Finite Group Databasebeta

← #35223199

Subgroups — #35223199

Order 20864.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782133C211yesmaximal, characteristic
2#23781726C211yesmaximal, characteristic, complemented
3#23781725C211yesmaximal, characteristic, complemented
4#23781725C211yesmaximal, characteristic, complemented
5#23781720C211yesmaximal, characteristic, complemented
6#23781719C211yesmaximal, characteristic, complemented
7#23781716C211yesmaximal, characteristic, complemented
8#23781715C211yesmaximal, characteristic, complemented
9#23781687C211yesmaximal, characteristic, complemented
10C4⋊C8×D163C211yesmaximal, characteristic, complemented
11#23781630C211yesmaximal, characteristic
12#23781630C211yesmaximal, characteristic
13#23781622C211yesmaximal, characteristic, complemented
14#23781619C211yesmaximal, characteristic, complemented
15#433×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579682C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104700C411yescharacteristic
22#19104700C411yescharacteristic
23#19104697C411yescharacteristic
24#19104697C411yescharacteristic
25(C22⋊D163)⋊C4C411yescharacteristic
26(C22⋊D163)⋊C4C411yescharacteristic
27(C22⋊D163)⋊C4C411yescharacteristic
28(C22⋊D163)⋊C4C411yescharacteristic
29#19104692C2211yescharacteristic
30#19104692C2211yescharacteristic
31C2×C4×Dic163C2211yescharacteristic
32C42×D163C2211yescharacteristic
33C42×D163C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35C4⋊(C163⋊C8)C2211yescharacteristic
36Dic163⋊C8C2211yescharacteristic
37Dic163⋊C8C2211yescharacteristic
38Dic163⋊C8C2211yescharacteristic
39Dic163⋊C8C2211yescharacteristic
40C22⋊(C163⋊C8)C2211yescharacteristic
41C22⋊(C163⋊C8)C2211yescharacteristic
42(C2×D163)⋊C8C2211yescharacteristic
43(C2×D163)⋊C8C2211yescharacteristic
44(C2×D163)⋊C8C2211yescharacteristic
45(C2×D163)⋊C8C2211yescharacteristic
46#19104646C2211yescharacteristic
47#19104645C2211yescharacteristic
48#19104645C2211yescharacteristic
49C4×C163⋊C8C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52#117×C163C2211yescharacteristic
53C4⋊C8×C163C2211yescharacteristic
54C4⋊C8×C163C2211yescharacteristic
55C22⋊C8×C163C2211yescharacteristic
56C22⋊C8×C163C2211yescharacteristic
57#97×C163C2211yescharacteristic
58C4×C8×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—22no—
61C2×Dic326C2×C411yescharacteristic
62C2×D652C2×C411yescharacteristic
63C2×C22⋊D163C2×C411yescharacteristic
64C2×C22⋊D163C2×C411yescharacteristic
65C22×Dic163C2×C411yescharacteristic
66C22×Dic163C2×C411yescharacteristic
67C2×C4×D163C2×C411yescharacteristic
68C2×C4×D163C2×C411yescharacteristic
69C2×C4×D163C2×C411yescharacteristic
70C2×C4×D163C2×C411yescharacteristic
71C2×C4×D163C2311yescharacteristic
72C2×C4×D163C2311yescharacteristic
73#14579682—22no—
74#14579682—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C8×D163—22no—
80C8×D163—22no—
81Dic163⋊C4C2×C411yescharacteristic
82Dic163⋊C4C2×C411yescharacteristic
83Dic163⋊C4C2×C411yescharacteristic
84Dic163⋊C4C2×C411yescharacteristic
85(C2×D163)⋊C4C2×C411yescharacteristic
86(C2×D163)⋊C4C2×C411yescharacteristic
87(C2×D163)⋊C4C2×C411yescharacteristic
88(C2×D163)⋊C4C2×C411yescharacteristic
89C4×Dic163C2×C411yescharacteristic
