Finite Group Databasebeta

← #35227314

Subgroups — #35227314

Order 20864.

340 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782390C211yesmaximal, characteristic, complemented
2#23782371C211yesmaximal, characteristic
3#23782032C211yesmaximal, characteristic, complemented
4#23782014C211yesmaximal, characteristic, complemented
5#23781974C211yesmaximal, characteristic, complemented
6#23781969C211yesmaximal, characteristic, complemented
7#23781958C211yesmaximal, characteristic, complemented
8#23781869C211yesmaximal, characteristic
9#23781869C211yesmaximal, characteristic
10#23781867C211yesmaximal, characteristic
11#23781793C211yesmaximal, characteristic, complemented
12#23781768C211yesmaximal, characteristic
13#23781767C211yesmaximal, characteristic
14#23781750C211yesmaximal, characteristic
15#491×C163C211yesmaximal, characteristic, complemented
16C2×#14579688C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19#19104734C2211yescharacteristic
20#19104733C2211yescharacteristic
21#19104733C2211yescharacteristic
22#19104726—21no—
23#19104726C2211yescharacteristic
24#19104725—21no—
25#19104725—21no—
26#19104715—21no—
27#19104713—21no—
28#19104713—21no—
29Q8×Dic163C2211yescharacteristic
30Q8×Dic163C2211yescharacteristic
31C4×Dic326C2211yescharacteristic
32C4×Dic326C2211yescharacteristic
33#19104700—21no—
34C4×D652C2211yescharacteristic
35#19104697C2211yescharacteristic
36#19104697C2211yescharacteristic
37C4⋊C4×D163—21no—
38#19104669C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43Dic163⋊C8C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C4⋊Q8×C163C2211yescharacteristic
53#125×C163C2211yescharacteristic
54#103×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326D411yescharacteristic
61C2×Dic326C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C4×D163—21no—
64C2×C4×D163—21no—
65C2×C4×D163—21no—
66#14579688—22no—
67#14579686—22no—
68#14579686—22no—
69C2×C163⋊C8C2311yescharacteristic
70C2×C163⋊C8C2311yescharacteristic
71C4⋊Dic163—21no—
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163D411yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4D411yescharacteristic
85Dic163⋊C4D411yescharacteristic
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163C2311yescharacteristic
93C4×Dic163C2311yescharacteristic
94C2×Q8×C163C2311yescharacteristic
95C2×Q8×C163C2311yescharacteristic
96C2×C8×C163C2311yescharacteristic
97C2×C8×C163C2311yescharacteristic
98C4⋊C4×C163—21no—
99C4⋊C4×C163—21no—
100C4⋊C4×C163—21no—
101C4⋊C4×C163—21no—
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C42×C163C2311yescharacteristic
105C22×D163—21no—
106Dic326—22no—
107Dic326—21no—
108Dic326—21no—
109Dic326—81no—
110D652—21no—
111D652#4112yes—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163C2×D411yescharacteristic
116C2×Dic163C2×D411yescharacteristic
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C4×D163—42no—
120C4×D163—81no—
121C4×D163—81no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124Q8×C163—21no—
125Q8×C163—21no—
126Q8×C163—21no—
127Q8×C163—21no—
128C2×C4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163C2×D411yescharacteristic
133C2×C4×C163C2×D411yescharacteristic
134C2×C4×C163C2411yescharacteristic
135C8×C163—21no—
136C8×C163—21no—
137C22×C163C22×D411yescharacteristic
138C2×D163—22no—
139C2×D163—41no—
140C2×D163—41no—
141C4×C163—21no—
142C4×C163—21no—
143C4×C163—41no—
144C4×C163—41no—
145C4×C163—41no—
146C4×C163—41no—
147C4×C163C2×#4111yescharacteristic
148C4×C1632−1+411yescharacteristic
149Dic163—21no—
150Dic163—21no—
151Dic163—21no—
152Dic163—41no—
153Dic163—41no—
154Dic163—41no—
155Dic163#13712yes—
156C2×C163#54811yescharacteristic
157C2×C163C2×#13711yescharacteristic
