Finite Group Databasebeta

← #35227266

Subgroups — #35227266

Order 20864.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782391C211yesmaximal, characteristic, complemented
2#23782336C211yesmaximal, characteristic
3#23782030C211yesmaximal, characteristic, complemented
4#23782028C211yesmaximal, characteristic, complemented
5#23782010C211yesmaximal, characteristic, complemented
6#23781968C211yesmaximal, characteristic, complemented
7#23781965C211yesmaximal, characteristic, complemented
8#23781944C211yesmaximal, characteristic, complemented
9#23781865C211yesmaximal, characteristic, complemented
10#23781863C211yesmaximal, characteristic, complemented
11#23781863C211yesmaximal, characteristic, complemented
12#23781793C211yesmaximal, characteristic, complemented
13#23781768C211yesmaximal, characteristic, complemented
14#23781750C211yesmaximal, characteristic, complemented
15#440×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×#14579686C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19#19104734—21no—
20#19104734C2211yescharacteristic
21#19104732C2211yescharacteristic
22#19104731C2211yescharacteristic
23#19104731C2211yescharacteristic
24#19104727—21no—
25#19104727C2211yescharacteristic
26#19104726—21no—
27#19104726C2211yescharacteristic
28#19104709—21no—
29#19104708—21no—
30#19104708—21no—
31Q8×Dic163C2211yescharacteristic
32Q8×Dic163C2211yescharacteristic
33C4×Dic326—21no—
34C4×Dic326C2211yescharacteristic
35#19104697C2211yescharacteristic
36#19104697C2211yescharacteristic
37#19104668C2211yescharacteristic
38#19104666C2211yescharacteristic
39C4⋊(C163⋊C8)C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42#19104657C2211yescharacteristic
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49#19104655C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59#14579699—44no—
60C2×Dic326—21no—
61C2×Dic326D411yescharacteristic
62C2×D652—21no—
63C2×D652—21no—
64C2×D652C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72#14579686—22no—
73#14579686—22no—
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163—21no—
79C4⋊Dic163D411yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87Dic163⋊C4D411yescharacteristic
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163C2311yescharacteristic
93C4×Dic163C2311yescharacteristic
94C2×Q8×C163C2311yescharacteristic
95C2×D4×C163C2311yescharacteristic
96C22×C4×C163—21no—
97SD16×C163—22no—
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C22⋊C4×C163—21no—
104C42×C163—21no—
105C42×C163C2311yescharacteristic
106C22×D163—21no—
107C22×D163—21no—
108Dic326—22no—
109Dic326—42no—
110D652—21no—
111D652—21no—
112D652—21no—
113D652—21no—
114D652—42no—
115D652—81no—
116C22⋊D163—42no—
117C22⋊D163—42no—
118C22⋊D163—81no—
119C22⋊D163—81no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—42no—
127C4×D163—42no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C23×C163—21no—
131Q8×C163—21no—
132Q8×C163#4112yescomplemented
133D4×C163—21no—
134D4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—42no—
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2411yescharacteristic
142C8×C163—21no—
143C8×C163—21no—
144C22×C163—41no—
145C22×C163—41no—
146C22×C163—41no—
147C22×C163C22×D411yescharacteristic
148C2×D163—41no—
149C2×D163—41no—
150C2×D163—41no—
151C2×D163—41no—
152C2×D163—41no—
153C2×D163—41no—
154C4×C163—22no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163C2×#4111yescharacteristic
160C4×C1632−1+411yescharacteristic
161Dic163—21no—
162Dic163—21no—
163Dic163—21no—
164Dic163—41no—
165Dic163—41no—
166Dic163AGL(1,ℤ/8)12yes—
167C2×C163—81no—
168C2×C163#54811yescharacteristic
169C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C163#57711yescharacteristic
171D163—81no—
172D163—81no—
173C163#336811yescharacteristic, complemented, Sylow
