×D163—21no—
64C2×C4×D163D411yescharacteristic
65C2×C4×D163D411yescharacteristic
66C2×C4×D163C2311yescharacteristic
67#14579689—22no—
68#14579688—22no—
69#14579684—22no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C8×D163—22no—
73C8×D163—22no—
74C4⋊Dic163—21no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92C4×Dic163—21no—
93C4×Dic163D411yescharacteristic
94C4×Dic163D411yescharacteristic
95C4×Dic163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99SD16×C163—22no—
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163—21no—
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C42×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163C2×D411yescharacteristic
110Dic326—21no—
111Dic326—21no—
112D652—21no—
113D652—21no—
114D652—42no—
115C22⋊D163—42no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C4×D163—22no—
123C4×D163—22no—
124C4×D163—21no—
125C4×D163—21no—
126C4×D163—42no—
127C163⋊C8—21no—
128C163⋊C8#4112yes—
129C23×C163—21no—
130Q8×C163—21no—
131Q8×C163—21no—
132D4×C163—21no—
133D4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—42no—
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2411yescharacteristic
141C8×C163—21no—
142C8×C163#4112yes—
143C22×C163—41no—
144C22×C163—41no—
145C22×C163—41no—
146C22×C163C22×D411yescharacteristic
147C2×D163—21no—
148C2×D163—21no—
149C2×D163—41no—
150C2×D163—41no—
151C2×D163—41no—
152C2×D163#13512yes—
153C4×C163—21no—
154C4×C163—41no—
155C4×C163—41no—
156C4×C163—41no—
157C4×C163#13512yes—
158C4×C163C2×#4111yescharacteristic
159C4×C163C2×#4111yescharacteristic
160Dic163—21no—
161Dic163—21no—
162Dic163—21no—
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166C2×C163—81no—
167C2×C163#53211yescharacteristic
168C2×C163C2×#13511yescharacteristic
169C2×C163C2×#13511yescharacteristic
170D163—22no—
171D163—81no—
172C163#316811yescharacteristic, complemented, Sylow
173#3168—1631nomaximal, Sylow
174#529—1631no—
175#529—1631no—
176#499—1631no—
177#495—1631no—
178#488—1631no—
179#488—1631no—
180#467—1631no—
181#466—1631no—
182#465—1631no—
183#465—1631no—
184C4×Q16—1631no—
185C4×SD16—1631no—
186C4×SD16D16311yescharacteristic, complemented
187C4×D8—1631no—
188C2×C4×C8—1631no—
189C2×Q16—1631no—
190C2×SD16—1631no—
191C2×SD16C2×D16311yescharacteristic
192C2×D8—1631no—
193C22×C8—1631no—
194C22×C8—1631no—
195#126—3261no—
196#126—3261no—
197#125—1631no—
198#125—1631no—
199#124—1631no—
200#124—1631no—
201#123—3261no—
202#123—3261no—
203C22⋊Q8—1631no—
204C22⋊Q8—1631no—
205#121—1631no—
206#121—1631no—
207C4×Q8—1631no—
208C4×Q8C2×D16311yescharacteristic
209C4×D4—3261no—
210C4×D4—3261no—
211C4×D4—1631no—
212C4×D4C2×D16311yescharacteristic
213#117—3261no—
214#117—3261no—
215C2×C42—1631no—
216#107—1631no—
217#107—1631no—
218#106—1631no—
219#106C2×D16311yescharacteristic
220#103—1631no—
221#103—1631no—
222#103—1631no—
223#103C2×D16311yescharacteristic
224#102—1631no—
225#102—1631no—
226#102—1631no—
227#102C2×D16311yescharacteristic
228C4×C8—1631no—
229C4×C8—1631no—
230C4×C8—1631no—
231C4×C8C2×D16311yescharacteristic
232C2×Q8—1631no—
233C2×Q8C22×D16311yescharacteristic
234C2×D4—3261no—
235C2×D4—3261no—
236C2×D4—1631no—
237C2×D4C22×D16311yescharacteristic
238C22×C4—21no—
239C22×C4—3261no—
240C22×C4—1631no—
241C22×C4—1631no—
242C22×C4—1631no—
243Q16—3262no—
244SD16—22no—
245SD16—3262no—
246D8—3262no—
247C2×C8—3262no—
248C2×C8—3262no—
249C2×C8—1631no—
250C2×C8—1631no—
251C2×C8C22×D16311yescharacteristic
252C2×C8C22×D16311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—3261no—
255C4⋊C4—3261no—
256C4⋊C4—3261no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—1631no—
262C4⋊C4—1631no—
263C4⋊C4C22×D16311yescharacteristic
264C4⋊C4C22×D16311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3261no—
267C22⋊C4—3261no—
268C22⋊C4—3261no—
269C22⋊C4—3261no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C42—21no—
276C42—3261no—
277C42—1631no—
278C42—1631no—
279C42—1631no—
280C42C22×D16311yescharacteristic
281C23—21no—
282C23—3261no—
283C23—1631no—
284Q8—21no—
285Q8—21no—
286Q8—3261no—
287Q8—3261no—
288D4—21no—
289D4—21no—
290D4—3261no—
291D4—3261no—
292D4—6522no—
293D4—6522no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—3262no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4—3262no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—6522no—
306C2×C4—1631no—
307C2×C4—1631no—
308C2×C4—1631no—
309C2×C4D4×D16311yescharacteristic
310C2×C4D4×D16311yescharacteristic
311C2×C4C23×D16311yescharacteristic
312C8—21no—
313C8—3261no—
314C8—1632no—
315C8#1457969912yes—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—3261no—
320C22—3261no—
321C22—6521no—
322C22—6521no—
323C22—6521no—
324C22—1632no—
325C22C2×D4×D16311yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—3261no—
331C4—3261no—
332C4—3261no—
333C4—6521no—
334C4—6521no—
335C4—6521no—
336C4#1910475612yes—
337C4#41×D16311yescharacteristic
338C4C2×#1457969911yescharacteristic
339C2—81no—
340C2—13041no—
341C2—3262no—
342C2#2378221211yescharacteristic
343C2#135×D16311yescharacteristic
344C2C2×#1910475611yescharacteristic
subgroupG/H
centerC2×C4D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC4×SD16×C163C2
socleC22×C163C22×D4
prime radical#23781900C2
hypercenterC4×SD16D163
hypocenterC163#3168
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #23782202C2semidirect
2. #23782197C2semidirect
3. #23782065C2semidirect
4. #23782046C2semidirect
5. #23782021C2semidirect
6. #23782018C2semidirect
7. #23781905C2semidirect
9. #23781895C2semidirect
10. #23781891C2semidirect
11. C4×#14579689C2semidirect
12. C4×#14579688C2semidirect
13. C4×#14579684C2semidirect
14. C4×C8×D163C2semidirect
15. C4×SD16×C163C2semidirect
172. C163#3168semidirect
186. C4×SD16D163semidirect

Maximal subgroups

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

subgroupindex
#237822022
#237821972
#237820652
#237820462
#237820212
#237820182
#237819052
#237819002
#237818952
#237818912
C4×#145796892
C4×#145796882
C4×#145796842
C4×C8×D1632
C4×SD16×C1632
#3168163