Finite Group Databasebeta

← #35223704

Subgroups — #35223704

Order 20864.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782193C211yesmaximal, characteristic
2#23781794C211yesmaximal, characteristic, complemented
3#23781792C211yesmaximal, characteristic, complemented
4#23781742C211yesmaximal, characteristic, complemented
5#23781741C211yesmaximal, characteristic, complemented
6#23781736C211yesmaximal, characteristic, complemented
7#23781736C211yesmaximal, characteristic, complemented
8#23781735C211yesmaximal, characteristic, complemented
9#23781734C211yesmaximal, characteristic, complemented
10#23781726C211yesmaximal, characteristic, complemented
11#23781725C211yesmaximal, characteristic, complemented
12#23781721C211yesmaximal, characteristic, complemented
13#23781690C211yesmaximal, characteristic, complemented
14#23781687C211yesmaximal, characteristic, complemented
15#424×C163C211yesmaximal, characteristic, complemented
16C2×#14579682C2211yescharacteristic
17C2×#14579682C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104730C411yescharacteristic
21#19104728C411yescharacteristic
22#19104725C411yescharacteristic
23#19104721C411yescharacteristic
24#19104720C411yescharacteristic
25#19104717C411yescharacteristic
26#19104713C411yescharacteristic
27#19104708C411yescharacteristic
28C4×Dic326C2211yescharacteristic
29C4×D652C2211yescharacteristic
30C4×C22⋊D163C2211yescharacteristic
31C4×C22⋊D163C2211yescharacteristic
32#19104692C2211yescharacteristic
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
49#19104645C2211yescharacteristic
50C4×C163⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×M16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54#117×C163C2211yescharacteristic
55C4⋊C8×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic
58C4⋊C8×C163C2211yescharacteristic
59C2×Dic326C2×C411yescharacteristic
60C2×D652C2×C411yescharacteristic
61C2×C22⋊D163C2×C411yescharacteristic
62C2×C22⋊D163C2×C411yescharacteristic
63C2×C4×D163C2×C411yescharacteristic
64C2×C4×D163C2×C411yescharacteristic
65C2×C4×D163C2311yescharacteristic
66C2×C4×D163C2311yescharacteristic
67#14579682—41no—
68#14579682—41no—
69#14579682—41no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C8×D163—41no—
75C4⋊Dic163C2×C411yescharacteristic
76C4⋊Dic163C2×C411yescharacteristic
77Dic163⋊C4C2×C411yescharacteristic
78Dic163⋊C4C2×C411yescharacteristic
79Dic163⋊C4C2×C411yescharacteristic
80Dic163⋊C4C2×C411yescharacteristic
81Dic163⋊C4C2×C411yescharacteristic
82Dic163⋊C4C2×C411yescharacteristic
83C22⋊Dic163C2×C411yescharacteristic
84C22⋊Dic163C2×C411yescharacteristic
85(C2×D163)⋊C4C2×C411yescharacteristic
86(C2×D163)⋊C4C2×C411yescharacteristic
87(C2×D163)⋊C4C2×C411yescharacteristic
88(C2×D163)⋊C4C2×C411yescharacteristic
89(C2×D163)⋊C4C2×C411yescharacteristic
90(C2×D163)⋊C4C2×C411yescharacteristic
91C4×Dic163C2×C411yescharacteristic
92C4×Dic163C2×C411yescharacteristic
93C4×Dic163C2311yescharacteristic
94C4×Dic163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96M16×C163—41no—
97C2×C8×C163—41no—
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—41no—
111D652—41no—
112C22⋊D163—41no—
113C22⋊D163—41no—
114C2×Dic163C22×C411yescharacteristic
115C2×Dic163C22×C411yescharacteristic
116C2×Dic163C22×C411yescharacteristic
117C2×Dic163C22×C411yescharacteristic
118C2×Dic163C22×C411yescharacteristic
119C2×Dic163C22×C411yescharacteristic
120C4×D163—21no—
121C4×D163—21no—
122C4×D163—21no—
123C4×D163—21no—
124C4×D163—41no—
125C4×D163—41no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C23×C163C22×C411yescharacteristic
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163C22×C411yescharacteristic
134C2×C4×C163C22×C411yescharacteristic
135C2×C4×C163C22×C411yescharacteristic
136C2×C4×C163C22×C411yescharacteristic
137C2×C4×C163C22×C411yescharacteristic
138C2×C4×C163C2411yescharacteristic
139C8×C163—21no—
140C8×C163—21no—
141C8×C163—21no—
142C8×C163—21no—
143C22×C163—21no—
144C22×C163—21no—
145C22×C163—21no—
146C22×C163C23×C411yescharacteristic
147C2×D163—21no—
148C2×D163—21no—
149C2×D163—21no—
150C2×D163—21no—
151C2×D163—21no—
152C2×D163—21no—
153C4×C163—21no—
154C4×C163—21no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163H(5,𝔽2)11yescharacteristic
159C4×C1632−1+411yescharacteristic
160Dic163—21no—
161Dic163—21no—
162Dic163—21no—
163Dic163—21no—
164Dic163—21no—
165Dic163—21no—