174#3368—1631nomaximal, Sylow
175#557—1631no—
176#550—1631no—
177#490—1631no—
178#490—1631no—
179#486—1631no—
180#478—1631no—
181#478—1631no—
182#474—1631no—
183#460—1631no—
184#460—1631no—
185#460—1631no—
186#446—1631no—
187#440—1631no—
188#440D16311yescharacteristic, complemented
189C4×SD16—1631no—
190C2×#41—3261no—
191C2×SD16—1631no—
192C2×SD16—1631no—
193C2×SD16C2×D16311yescharacteristic
194C4⋊Q8—3261no—
195C4⋊Q8—1631no—
196#127—1631no—
197#127—1631no—
198#127—1631no—
199#125—3261no—
200#125—3261no—
201#125—1631no—
202#125—1631no—
203#121—3261no—
204#121—3261no—
205#121—3261no—
206C4×Q8—3261no—
207C4×Q8—1631no—
208C4×Q8—1631no—
209C4×Q8—1631no—
210C4×Q8C2×D16311yescharacteristic
211C4×D4—1631no—
212C4×D4—1631no—
213C4×D4C2×D16311yescharacteristic
214#107—1631no—
215#107C2×D16311yescharacteristic
216#106—1631no—
217C4⋊C8—1631no—
218C4⋊C8—1631no—
219C4⋊C8—1631no—
220#103C2×D16311yescharacteristic
221#102—1631no—
222#102—1631no—
223#102—1631no—
224#102—1631no—
225#102—1631no—
226#102—1631no—
227#102—1631no—
228#102—1631no—
229#102C2×D16311yescharacteristic
230#97—1631no—
231#97C2×D16311yescharacteristic
232C4×C8—1631no—
233#41—6524no—
234C2×Q8—3261no—
235C2×Q8—1631no—
236C2×Q8C22×D16311yescharacteristic
237C2×D4—3261no—
238C2×D4—3261no—
239C2×D4—3261no—
240C2×D4—3261no—
241C2×D4—3261no—
242C2×D4—3261no—
243C2×D4—1631no—
244C2×D4—1631no—
245C2×D4C22×D16311yescharacteristic
246C22×C4—21no—
247C22×C4—3261no—
248C22×C4—3261no—
249SD16—22no—
250SD16—3262no—
251SD16—3262no—
252C2×C8—1631no—
253C2×C8—1631no—
254C2×C8C22×D16311yescharacteristic
255C2×C8C22×D16311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4—1631no—
268C4⋊C4—1631no—
269C4⋊C4C22×D16311yescharacteristic
270C4⋊C4C22×D16311yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C42—21no—
275C42—3261no—
276C42—3261no—
277C42—1631no—
278C42—1631no—
279C42C22×D16311yescharacteristic
280C23—21no—
281C23—3261no—
282C23—3261no—
283Q8—21no—
284Q8—3262no—
285Q8—6522no—
286Q8#1457969912yescomplemented
287D4—21no—
288D4—21no—
289D4—13041no—
290D4—13041no—
291D4—13041no—
292D4—3261no—
293D4—3261no—
294D4—3261no—
295D4—3261no—
296D4—6522no—
297D4—6522no—
298D4—6522no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—6522no—
306C2×C4—6522no—
307C2×C4—1631no—
308C2×C4—1631no—
309C2×C4—1631no—
310C2×C4—1631no—
311C2×C4D4×D16311yescharacteristic
312C2×C4D4×D16311yescharacteristic
313C2×C4C23×D16311yescharacteristic
314C8—21no—
315C8—21no—
316C8—3261no—
317C8—3261no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—6521no—
322C22—6521no—
323C22—6521no—
324C22—6521no—
325C22—6521no—
326C22—6521no—
327C22C2×D4×D16311yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—3261no—
334C4—3261no—
335C4—3261no—
336C4—6521no—
337C4—6521no—
338C4—1632no—
339C4C2×#1457969911yescharacteristic
340C4#1910478211yescharacteristic
341C2—81no—
342C2—13041no—
343C2—13041no—
344C2#2378232411yescharacteristic
345C2AGL(1,ℤ/8)×D16311yescharacteristic
346C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#440×C163C2
socleC22×C163C22×D4
prime radical#23782336C2
hypercenter#440D163
hypocenterC163#3368
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #23782391C2semidirect
3. #23782030C2semidirect
4. #23782028C2semidirect
5. #23782010C2semidirect
6. #23781968C2semidirect
7. #23781965C2semidirect
8. #23781944C2semidirect
9. #23781865C2semidirect
10. #23781863C2semidirect
11. #23781863C2semidirect
12. #23781793C2semidirect
13. #23781768C2semidirect
14. #23781750C2semidirect
15. #440×C163C2semidirect
132. Q8×C163#41semidirect
173. C163#3368semidirect
188. #440D163semidirect
286. Q8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823912
#237823362
#237820302
#237820282
#237820102
#237819682
#237819652
#237819442
#237818652
#237818632
#237817932
#237817682
#237817502
#440×C1632
#3368163