Finite Group Databasebeta

← #35226377

Subgroups — #35226377

Order 20864.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782357C211yesmaximal, characteristic, complemented
2#23782306C211yesmaximal, characteristic
3#23782021C211yesmaximal, characteristic, complemented
4#23781993C211yesmaximal, characteristic, complemented
5#23781992C211yesmaximal, characteristic, complemented
6#23781968C211yesmaximal, characteristic, complemented
7#23781897C211yesmaximal, characteristic, complemented
8#23781889C211yesmaximal, characteristic, complemented
9#23781826C211yesmaximal, characteristic, complemented
10#23781815C211yesmaximal, characteristic, complemented
11#23781750C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic, complemented
13#23781660C211yesmaximal, characteristic, complemented
14C2×#19104657C211yesmaximal, characteristic, complemented
15#467×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic, complemented
21#19104730—21no—
22#19104726—21no—
23#19104726C2211yescharacteristic, complemented
24#19104724C2211yescharacteristic
25#19104720—21no—
26#19104719—21no—
27#19104711C2211yescharacteristic, complemented
28#19104710—21no—
29#19104710C2211yescharacteristic
30#19104705—21no—
31Q8×Dic163C2211yescharacteristic, complemented
32#19104697C2211yescharacteristic
33C4×C22⋊D163C2211yescharacteristic
34#19104693C2211yescharacteristic, complemented
35#19104691—21no—
36C2×C4⋊Dic163C2211yescharacteristic
37C22⋊C4×D163—21no—
38#19104666C2211yescharacteristic, complemented
39#19104666C2211yescharacteristic, complemented
40Dic163⋊C8C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×Q16×C163C2211yescharacteristic, complemented
53C22×C8×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#103×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579700—22no—
61#14579700—41no—
62C2×D652—22no—
63C2×D652—21no—
64C2×D652—21no—
65C2×D652—21no—
66C2×D652—41no—
67C2×D652C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163D411yescharacteristic
70C22×Dic163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579686—22no—
75#14579686—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C8×D163—41no—
79C4⋊Dic163—21no—
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163D411yescharacteristic
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C2×Q8×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101Q16×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117D652—21no—
118D652—21no—
119D652—21no—
120D652—42no—
121D652—42no—
122D652—81no—
123D652#4112yes—
124C22⋊D163—22no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—42no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—42no—
134C4×D163—81no—
135C163⋊C8—21no—
136C163⋊C8—21no—
137C23×C163C2×D411yescharacteristic
138Q8×C163—21no—
139Q8×C163—21no—
140Q8×C163—21no—
141Q8×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2411yescharacteristic
150C8×C163—21no—
151C8×C163—21no—
152C22×C163—21no—
153C22×C163—21no—
154C22×C163#13512yes—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—22no—
158C2×D163—21no—
159C2×D163—21no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—84no—
166C4×C163—21no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163C2×#4111yescharacteristic
172C4×C163H(5,𝔽2)11yescharacteristic
173Dic163—21no—
174Dic163—21no—
175Dic163—21no—
176Dic163—41no—
177Dic163—41no—
178C2×C163—22no—
179C2×C163#54611yescharacteristic
180C2×C163C2×#13511yescharacteristic
181C2×C163#57711yescharacteristic
182D163—41no—
183D163—42no—
184D163—81no—
185C163#329611yescharacteristic, complemented, Sylow
186#3296—1631nomaximal, Sylow
187#552—1631no—
188#546—1631no—
189#488—1631no—
190#483—1631no—
191#482—1631no—
192#478—1631no—
193#467D16311yescharacteristic, complemented
194#465—1631no—
195#464—1631no—
196#452—1631no—
197#450—1631no—
198C4×SD16—1631no—
199C8×D4—1631no—
200#416—1631no—
201C2×#102—1631no—
202C2×#41—1631no—
203C22×D4—1631no—
204C2×Q16C2×D16311yescharacteristic, complemented
205C2×SD16—1631no—
206C2×SD16—1631no—
207C22×C8—1631no—
208C22×C8C2×D16311yescharacteristic
209#126—3261no—
210#125—3261no—
211#125—1631no—