166C2×C163—41no—
167C2×C163#52011yescharacteristic
168C2×C163#56811yescharacteristic
169C2×C163#56811yescharacteristic
170D163—41no—
171D163—41no—
172C163#299511yescharacteristic, complemented, Sylow
173#2995—1631nomaximal, Sylow
174#529—1631no—
175#446—1631no—
176#446—1631no—
177#436—1631no—
178#436—1631no—
179#435—1631no—
180#435—1631no—
181#435—1631no—
182#435—1631no—
183#433—1631no—
184#433—1631no—
185#432—1631no—
186#424—1631no—
187#424D16311yescharacteristic, complemented
188#423—1631no—
189C2×M16—1631no—
190C2×M16—1631no—
191C2×M16—1631no—
192C2×M16C2×D16311yescharacteristic
193C22×C8—1631no—
194C22×C8C2×D16311yescharacteristic
195#126—1631no—
196#126—1631no—
197#125—1631no—
198#124—1631no—
199#123—1631no—
200#123—1631no—
201C22⋊Q8—1631no—
202#121—1631no—
203C4×Q8—1631no—
204C4×D4—1631no—
205C4×D4—1631no—
206C4×D4—1631no—
207#117—1631no—
208#117C2×D16311yescharacteristic
209C2×C42—1631no—
210C4⋊C8—1631no—
211C4⋊C8—1631no—
212C4⋊C8—1631no—
213C4⋊C8—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216C4⋊C8C2×D16311yescharacteristic
217C4⋊C8C2×D16311yescharacteristic
218C4⋊C8C2×D16311yescharacteristic
219C4⋊C8C2×D16311yescharacteristic
220C22⋊C8—1631no—
221C22⋊C8—1631no—
222C22⋊C8—1631no—
223C22⋊C8—1631no—
224C22⋊C8—1631no—
225C22⋊C8—1631no—
226#97—1631no—
227#97—1631no—
228#97—1631no—
229#97—1631no—
230C4×C8—1631no—
231C4×C8—1631no—
232C2×Q8—1631no—
233C2×D4—1631no—
234C2×D4—1631no—
235C2×D4—1631no—
236C22×C4—1631no—
237C22×C4—1631no—
238C22×C4—1631no—
239C22×C4—1631no—
240C22×C4C22×D16311yescharacteristic
241M16—41no—
242M16—6521no—
243M16—6521no—
244M16—6521no—
245C2×C8—41no—
246C2×C8—6521no—
247C2×C8—1631no—
248C2×C8—1631no—
249C2×C8—1631no—
250C2×C8—1631no—
251C2×C8C22×D16311yescharacteristic
252C2×C8C22×D16311yescharacteristic
253C2×C8C22×D16311yescharacteristic
254C2×C8C22×D16311yescharacteristic
255C4⋊C4—1631no—
256C4⋊C4—1631no—
257C4⋊C4—1631no—
258C4⋊C4—1631no—
259C4⋊C4—1631no—
260C4⋊C4—1631no—
261C4⋊C4—1631no—
262C4⋊C4—1631no—
263C4⋊C4C4×D16311yescharacteristic
264C4⋊C4C4×D16311yescharacteristic
265C22⋊C4—1631no—
266C22⋊C4—1631no—
267C22⋊C4—1631no—
268C22⋊C4—1631no—
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—
279C42C22×D16311yescharacteristic
280C42C22×D16311yescharacteristic
281C23—1631no—
282C23—1631no—
283C23C2×C4×D16311yescharacteristic
284Q8—6521no—
285D4—6521no—
286D4—6521no—
287D4—6521no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—3261no—
291C2×C4—3261no—
292C2×C4—3261no—
293C2×C4—3261no—
294C2×C4—6521no—
295C2×C4—6521no—
296C2×C4—1631no—
297C2×C4—1631no—
298C2×C4—1631no—
299C2×C4—1631no—
300C2×C4—1631no—
301C2×C4—1631no—
302C2×C4C2×C4×D16311yescharacteristic
303C2×C4C2×C4×D16311yescharacteristic
304C2×C4C2×C4×D16311yescharacteristic
305C2×C4C2×C4×D16311yescharacteristic
306C2×C4C2×C4×D16311yescharacteristic
307C2×C4C23×D16311yescharacteristic
308C8—21no—
309C8—21no—
310C8—21no—
311C8—21no—
312C8—3261no—
313C8—3261no—
314C8—3261no—
315C8—3261no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—3261no—
320C22—3261no—
321C22—3261no—
322C22—3261no—
323C22—3261no—
324C22—3261no—
325C22C22×C4×D16311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—3261no—
332C4—3261no—
333C4—3261no—
334C4—3261no—
335C4—3261no—
336C4—3261no—
337C4#1910478011yescharacteristic
338C4#1910478311yescharacteristic
339C2—41no—
340C2—6521no—
341C2—6521no—
342C2#2378214611yescharacteristic
343C2#2378243011yescharacteristic
344C2#2378243011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D163
commutator, socleC22×C163C23×C4
Fitting#424×C163C2
prime radical#19104708C4
cosocleC2×C4×C163C24
hypercenter#424D163
hypocenterC163#2995
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #23781794C2semidirect
3. #23781792C2semidirect
4. #23781742C2semidirect
5. #23781741C2semidirect
6. #23781736C2semidirect
7. #23781736C2semidirect
8. #23781735C2semidirect
9. #23781734C2semidirect
10. #23781726C2semidirect
11. #23781725C2semidirect
12. #23781721C2semidirect
13. #23781690C2semidirect
14. #23781687C2semidirect
15. #424×C163C2semidirect
172. C163#2995semidirect
187. #424D163semidirect

Maximal subgroups

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

subgroupindex
#237821932
#237817942
#237817922
#237817422
#237817412
#237817362
#237817352
#237817342
#237817262
#237817252
#237817212
#237816902
#237816872
#424×C1632
#2995163