Finite Group Databasebeta

← #35224832

Subgroups — #35224832

Order 20864.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782275C211yesmaximal, characteristic
2#23782142C211yesmaximal, characteristic
3#23782006C211yesmaximal, characteristic
4#23781975C211yesmaximal, characteristic, complemented
5#23781971C211yesmaximal, characteristic, complemented
6#23781874C211yesmaximal, characteristic, complemented
7#23781874C211yesmaximal, characteristic, complemented
8#23781823C211yesmaximal, characteristic
9#23781820C211yesmaximal, characteristic
10#23781771C211yesmaximal, characteristic, complemented
11#23781769C211yesmaximal, characteristic, complemented
12#23781759C211yesmaximal, characteristic, complemented
13#23781757C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic
15#485×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C2×Dic652C2211yescharacteristic
18C2×Dic652C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×#14579689C2211yescharacteristic
21#19104733C2211yescharacteristic
22#19104728—21no—
23#19104728—21no—
24#19104726—21no—
25#19104726C2211yescharacteristic
26#19104725—21no—
27#19104714C2211yescharacteristic
28Q8×Dic163C2211yescharacteristic
29C4×Dic326—21no—
30C4×Dic326—21no—
31#19104700C2211yescharacteristic
32#19104700C2211yescharacteristic
33#19104700C2211yescharacteristic
34#19104700C2211yescharacteristic
35#19104700C2211yescharacteristic
36#19104694C2211yescharacteristic
37#19104691—21no—
38#19104691—21no—
39#19104668C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C22⋊Q8×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×Dic326—22no—
60C2×Dic326—22no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63C2×Dic326C2311yescharacteristic
64C2×Dic326C2311yescharacteristic
65C22×Dic163—21no—
66Dic652—22no—
67Dic652—22no—
68#14579689—22no—
69#14579689—22no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163D411yescharacteristic
75C4⋊Dic163D411yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163D411yescharacteristic
88C22⋊Dic163D411yescharacteristic
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C4⋊C4×C163—21no—
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163—21no—
108Dic326—24no—
109Dic326—21no—
110Dic326—21no—
111Dic326—21no—
112Dic326—21no—
113Dic326#4112yes—
114Dic326#4112yes—
115C2×Dic163—24no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C163⋊C8—21no—
125C163⋊C8—21no—
126C23×C163C2×D411yescharacteristic
127Q8×C163—21no—
128Q8×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2411yescharacteristic
137C8×C163—21no—
138C8×C163—21no—
139C22×C163—21no—
140C22×C163—21no—
141C22×C163—21no—
142C22×C163C22×D411yescharacteristic
143C4×C163—21no—
144C4×C163—41no—
145C4×C163—41no—
146C4×C163—41no—
147C4×C163—41no—
148C4×C163C2×#4111yescharacteristic
149C4×C163C2×#4111yescharacteristic
150Dic163—22no—
151Dic163—22no—
152Dic163—21no—
153Dic163—21no—
154Dic163—21no—
155Dic163—21no—
156Dic163—41no—
157Dic163—41no—
158C2×C163—41no—
159C2×C163#52511yescharacteristic
160C2×C163#57811yescharacteristic
161C2×C163#57811yescharacteristic
162C163#312711yescharacteristic, complemented, Sylow
163#3127—1631nomaximal, Sylow
164#541—1631no—
165#519—1631no—
166#485—1631no—
167#485D16311yescharacteristic
168#479—1631no—
169#479—1631no—
170#462—1631no—
171#462—1631no—
172#451—1631no—