90C4×Dic163C2×C411yescharacteristic
91C4×Dic163C2×C411yescharacteristic
92C4×Dic163C2×C411yescharacteristic
93C4×Dic163C2×C411yescharacteristic
94C4×Dic163C2×C411yescharacteristic
95C4×Dic163C2311yescharacteristic
96C4×Dic163C2311yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163C2×C411yescharacteristic
103C4⋊C4×C163C2×C411yescharacteristic
104C22⋊C4×C163C2×C411yescharacteristic
105C22⋊C4×C163C2×C411yescharacteristic
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163C22×C411yescharacteristic
109C22×D163C22×C411yescharacteristic
110Dic326—22no—
111D652—22no—
112C22⋊D163—22no—
113C22⋊D163—22no—
114C2×Dic163—22no—
115C2×Dic163—41no—
116C2×Dic163C22×C411yescharacteristic
117C2×Dic163C22×C411yescharacteristic
118C2×Dic163C22×C411yescharacteristic
119C2×Dic163C22×C411yescharacteristic
120C2×Dic163C22×C411yescharacteristic
121C2×Dic163C22×C411yescharacteristic
122C4×D163—21no—
123C4×D163—21no—
124C4×D163—22no—
125C4×D163—22no—
126C4×D163—41no—
127C4×D163—41no—
128C4×D163#4112yes—
129C4×D163#4112yes—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C163⋊C8—21no—
134C23×C163C22×C411yescharacteristic
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163C22×C411yescharacteristic
138C2×C4×C163C22×C411yescharacteristic
139C2×C4×C163C22×C411yescharacteristic
140C2×C4×C163C22×C411yescharacteristic
141C2×C4×C163C22×C411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C8×C163—21no—
146C8×C163—21no—
147C22×C163—21no—
148C22×C163—21no—
149C22×C163—21no—
150C22×C163C23×C411yescharacteristic
151C2×D163—21no—
152C2×D163—21no—
153C2×D163—21no—
154C2×D163—21no—
155C2×D163—21no—
156C2×D163#11712yes—
157C4×C163—21no—
158C4×C163—21no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163C2×#4111yescharacteristic
163C4×C163C2×#4111yescharacteristic
164Dic163—21no—
165Dic163—21no—
166Dic163—21no—
167Dic163—21no—
168Dic163#11712yes—
169Dic163#13112yes—
170C2×C163—41no—
171C2×C163C2×#11711yescharacteristic
172C2×C163C2×#13111yescharacteristic
173C2×C163#56811yescharacteristic
174D163—22no—
175D163—41no—
176C163#292511yescharacteristic, complemented, Sylow
177#2925—1631nomaximal, Sylow
178C4×#41—1631no—
179#433—1631no—
180#433—1631no—
181#433—1631no—
182#433D16311yescharacteristic, complemented
183#432—1631no—
184#432—1631no—
185#431—1631no—
186#431—1631no—
187#423—1631no—
188C2×C4⋊C8—1631no—
189#408—1631no—
190#408—1631no—
191#405—1631no—
192#405—1631no—
193C2×#41—1631no—
194C2×M16—1631no—
195C2×M16—1631no—
196C22×C8—1631no—
197C22×C8—1631no—
198C4×Q8—1631no—
199C4×Q8—1631no—
200C4×D4—1631no—
201C4×D4—1631no—
202C4×D4—1631no—
203C4×D4—1631no—
204C4×D4—1631no—
205C4×D4—1631no—
206#117—1631no—
207#117—1631no—
208#117C2×D16311yescharacteristic
209C2×C42—1631no—
210C2×C42—1631no—
211C2×C42—1631no—
212C4⋊C8—1631no—
213C4⋊C8—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216C4⋊C8—1631no—
217C4⋊C8—1631no—
218C4⋊C8C2×D16311yescharacteristic
219C4⋊C8C2×D16311yescharacteristic
220C22⋊C8—1631no—
221C22⋊C8—1631no—
222C22⋊C8—1631no—
223C22⋊C8—1631no—
224C22⋊C8—1631no—
225C22⋊C8—1631no—
226C22⋊C8C2×D16311yescharacteristic
227C22⋊C8C2×D16311yescharacteristic
228#97—1631no—
229#97—1631no—
230#97—1631no—
231#97C2×D16311yescharacteristic
232C4×C8—1631no—
