Finite Group Databasebeta

← #35224322

Subgroups — #35224322

Order 20864.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782218C211yesmaximal, characteristic, complemented
2#23782133C211yesmaximal, characteristic
3#23781975C211yesmaximal, characteristic, complemented
4#23781971C211yesmaximal, characteristic, complemented
5#23781968C211yesmaximal, characteristic, complemented
6#23781964C211yesmaximal, characteristic, complemented
7#23781959C211yesmaximal, characteristic, complemented
8#23781954C211yesmaximal, characteristic, complemented
9#23781949C211yesmaximal, characteristic, complemented
10#23781944C211yesmaximal, characteristic, complemented
11#23781701C211yesmaximal, characteristic, complemented
12C2×Dic163⋊C8C211yesmaximal, characteristic, complemented
13#23781662C211yesmaximal, characteristic
14#23781659C211yesmaximal, characteristic
15#427×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×C163⋊C8C2211yescharacteristic
18#19104734C2211yescharacteristic
19#19104734C2211yescharacteristic
20#19104726C2211yescharacteristic
21#19104726C2211yescharacteristic
22#19104712—21no—
23#19104712—21no—
24C4×Dic326—21no—
25C4×Dic326—21no—
26#19104700C2211yescharacteristic
27#19104700C2211yescharacteristic
28#19104697C2211yescharacteristic
29#19104697C2211yescharacteristic
30(C22⋊D163)⋊C4—21no—
31(C22⋊D163)⋊C4—21no—
32#19104694C2211yescharacteristic
33#19104692—21no—
34C2×C4×Dic163C2211yescharacteristic
35C42×D163—21no—
36#19104668C2211yescharacteristic
37#19104668C2211yescharacteristic
38#19104666C2211yescharacteristic
39#19104666C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43Dic163⋊C8C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104655C2211yescharacteristic
51#19104655C2211yescharacteristic
52C22×C8×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic
54#117×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#106×C163C2211yescharacteristic
58#106×C163C2211yescharacteristic
59#14579699—22no—
60#14579699Q814yescomplemented
61C2×Dic326—21no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C22×Dic163D411yescharacteristic
67C22×Dic163D411yescharacteristic
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C163⋊C8—22no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163—21no—
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84C22⋊Dic163—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87C4×Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163D411yescharacteristic
91C4×Dic163D411yescharacteristic
92C4×Dic163C2311yescharacteristic
93C4×Dic163C2311yescharacteristic
94C22×C4×C163C2311yescharacteristic
95C2×C8×C163—22no—
96C2×C8×C163C2311yescharacteristic
97C2×C8×C163C2311yescharacteristic
98C4⋊C4×C163C2311yescharacteristic
99C4⋊C4×C163C2311yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C22⋊C4×C163—21no—
103C22⋊C4×C163—21no—
104C42×C163—21no—
105C42×C163—21no—
106C22×D163—21no—
107Dic326—21no—
108Dic326—42no—
109Dic326—42no—
110Dic326C2×Q812yes—
111D652—21no—
112D652C2×Q812yes—
113C22⋊D163—24no—
114C2×Dic163—21no—
115C2×Dic163—22no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—41no—
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—24no—
124C4×D163—42no—
125C4×D163—42no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C23×C163C2×D411yescharacteristic
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163C2×D411yescharacteristic
134C2×C4×C163C2×Q812yes—
135C2×C4×C163#4112yes—
136C2×C4×C163C2411yescharacteristic
137C8×C163—21no—
138C8×C163—21no—
139C22×C163—21no—
140C22×C163—21no—
141C22×C163C22⋊Q812yes—
142C22×C163C22×D411yescharacteristic
143C2×D163—22no—
144C2×D163—41no—
145C2×D163—41no—
146C4×C163—41no—
147C4×C163—41no—
148C4×C163—41no—
149C4×C163—41no—
150C4×C163C22⋊Q812yes—
151C4×C163C22×Q811yescharacteristic
152C4×C163C2×#4111yescharacteristic
153Dic163—22no—
154Dic163—21no—
155Dic163—21no—
156Dic163—21no—
157Dic163—41no—
158Dic163—41no—
159Dic163#13512yes—
160C2×C163—22no—
161C2×C163C2×C22⋊Q811yescharacteristic
162C2×C163C2×#13511yescharacteristic
163C2×C163#57511yescharacteristic
164D163—42no—