212#124—1631no—
213#123—3261no—
214#123—3261no—
215C22⋊Q8—1631no—
216C22⋊Q8C2×D16311yescharacteristic
217C22⋊Q8C2×D16311yescharacteristic
218#121—3261no—
219#121—1631no—
220#120—3261no—
221C4×Q8—1631no—
222C4×D4—1631no—
223C4×D4—1631no—
224#117—3261no—
225#117—1631no—
226C2×C4⋊C4—1631no—
227C2×C22⋊C4—3261no—
228#107C2×D16311yescharacteristic, complemented
229#106—1631no—
230#106—1631no—
231C4⋊C8—1631no—
232#103—1631no—
233#103—1631no—
234#103C2×D16311yescharacteristic, complemented
235#103C2×D16311yescharacteristic, complemented
236#102—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242C22⋊C8—1631no—
243C22⋊C8—1631no—
244C4×C8—1631no—
245C24—3261no—
246#41—3262no—
247#41—6521no—
248C2×Q8C22×D16311yescharacteristic
249C2×Q8C22×D16311yescharacteristic
250C2×D4—3261no—
251C2×D4—3262no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—6521no—
255C2×D4—1631no—
256C2×D4—1631no—
257C2×D4—1631no—
258C22×C4—3261no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—1631no—
262C22×C4C22×D16311yescharacteristic
263Q16—41no—
264SD16—3262no—
265SD16—6521no—
266C2×C8—41no—
267C2×C8—6521no—
268C2×C8—1631no—
269C2×C8—1631no—
270C2×C8C22×D16311yescharacteristic
271C2×C8C22×D16311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—3261no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—1631no—
280C4⋊C4—1631no—
281C4⋊C4—1631no—
282C4⋊C4C22×D16311yescharacteristic
283C4⋊C4C22×D16311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—3261no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—1631no—
293C22⋊C4—1631no—
294C42—3261no—
295C42—3261no—
296C42—1631no—
297C23—13041no—
298C23—3261no—
299C23—3261no—
300C23—6522no—
301C23—6522no—
302C23—1631no—
303C23D4×D16311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—13041no—
309D4—3262no—
310D4—3261no—
311D4—3261no—
312D4—3261no—
313D4—6522no—
314D4—6522no—
315D4—1632no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—13041no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—6522no—
328C2×C4—6522no—
329C2×C4—1631no—
330C2×C4—1631no—
331C2×C4—1631no—
332C2×C4D4×D16311yescharacteristic
333C2×C4C23×D16311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3261no—
337C8—3261no—
338C22—21no—
339C22—21no—
340C22—13044no—
341C22—3261no—
342C22—3261no—
343C22—3261no—
344C22—3262no—
345C22—6521no—
346C22—6521no—
347C22—6521no—
348C22—6521no—
349C22—6521no—
350C22#1910475812yes—
351C22C2×D4×D16311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3261no—
358C4—3261no—
359C4—3261no—
360C4—6521no—
361C4—6521no—
362C4#41×D16311yescharacteristic
363C4#1910478011yescharacteristic
364C2—22no—
365C2—13041no—
366C2—6522no—
367C2—6521no—
368C2#2378230411yescharacteristic
369C2C2×#1910475811yescharacteristic
370C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#467×C163C2
socleC22×C163C22×D4
prime radical#23782306C2
hypercenter#467D163
hypocenterC163#3296
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #23782357C2semidirect
3. #23782021C2semidirect
4. #23781993C2semidirect
5. #23781992C2semidirect
6. #23781968C2semidirect
7. #23781897C2semidirect
8. #23781889C2semidirect
9. #23781826C2semidirect
10. #23781815C2semidirect
11. #23781750C2semidirect
12. C8×C22⋊D163C2semidirect
13. #23781660C2semidirect
14. C2×#19104657C2semidirect
15. #467×C163C2semidirect
20. C2×C8×D163C22semidirect
23. #19104726C22semidirect
27. #19104711C22semidirect
31. Q8×Dic163C22semidirect
34. #19104693C22semidirect
38. #19104666C22semidirect
39. #19104666C22semidirect
40. Dic163⋊C8C22semidirect
41. #19104659C22semidirect
42. #19104659C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
52. C2×Q16×C163C22semidirect
56. #107×C163C22semidirect
57. #103×C163C22semidirect
58. #103×C163C22semidirect
185. C163#3296semidirect
193. #467D163semidirect
204. C2×Q16C2×D163semidirect
228. #107C2×D163semidirect
234. #103C2×D163semidirect
235. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823572
#237823062
#237820212
#237819932
#237819922
#237819682
#237818972
#237818892
#237818262
#237818152
#237817502
C8×C22⋊D1632
#237816602
C2×#191046572
#467×C1632
#3296163