233C4×C8—1631no—
234C4×C8—1631no—
235C4×C8C2×D16311yescharacteristic
236#41—3262no—
237#41—3262no—
238C2×Q8—1631no—
239C2×D4—1631no—
240C2×D4—1631no—
241C2×D4—1631no—
242C22×C4—1631no—
243C22×C4—1631no—
244C22×C4—1631no—
245C22×C4—1631no—
246C22×C4—1631no—
247C22×C4—1631no—
248C22×C4—1631no—
249C22×C4—1631no—
250C22×C4C22×D16311yescharacteristic
251M16—3262no—
252M16—3262no—
253C2×C8—3262no—
254C2×C8—3262no—
255C2×C8—1631no—
256C2×C8—1631no—
257C2×C8—1631no—
258C2×C8—1631no—
259C2×C8C22×D16311yescharacteristic
260C2×C8C22×D16311yescharacteristic
261C2×C8C22×D16311yescharacteristic
262C2×C8C22×D16311yescharacteristic
263C4⋊C4—1631no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4C4×D16311yescharacteristic
268C4⋊C4C4×D16311yescharacteristic
269C22⋊C4—1631no—
270C22⋊C4—1631no—
271C22⋊C4—1631no—
272C22⋊C4—1631no—
273C22⋊C4C4×D16311yescharacteristic
274C22⋊C4C4×D16311yescharacteristic
275C42—1631no—
276C42—1631no—
277C42—1631no—
278C42—1631no—
279C42—1631no—
280C42—1631no—
281C42—1631no—
282C42—1631no—
283C42C22×D16311yescharacteristic
284C42C22×D16311yescharacteristic
285C23—1631no—
286C23—1631no—
287C23C2×C4×D16311yescharacteristic
288Q8—3262no—
289D4—3262no—
290D4—3262no—
291D4—3262no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—3262no—
295C2×C4—3262no—
296C2×C4—3262no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—6521no—
300C2×C4—6521no—
301C2×C4—6521no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4—1631no—
305C2×C4—1631no—
306C2×C4—1631no—
307C2×C4—1631no—
308C2×C4—1632no—
309C2×C4—1632no—
310C2×C4C2×C4×D16311yescharacteristic
311C2×C4C2×C4×D16311yescharacteristic
312C2×C4C2×C4×D16311yescharacteristic
313C2×C4C2×C4×D16311yescharacteristic
314C2×C4C2×C4×D16311yescharacteristic
315C2×C4C23×D16311yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C8—3261no—
321C8—3261no—
322C8—3261no—
323C8—3261no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—3261no—
328C22—3261no—
329C22—3261no—
330C22—3261no—
331C22—3261no—
332C22—1632no—
333C22C22×C4×D16311yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—3261no—
340C4—3261no—
341C4—3261no—
342C4—3261no—
343C4—1632no—
344C4—1632no—
345C4#41×D16311yescharacteristic
346C4#41×D16311yescharacteristic
347C2—41no—
348C2—3262no—
349C2—6521no—
350C2#117×D16311yescharacteristic
351C2#131×D16311yescharacteristic
352C2#2378243011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D163
commutator, socleC22×C163C23×C4
Fitting#433×C163C2
prime radicalC2×#14579699C22
cosocleC2×C4×C163C24
hypercenter#433D163
hypocenterC163#2925
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #23781726C2semidirect
3. #23781725C2semidirect
4. #23781725C2semidirect
5. #23781720C2semidirect
6. #23781719C2semidirect
7. #23781716C2semidirect
8. #23781715C2semidirect
9. #23781687C2semidirect
10. C4⋊C8×D163C2semidirect
13. #23781622C2semidirect
14. #23781619C2semidirect
15. #433×C163C2semidirect
176. C163#2925semidirect
182. #433D163semidirect

Maximal subgroups

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

subgroupindex
#237821332
#237817262
#237817252
#237817202
#237817192
#237817162
#237817152
#237816872
C4⋊C8×D1632
#237816302
#237816222
#237816192
#433×C1632
#2925163