Finite Group Databasebeta

← #35227277

Subgroups — #35227277

Order 20864.

338 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
1Q8×Dic326C211yesmaximal, characteristic, complemented
2#23782387C211yesmaximal, characteristic
3#23782026C211yesmaximal, characteristic
4#23782025C211yesmaximal, characteristic
5#23782014C211yesmaximal, characteristic, complemented
6#23781954C211yesmaximal, characteristic
7#23781950C211yesmaximal, characteristic, complemented
8#23781949C211yesmaximal, characteristic, complemented
9#23781889C211yesmaximal, characteristic
10#23781887C211yesmaximal, characteristic
11#23781873C211yesmaximal, characteristic, complemented
12#23781793C211yesmaximal, characteristic, complemented
13#23781769C211yesmaximal, characteristic, complemented
14#23781750C211yesmaximal, characteristic
15#441×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18#19104735—21no—
19#19104735C2211yescharacteristic
20#19104734—21no—
21#19104734—21no—
22#19104734C2211yescharacteristic
23#19104734C2211yescharacteristic
24#19104733C2211yescharacteristic
25#19104730—21no—
26#19104729—21no—
27#19104724C2211yescharacteristic
28#19104723C2211yescharacteristic
29#19104713—21no—
30Q8×Dic163C2211yescharacteristic
31Q8×Dic163C2211yescharacteristic
32C4×Dic326—21no—
33C4×Dic326C2211yescharacteristic
34#19104700C2211yescharacteristic
35#19104697C2211yescharacteristic
36#19104692—21no—
37#19104666C2211yescharacteristic
38#19104666C2211yescharacteristic
39C4⋊(C163⋊C8)C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104655C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×Q16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×Dic326D411yescharacteristic
63C2×Dic326C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C4×D163—21no—
66#14579689—22no—
67#14579686—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163—21no—
71C4⋊Dic163—21no—
72C4⋊Dic163—21no—
73C4⋊Dic163D411yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4D411yescharacteristic
82Dic163⋊C4D411yescharacteristic
83(C2×D163)⋊C4—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93C2×Q8×C163C2311yescharacteristic
94C2×Q8×C163C2311yescharacteristic
95Q16×C163—22no—
96C2×C8×C163C2311yescharacteristic
97C2×C8×C163C2311yescharacteristic
98C4⋊C4×C163—21no—
99C4⋊C4×C163—21no—
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C42×C163—21no—
103C42×C163—21no—
104C42×C163C2311yescharacteristic
105C22×D163—21no—
106Dic326—22no—
107Dic326—21no—
108Dic326—21no—
109Dic326—81no—
110Dic326—81no—
111Dic326—81no—
112D652—21no—
113D652—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C4×D163—42no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124Q8×C163—21no—
125Q8×C163—21no—
126Q8×C163—21no—
127Q8×C163#4112yes—
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—41no—
139C2×D163—41no—
140C2×D163—41no—
141C4×C163—22no—
142C4×C163—21no—
143C4×C163—21no—
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—81no—
160C163#336911yescharacteristic, complemented, Sylow
161#3369—1631nomaximal, Sylow
162Q82—1631no—
163#556—1631no—
164#489—1631no—
165#489—1631no—
166#487—1631no—
167#476—1631no—
168#475—1631no—
169#475—1631no—
170#464—1631no—
171#464—1631no—
172#462—1631no—
173#446—1631no—
174#441—1631no—
175#441D16311yescharacteristic, complemented
176C4×SD16—1631no—
177C2×Q16—1631no—
178C2×Q16C2×D16311yescharacteristic
179C2×SD16—1631no—
180C4⋊Q8—3261no—
181C4⋊Q8—3261no—
182C4⋊Q8—3261no—
183C4⋊Q8—1631no—
184C4⋊Q8—1631no—
185C4⋊Q8—1631no—
186C4⋊Q8—1631no—
187#126—3261no—
188#126—3261no—
189#124—1631no—
190#124—1631no—
191C22⋊Q8—3261no—
192C4×Q8—3261no—
193C4×Q8—1631no—
194C4×Q8—1631no—
195C4×Q8—1631no—
196C4×Q8—1631no—
197C4×Q8C2×D16311yescharacteristic
198C4×Q8C2×D16311yescharacteristic
199C4×D4—1631no—
200#117—3261no—
201#106—1631no—
202#106—1631no—
203#106C2×D16311yescharacteristic
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#102—1631no—
215#102—1631no—
216#102—1631no—
217#97—1631no—
218#97C2×D16311yescharacteristic
219C4×C8—1631no—
220C2×Q8—3261no—
221C2×Q8—3261no—
222C2×Q8—3261no—
223C2×Q8—1631no—
224C2×Q8—1631no—
225C2×Q8C22×D16311yescharacteristic
226C2×Q8C22×D16311yescharacteristic
227C2×D4—1631no—
228C22×C4—3261no—
229Q16—22no—
230Q16—3262no—
231SD16—3262no—
232C2×C8—1631no—
233C2×C8—1631no—
234C2×C8C22×D16311yescharacteristic
235C2×C8C22×D16311yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—3261no—
239C4⋊C4—3261no—
240C4⋊C4—3261no—
241C4⋊C4—3261no—
242C4⋊C4—3261no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—1631no—
247C4⋊C4—1631no—
248C4⋊C4—1631no—
249C4⋊C4—1631no—
250C4⋊C4—1631no—
251C4⋊C4C22×D16311yescharacteristic
252C4⋊C4C22×D16311yescharacteristic
253C22⋊C4—3261no—
254C22⋊C4—3261no—
255C22⋊C4—3261no—
256C22⋊C4—3261no—
257C22⋊C4—3261no—
258C42—21no—
259C42—21no—
260C42—3261no—
261C42—3261no—
262C42—3261no—
263C42—1631no—
264C42—1631no—
265C42C22×D16311yescharacteristic
266C23—3261no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—13041no—
271Q8—13041no—
272Q8—13041no—
273Q8—3262no—
274Q8—3261no—
275Q8—3261no—
276Q8#1457969912yes—
277D4—3261no—
278D4—3261no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
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—6521no—
299C22—6521no—
300C22—6521no—
301C22C2×D4×D16311yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—3261no—
309C4—3261no—
310C4—3261no—
311C4—6521no—
312C4—6521no—
313C4—6521no—
314C4—1632no—
315C4C2×#1457969911yescharacteristic
316C4#1910478211yescharacteristic
317C2—13041no—
318C2#2378232411yescharacteristic
319C2#137×D16311yescharacteristic
320C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#441×C163C2
socleC22×C163C22×D4
prime radicalC2×D652C23
hypercenter#441D163
hypocenterC163#3369
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. Q8×Dic326C2semidirect
5. #23782014C2semidirect
7. #23781950C2semidirect
8. #23781949C2semidirect
11. #23781873C2semidirect
12. #23781793C2semidirect
13. #23781769C2semidirect
15. #441×C163C2semidirect
160. C163#3369semidirect
175. #441D163semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3262
#237823872
#237820262
#237820252
#237820142
#237819542
#237819502
#237819492
#237818892
#237818872
#237818732
#237817932
#237817692
#237817502
#441×C1632
#3369163