165C163#307611yescharacteristic, complemented, Sylow
166#3076—1631nomaximal, Sylow
167#533—1631no—
168C4×#41—1631no—
169#479—1631no—
170#479—1631no—
171#478—1631no—
172#478—1631no—
173#477—1631no—
174#476—1631no—
175#475—1631no—
176#474—1631no—
177#427—1631no—
178#427D16311yescharacteristic, complemented
179C2×C4⋊C8—1631no—
180#416—1631no—
181#416—1631no—
182C2×#41—1631no—
183C22×C8—1631no—
184C22×C8C2×D16311yescharacteristic
185C4⋊Q8—1631no—
186C4⋊Q8—1631no—
187#125—1631no—
188#125—1631no—
189C22⋊Q8—3261no—
190C22⋊Q8—3261no—
191C4×Q8—3261no—
192C4×Q8—3261no—
193C4×Q8—1631no—
194C4×Q8—1631no—
195C4×D4—3261no—
196C4×D4—3261no—
197C4×D4—1631no—
198C4×D4—1631no—
199#117—3261no—
200#117—1631no—
201#117C2×D16311yescharacteristic
202#117C2×D16311yescharacteristic
203C2×C42—3261no—
204C2×C42—1631no—
205#107—1631no—
206#107—1631no—
207#107C2×D16311yescharacteristic
208#107C2×D16311yescharacteristic
209#106—1631no—
210#106—1631no—
211#106C2×D16311yescharacteristic
212#106C2×D16311yescharacteristic
213C4⋊C8—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216C4⋊C8—1631no—
217#103—1631no—
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#102—1631no—
222#102—1631no—
223#102—1631no—
224#102—1631no—
225#41—3262no—
226#41—1634no—
227C2×Q8—3261no—
228C2×Q8—3261no—
229C2×Q8—1631no—
230C2×D4—3261no—
231C2×D4—1631no—
232C22×C4—3261no—
233C22×C4—3261no—
234C22×C4—3261no—
235C22×C4—1631no—
236C22×C4—1631no—
237C22×C4C22×D16311yescharacteristic
238C2×C8—22no—
239C2×C8—3262no—
240C2×C8—1631no—
241C2×C8—1631no—
242C2×C8C22×D16311yescharacteristic
243C2×C8C22×D16311yescharacteristic
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—3261no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—3261no—
252C4⋊C4—1631no—
253C4⋊C4—1631no—
254C4⋊C4C22×D16311yescharacteristic
255C4⋊C4C22×D16311yescharacteristic
256C4⋊C4C22×D16311yescharacteristic
257C4⋊C4C22×D16311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—3261no—
261C22⋊C4—3261no—
262C22⋊C4—3261no—
263C42—21no—
264C42—21no—
265C42—3261no—
266C42—3261no—
267C42—3261no—
268C42—1631no—
269C42—1631no—
270C42—1631no—
271C42—1631no—
272C23—3261no—
273C23D4×D16311yescharacteristic
274Q8—3261no—
275Q8—6522no—
276Q8—6522no—
277Q8—1632no—
278D4—3264no—
279D4—3261no—
280D4—1632no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—3261no—
286C2×C4—3262no—
287C2×C4—3264no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—6522no—
291C2×C4—6522no—
292C2×C4—6521no—
293C2×C4—1631no—
294C2×C4—1631no—
295C2×C4—1631no—
296C2×C4—1631no—
297C2×C4D4×D16311yescharacteristic
298C2×C4Q8×D16312yes—
299C2×C4#1457969912yes—
300C2×C4C23×D16311yescharacteristic
301C8—21no—
302C8—21no—
303C8—3261no—
304C8—3261no—
305C22—21no—
306C22—21no—
307C22—3262no—
308C22—6521no—
309C22—6521no—
310C22#1910471312yes—
311C22C2×D4×D16311yescharacteristic
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—3262no—
317C4—3261no—
318C4—3261no—
319C4—3261no—
320C4—6521no—
321C4—6521no—
322C4—1632no—
323C4#1910471312yes—
324C4C2×Q8×D16311yescharacteristic
325C4C2×#1457969911yescharacteristic
326C2—22no—
327C2—6522no—
328C2C2×#1910471311yescharacteristic
329C2#135×D16311yescharacteristic
330C2#2378246211yescharacteristic
subgroupG/H
centerC2×C4D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#427×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579699C22
hypercenter#427D163
hypocenterC163#3076
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #23782218C2semidirect
3. #23781975C2semidirect
4. #23781971C2semidirect
5. #23781968C2semidirect
6. #23781964C2semidirect
7. #23781959C2semidirect
8. #23781954C2semidirect
9. #23781949C2semidirect
10. #23781944C2semidirect
11. #23781701C2semidirect
12. C2×Dic163⋊C8C2semidirect
15. #427×C163C2semidirect
60. #14579699Q8semidirect
165. C163#3076semidirect
178. #427D163semidirect

Maximal subgroups

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

subgroupindex
#237822182
#237821332
#237819752
#237819712
#237819682
#237819642
#237819592
#237819542
#237819492
#237819442
#237817012
C2×Dic163⋊C82
#237816622
#237816592
#427×C1632
#3076163