Finite Group Databasebeta

← #35227224

Subgroups — #35227224

Order 20864.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782392C211yesmaximal, characteristic, complemented
2#23782387C211yesmaximal, characteristic
3#23782022C211yesmaximal, characteristic
4#23782021C211yesmaximal, characteristic
5#23782014C211yesmaximal, characteristic, complemented
6#23781975C211yesmaximal, characteristic, complemented
7#23781944C211yesmaximal, characteristic
8#23781890C211yesmaximal, characteristic
9#23781888C211yesmaximal, characteristic
10#23781874C211yesmaximal, characteristic, complemented
11#23781789C211yesmaximal, characteristic, complemented
12#23781759C211yesmaximal, characteristic, complemented
13C4×#14579689C211yesmaximal, characteristic, complemented
14#23781752C211yesmaximal, characteristic
15#479×C163C211yesmaximal, characteristic, complemented
16C2×Dic652C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19#19104734—21no—
20#19104734C2211yescharacteristic
21#19104733C2211yescharacteristic
22#19104730—21no—
23#19104729—21no—
24#19104726C2211yescharacteristic
25#19104725—21no—
26#19104725—21no—
27#19104724C2211yescharacteristic
28#19104723C2211yescharacteristic
29#19104713—21no—
30C4×Dic326C2211yescharacteristic
31C4×Dic326C2211yescharacteristic
32#19104700—21no—
33#19104700C2211yescharacteristic
34#19104700C2211yescharacteristic
35#19104700C2211yescharacteristic
36#19104697C2211yescharacteristic
37#19104692—21no—
38#19104668C2211yescharacteristic
39Dic163⋊C8C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41#19104660C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104658C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C4×C163⋊C8C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52#125×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326D411yescharacteristic
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63C2×D652C2311yescharacteristic
64C2×C4×D163—21no—
65Dic652—22no—
66#14579689—22no—
67#14579688—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163—21no—
71C4⋊Dic163D411yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73Dic163⋊C4—21no—
74Dic163⋊C4—21no—
75Dic163⋊C4—21no—
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×Dic163—21no—
92C4×Dic163C2311yescharacteristic
93C4×Dic163C2311yescharacteristic
94C2×Q8×C163C2311yescharacteristic
95C2×C8×C163C2311yescharacteristic
96C2×C8×C163C2311yescharacteristic
97C4⋊C4×C163—21no—
98C4⋊C4×C163—21no—
99C4⋊C4×C163—21no—
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C42×C163—21no—
104C42×C163C2311yescharacteristic
105C22×D163—21no—
106Dic326—21no—
107Dic326—22no—
108Dic326—21no—
109Dic326—21no—
110Dic326—42no—
111Dic326#4112yes—
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—
126C2×C4×C163—21no—
127C2×C4×C163—21no—
128C2×C4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163C2×D411yescharacteristic
131C2×C4×C163C2×D411yescharacteristic
132C2×C4×C163C2411yescharacteristic
133C8×C163—21no—
134C8×C163—21no—
135C22×C163C22×D411yescharacteristic
136C2×D163—41no—
137C2×D163—41no—
138C2×D163—41no—
139C4×C163—21no—
140C4×C163—21no—
141C4×C163—41no—
142C4×C163—41no—
143C4×C163—41no—
144C4×C163—41no—
145C4×C163C2×#4111yescharacteristic
146C4×C1632−1+411yescharacteristic
147Dic163—22no—
148Dic163—21no—
149Dic163—21no—
150Dic163—21no—
151Dic163—41no—
152Dic163—41no—
153Dic163#13512yes—
154C2×C163#54811yescharacteristic
155C2×C163C2×#13511yescharacteristic
156C2×C163#57811yescharacteristic
157D163—81no—
158C163#336411yescharacteristic, complemented, Sylow
159#3364—1631nomaximal, Sylow
160#557—1631no—
