Finite Group Databasebeta

← #35225100

Subgroups — #35225100

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#23782199C211yesmaximal, characteristic, complemented
2#23782064C211yesmaximal, characteristic, complemented
3#23782047C211yesmaximal, characteristic, complemented
4#23782022C211yesmaximal, characteristic, complemented
5#23782019C211yesmaximal, characteristic, complemented
6#23781904C211yesmaximal, characteristic, complemented
7#23781899C211yesmaximal, characteristic
8#23781896C211yesmaximal, characteristic, complemented
9#23781892C211yesmaximal, characteristic, complemented
10C4×#14579689C211yesmaximal, characteristic, complemented
11C4×#14579686C211yesmaximal, characteristic, complemented
12C4×#14579687C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14C2×C4×C163⋊C8C211yesmaximal, characteristic, complemented
15#529×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C22×C163⋊C8C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22#19104729—21no—
23#19104727C2211yescharacteristic
24#19104723C2211yescharacteristic
25#19104718—21no—
26#19104714C2211yescharacteristic
27#19104709C2211yescharacteristic
28C4×Dic326C2211yescharacteristic
29C4×D652C2211yescharacteristic
30C4×C22⋊D163—21no—
31#19104692—21no—
32#19104668C2211yescharacteristic
33#19104668C2211yescharacteristic
34#19104666C2211yescharacteristic
35#19104666C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104658C2211yescharacteristic
39#19104658C2211yescharacteristic
40#19104656C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44C4×C163⋊C8C2211yescharacteristic
45C4×C163⋊C8C2211yescharacteristic
46C4×C163⋊C8C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48#126×C163—21no—
49#125×C163C2211yescharacteristic
50#124×C163C2211yescharacteristic
51#123×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53#121×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55C4×D4×C163—21no—
56C4×D4×C163C2211yescharacteristic
57#117×C163—21no—
58C2×C42×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C2×D652C2311yescharacteristic
61C2×C22⋊D163—21no—
62C2×C4×D163—21no—
63#14579689—22no—
64#14579687—22no—
65#14579686—22no—
66#14579684—22no—
67C2×C163⋊C8—22no—
68C2×C163⋊C8—22no—
69C2×C163⋊C8C2311yescharacteristic
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C22⋊Dic163—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84C4×Dic163—21no—
85C2×Q8×C163C2311yescharacteristic
86C2×D4×C163—21no—
87C2×D4×C163C2311yescharacteristic
88C22×C4×C163—21no—
89C22×C4×C163D411yescharacteristic
90C22×C4×C163D411yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C4⋊C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163C2311yescharacteristic
97C4⋊C4×C163C2311yescharacteristic
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C42×C163—21no—
104C42×C163D411yescharacteristic
105C42×C163D411yescharacteristic
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109Dic326—21no—
110Dic326—21no—
111D652—21no—
112D652—21no—
113C22⋊D163—42no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C4×D163—42no—
118C163⋊C8—21no—
119C163⋊C8—21no—
120C163⋊C8#4112yes—
121C163⋊C8#4112yes—
122C23×C163—21no—
123C23×C163C2×D411yescharacteristic
124Q8×C163—21no—
125Q8×C163—21no—
126D4×C163—21no—
127D4×C163—21no—
128D4×C163—42no—
129C2×C4×C163—22no—
130C2×C4×C163—22no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—42no—
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C22×C163—21no—
144C22×C163—21no—
145C22×C163—41no—
146C22×C163—41no—
147C22×C163—41no—
148C22×C163#13512yes—
149C22×C163C22×D411yescharacteristic
150C2×D163—41no—
151C2×D163—41no—
152C2×D163—41no—
153C4×C163—21no—
154C4×C163—21no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163#13512yes—
161C4×C163C2×#4111yescharacteristic
162C4×C163C2×#4111yescharacteristic
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166C2×C163—22no—
167C2×C163—81no—