158C2×C163#57711yescharacteristic
159D163—42no—
160C163#337211yescharacteristic, complemented, Sylow
161#3372—1631nomaximal, Sylow
162#557—1631no—
163#554—1631no—
164#491—1631no—
165#491D16311yescharacteristic, complemented
166#487—1631no—
167#479—1631no—
168#479—1631no—
169#477—1631no—
170#461—1631no—
171#461—1631no—
172#461—1631no—
173#446—1631no—
174#440—1631no—
175#440—1631no—
176C4×SD16—1631no—
177C2×SD16—1631no—
178C2×SD16—1631no—
179C2×SD16—1631no—
180C4⋊Q8—1631no—
181C4⋊Q8—1631no—
182C4⋊Q8—1631no—
183C4⋊Q8C2×D16311yescharacteristic
184#125—3261no—
185#125—3261no—
186#125—3261no—
187#125—1631no—
188#125C2×D16311yescharacteristic
189C22⋊Q8—3261no—
190C22⋊Q8—3261no—
191C22⋊Q8—3261no—
192C4×Q8—3261no—
193C4×Q8—1631no—
194C4×Q8—1631no—
195C4×Q8—1631no—
196C4×Q8—1631no—
197C4×D4—1631no—
198C4×D4—1631no—
199C4×D4—1631no—
200C2×C4⋊C4—3261no—
201#107—1631no—
202#107—1631no—
203#106—1631no—
204C4⋊C8—1631no—
205C4⋊C8—1631no—
206C4⋊C8—1631no—
207#103—1631no—
208#103—1631no—
209#103—1631no—
210#103—1631no—
211#103—1631no—
212#103C2×D16311yescharacteristic
213#103C2×D16311yescharacteristic
214#103C2×D16311yescharacteristic
215#103C2×D16311yescharacteristic
216#102—1631no—
217#97—1631no—
218#97C2×D16311yescharacteristic
219C4×C8—1631no—
220C2×Q8—3261no—
221C2×Q8—1631no—
222C2×Q8—1631no—
223C2×Q8C22×D16311yescharacteristic
224C2×Q8C22×D16311yescharacteristic
225C2×D4—1631no—
226C22×C4—3261no—
227C22×C4—3261no—
228C22×C4—3261no—
229SD16—3262no—
230SD16—3262no—
231SD16—3262no—
232C2×C8—1631no—
233C2×C8—1631no—
234C2×C8C22×D16311yescharacteristic
235C2×C8C22×D16311yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—3261no—
241C4⋊C4—3261no—
242C4⋊C4—3261no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—3261no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—1631no—
251C4⋊C4—1631no—
252C4⋊C4—1631no—
253C4⋊C4—1631no—
254C4⋊C4—1631no—
255C4⋊C4C22×D16311yescharacteristic
256C4⋊C4C22×D16311yescharacteristic
257C22⋊C4—3261no—
258C22⋊C4—3261no—
259C22⋊C4—3261no—
260C42—3261no—
261C42—3261no—
262C42—3261no—
263C42—1631no—
264C42—1631no—
265C42C22×D16311yescharacteristic
266C23—3261no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—21no—
271Q8—13041no—
272Q8—3262no—
273Q8—3261no—
274Q8—3261no—
275D4—3261no—
276D4—1632no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—13041no—
282C2×C4—13041no—
283C2×C4—3261no—
284C2×C4—3261no—
285C2×C4—3261no—
286C2×C4—6522no—
287C2×C4—1631no—
288C2×C4—1631no—
289C2×C4—1631no—
290C2×C4—1631no—
291C2×C4D4×D16311yescharacteristic
292C2×C4D4×D16311yescharacteristic
293C2×C4C23×D16311yescharacteristic
294C8—21no—
295C8—21no—
296C8—3261no—
297C8—3261no—
298C22—3262no—
299C22—6521no—
300C22—6521no—
301C22C2×D4×D16311yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—3261no—
309C4—3261no—
310C4—3261no—
311C4—6521no—
312C4—6521no—
313C4—6521no—
314C4—1632no—
315C4#41×D16311yescharacteristic
316C4#1910478211yescharacteristic
317C2—6522no—
318C2#2378232311yescharacteristic
319C2#137×D16311yescharacteristic
320C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#491×C163C2
socleC22×C163C22×D4
prime radicalC2×D652C23
hypercenter#491D163
hypocenterC163#3372
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #23782390C2semidirect
3. #23782032C2semidirect
4. #23782014C2semidirect
5. #23781974C2semidirect
6. #23781969C2semidirect
7. #23781958C2semidirect
11. #23781793C2semidirect
15. #491×C163C2semidirect
160. C163#3372semidirect
165. #491D163semidirect

Maximal subgroups

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

subgroupindex
#237823902
#237823712
#237820322
#237820142
#237819742
#237819692
#237819582
#237818692
#237818672
#237817932
#237817682
#237817672
#237817502
#491×C1632
#3372163