161#556—1631no—
162#488—1631no—
163#488—1631no—
164#487—1631no—
165#479—1631no—
166#479D16311yescharacteristic, complemented
167#474—1631no—
168#464—1631no—
169#464—1631no—
170#462—1631no—
171C8×Q8—1631no—
172C4×Q16—1631no—
173C4×Q16—1631no—
174C4×SD16—1631no—
175C2×Q16—1631no—
176C2×Q16—1631no—
177C2×SD16—1631no—
178C4⋊Q8—3261no—
179C4⋊Q8—1631no—
180C4⋊Q8—1631no—
181#126—3261no—
182#126—3261no—
183#125—3261no—
184#125—3261no—
185#125—1631no—
186#125C2×D16311yescharacteristic
187#124—1631no—
188#124—1631no—
189C22⋊Q8—3261no—
190C4×Q8—3261no—
191C4×Q8—1631no—
192C4×Q8—1631no—
193C4×Q8—1631no—
194C4×Q8—1631no—
195C4×Q8—1631no—
196C4×Q8C2×D16311yescharacteristic
197C4×D4—1631no—
198#117—3261no—
199#107—1631no—
200#107C2×D16311yescharacteristic
201#106C2×D16311yescharacteristic
202C4⋊C8—1631no—
203C4⋊C8—1631no—
204C4⋊C8C2×D16311yescharacteristic
205#103—1631no—
206#103—1631no—
207#103—1631no—
208#103—1631no—
209#103—1631no—
210#103C2×D16311yescharacteristic
211#103C2×D16311yescharacteristic
212#102—1631no—
213#102—1631no—
214#102—1631no—
215C4×C8—1631no—
216C4×C8—1631no—
217C4×C8—1631no—
218C2×Q8—3261no—
219C2×Q8—1631no—
220C2×Q8—1631no—
221C2×Q8—1631no—
222C2×Q8C22×D16311yescharacteristic
223C2×D4—1631no—
224C22×C4—3261no—
225Q16—3262no—
226Q16—3262no—
227SD16—3262no—
228C2×C8—1631no—
229C2×C8—1631no—
230C2×C8C22×D16311yescharacteristic
231C2×C8C22×D16311yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—3261no—
236C4⋊C4—3261no—
237C4⋊C4—3261no—
238C4⋊C4—3261no—
239C4⋊C4—3261no—
240C4⋊C4—3261no—
241C4⋊C4—3261no—
242C4⋊C4—3261no—
243C4⋊C4—3261no—
244C4⋊C4—1631no—
245C4⋊C4—1631no—
246C4⋊C4—1631no—
247C4⋊C4—1631no—
248C4⋊C4C22×D16311yescharacteristic
249C4⋊C4C22×D16311yescharacteristic
250C4⋊C4C22×D16311yescharacteristic
251C22⋊C4—3261no—
252C22⋊C4—3261no—
253C22⋊C4—3261no—
254C22⋊C4—3261no—
255C22⋊C4—3261no—
256C42—21no—
257C42—3261no—
258C42—3261no—
259C42—3261no—
260C42—3261no—
261C42—1631no—
262C42—1631no—
263C42C22×D16311yescharacteristic
264C23—3261no—
265Q8—21no—
266Q8—21no—
267Q8—3262no—
268Q8—3261no—
269Q8—3261no—
270Q8—3261no—
271Q8—6522no—
272Q8—1632no—
273D4—3261no—
274D4—3261no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—3261no—
280C2×C4—3261no—
281C2×C4—3261no—
282C2×C4—6522no—
283C2×C4—1631no—
284C2×C4—1631no—
285C2×C4—1631no—
286C2×C4—1631no—
287C2×C4D4×D16311yescharacteristic
288C2×C4D4×D16311yescharacteristic
289C2×C4C23×D16311yescharacteristic
290C8—21no—
291C8—21no—
292C8—3261no—
293C8—3261no—
294C22—6521no—
295C22—6521no—
296C22—6521no—
297C22C2×D4×D16311yescharacteristic
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—3261no—
305C4—3261no—
306C4—3261no—
307C4—3262no—
308C4—6521no—
309C4—6521no—
310C4—1632no—
311C4#41×D16311yescharacteristic
312C4#1910478311yescharacteristic
313C2—13041no—
314C2#2378232511yescharacteristic
315C2#135×D16311yescharacteristic
316C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#479×C163C2
socleC22×C163C22×D4
prime radicalC2×D652C23
hypercenter#479D163
hypocenterC163#3364
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #23782392C2semidirect
5. #23782014C2semidirect
6. #23781975C2semidirect
10. #23781874C2semidirect
11. #23781789C2semidirect
12. #23781759C2semidirect
13. C4×#14579689C2semidirect
15. #479×C163C2semidirect
158. C163#3364semidirect
166. #479D163semidirect

Maximal subgroups

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

subgroupindex
#237823922
#237823872
#237820222
#237820212
#237820142
#237819752
#237819442
#237818902
#237818882
#237818742
#237817892
#237817592
C4×#145796892
#237817522
#479×C1632
#3364163