168C2×C163#53211yescharacteristic
169C2×C163C2×#13511yescharacteristic
170C2×C163C2×#13511yescharacteristic
171D163—81no—
172C163#316811yescharacteristic, complemented, Sylow
173#3168—1631nomaximal, Sylow
174#529—1631no—
175#529D16311yescharacteristic, complemented
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×SD16—1631no—
187C4×D8—1631no—
188C2×C4×C8—1631no—
189C2×Q16—1631no—
190C2×SD16—1631no—
191C2×SD16—1631no—
192C2×D8—1631no—
193C22×C8—1631no—
194C22×C8—1631no—
195#126—21no—
196#126—3261no—
197#125—1631no—
198#125C2×D16311yescharacteristic
199#124—1631no—
200#124C2×D16311yescharacteristic
201#123—21no—
202#123—3261no—
203C22⋊Q8—1631no—
204C22⋊Q8C2×D16311yescharacteristic
205#121—1631no—
206#121C2×D16311yescharacteristic
207C4×Q8—1631no—
208C4×Q8C2×D16311yescharacteristic
209C4×D4—21no—
210C4×D4—3261no—
211C4×D4—1631no—
212C4×D4C2×D16311yescharacteristic
213#117—21no—
214#117—3261no—
215C2×C42C2×D16311yescharacteristic
216#107—1631no—
217#107—1631no—
218#106—1631no—
219#106—1631no—
220#103—1631no—
221#103—1631no—
222#103—1631no—
223#103—1631no—
224#102—1631no—
225#102—1631no—
226#102—1631no—
227#102—1631no—
228C4×C8—1631no—
229C4×C8—1631no—
230C4×C8—1631no—
231C4×C8—1631no—
232C2×Q8—1631no—
233C2×Q8C22×D16311yescharacteristic
234C2×D4—21no—
235C2×D4—3261no—
236C2×D4—1631no—
237C2×D4C22×D16311yescharacteristic
238C22×C4—21no—
239C22×C4—3261no—
240C22×C4C22⋊D16311yescharacteristic
241C22×C4C22⋊D16311yescharacteristic
242C22×C4C22×D16311yescharacteristic
243Q16—3262no—
244SD16—3262no—
245SD16—3262no—
246D8—3262no—
247C2×C8—3262no—
248C2×C8—3262no—
249C2×C8—1631no—
250C2×C8—1631no—
251C2×C8—1631no—
252C2×C8—1631no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
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—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C42—21no—
276C42—3261no—
277C42C22⋊D16311yescharacteristic
278C42C22⋊D16311yescharacteristic
279C42C22×D16311yescharacteristic
280C42C22×D16311yescharacteristic
281C23—21no—
282C23—3261no—
283C23C2×C22⋊D16311yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—3261no—
287Q8—3261no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—3261no—
292D4—3261no—
293D4—6522no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—6522no—
306C2×C4C2×C22⋊D16311yescharacteristic
307C2×C4C2×C22⋊D16311yescharacteristic
308C2×C4C2×C22⋊D16311yescharacteristic
309C2×C4C2×C22⋊D16311yescharacteristic
310C2×C4C2×C22⋊D16311yescharacteristic
311C2×C4C23×D16311yescharacteristic
312C8—3261no—
313C8—3261no—
314C8—1632no—
315C8—1632no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—6521no—
322C22—6521no—
323C22—6521no—
324C22#1910475512yes—
325C22C22×C22⋊D16311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—6521no—
334C4—6521no—
335C4—6521no—
336C4#1910475512yes—
337C4#41×D16311yescharacteristic
338C4#41×D16311yescharacteristic
339C2—22no—
340C2—81no—
341C2—13041no—
342C2#2378221111yescharacteristic
343C2C2×#1910475511yescharacteristic
344C2C2×#1910475511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#529×C163C2
socleC22×C163C22×D4
prime radical#23781899C2
hypercenter#529D163
hypocenterC163#3168
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #23782199C2semidirect
2. #23782064C2semidirect
3. #23782047C2semidirect
4. #23782022C2semidirect
5. #23782019C2semidirect
6. #23781904C2semidirect
8. #23781896C2semidirect
9. #23781892C2semidirect
10. C4×#14579689C2semidirect
11. C4×#14579686C2semidirect
12. C4×#14579687C2semidirect
13. C4×#14579684C2semidirect
14. C2×C4×C163⋊C8C2semidirect
15. #529×C163C2semidirect
172. C163#3168semidirect
175. #529D163semidirect

Maximal subgroups

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

subgroupindex
#237821992
#237820642
#237820472
#237820222
#237820192
#237819042
#237818992
#237818962
#237818922
C4×#145796892
C4×#145796862
C4×#145796872
C4×#145796842
C2×C4×C163⋊C82
#529×C1632
#3168163