173#451—1631no—
174#441—1631no—
175#441—1631no—
176C4×Q16—1631no—
177C4×Q16—1631no—
178#433—1631no—
179C22×Q8—1631no—
180C2×Q16—1631no—
181C2×Q16—1631no—
182C2×Q16—1631no—
183C2×Q16—1631no—
184C4⋊Q8—1631no—
185#126—3261no—
186#126—3261no—
187#125—3261no—
188#125—3261no—
189#125—1631no—
190C22⋊Q8—1631no—
191C22⋊Q8C2×D16311yescharacteristic
192C4×Q8—3261no—
193C4×Q8—3261no—
194C4×Q8—1631no—
195C4×Q8—1631no—
196C4×Q8—1631no—
197C4×Q8—1631no—
198C4×Q8—1631no—
199C4×Q8—1631no—
200#117—3261no—
201#117—3261no—
202#117—1631no—
203#117C2×D16311yescharacteristic
204#107—1631no—
205#107C2×D16311yescharacteristic
206#106—1631no—
207#106C2×D16311yescharacteristic
208C4⋊C8—1631no—
209C4⋊C8—1631no—
210#103—1631no—
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#103—1631no—
216#103C2×D16311yescharacteristic
217#103C2×D16311yescharacteristic
218C22⋊C8—1631no—
219C22⋊C8C2×D16311yescharacteristic
220#97—1631no—
221C4×C8—1631no—
222C2×Q8—3262no—
223C2×Q8—3262no—
224C2×Q8—3261no—
225C2×Q8—1631no—
226C2×Q8—1631no—
227C2×Q8—1631no—
228C2×Q8C22×D16311yescharacteristic
229C22×C4—3261no—
230C22×C4C22×D16311yescharacteristic
231Q16—3262no—
232Q16—3262no—
233Q16—3262no—
234Q16—3262no—
235C2×C8—1631no—
236C2×C8—1631no—
237C2×C8C22×D16311yescharacteristic
238C2×C8C22×D16311yescharacteristic
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—3261no—
251C4⋊C4—1631no—
252C4⋊C4—1631no—
253C4⋊C4—1631no—
254C4⋊C4C22×D16311yescharacteristic
255C4⋊C4C22×D16311yescharacteristic
256C4⋊C4C22×D16311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—3261no—
260C22⋊C4—1631no—
261C22⋊C4—1631no—
262C42—21no—
263C42—3261no—
264C42—3261no—
265C42—3261no—
266C42—3261no—
267C42—3261no—
268C42—3261no—
269C42—1631no—
270C42—1631no—
271C23D4×D16311yescharacteristic
272Q8—21no—
273Q8—21no—
274Q8—3264no—
275Q8—3261no—
276Q8—3261no—
277Q8—3261no—
278Q8—3261no—
279Q8—1632no—
280Q8—1632no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—3264no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—3261no—
291C2×C4—3261no—
292C2×C4—1631no—
293C2×C4—1631no—
294C2×C4—1631no—
295C2×C4—1631no—
296C2×C4D4×D16311yescharacteristic
297C2×C4C23×D16311yescharacteristic
298C8—21no—
299C8—21no—
300C8—3261no—
301C8—3261no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×D4×D16311yescharacteristic
306C4—21no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—3262no—
312C4—3262no—
313C4—3261no—
314C4—3261no—
315C4—3261no—
316C4—3261no—
317C4—6521no—
318C4—6521no—
319C4#41×D16311yescharacteristic
320C4#41×D16311yescharacteristic
321C2—41no—
322C2#2378217311yescharacteristic
323C2#2378247911yescharacteristic
324C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#485×C163C2
socleC22×C163C22×D4
prime radicalC23×C163C2×D4
hypercenter#485D163
hypocenterC163#3127
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
4. #23781975C2semidirect
5. #23781971C2semidirect
6. #23781874C2semidirect
7. #23781874C2semidirect
10. #23781771C2semidirect
11. #23781769C2semidirect
12. #23781759C2semidirect
13. #23781757C2semidirect
162. C163#3127semidirect

Maximal subgroups

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

subgroupindex
#237822752
#237821422
#237820062
#237819752
#237819712
#237818742
#237818232
#237818202
#237817712
#237817692
#237817592
#237817572
#237817232
#485×C1632
#3127163