Finite Group Databasebeta

← #35226365

Subgroups — #35226365

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#23782311C211yesmaximal, characteristic
2#23782022C211yesmaximal, characteristic, complemented
3#23781995C211yesmaximal, characteristic, complemented
4#23781989C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781904C211yesmaximal, characteristic, complemented
7#23781896C211yesmaximal, characteristic, complemented
8#23781887C211yesmaximal, characteristic, complemented
9#23781825C211yesmaximal, characteristic, complemented
10#23781815C211yesmaximal, characteristic, complemented
11C4×#14579686C211yesmaximal, characteristic, complemented
12D4×C163⋊C8C211yesmaximal, characteristic, complemented
13#23781659C211yesmaximal, characteristic, complemented
14C2×#19104655C211yesmaximal, characteristic, complemented
15#552×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579689C2211yescharacteristic, complemented
19C2×#14579686C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22C22×C163⋊C8C2211yescharacteristic, complemented
23#19104723C2211yescharacteristic
24#19104720—21no—
25#19104714C2211yescharacteristic
26#19104709C2211yescharacteristic
27#19104708—21no—
28#19104703—21no—
29C4×D652C2211yescharacteristic
30C22⋊C4×D163—21no—
31#19104668C2211yescharacteristic, complemented
32#19104666C2211yescharacteristic, complemented
33#19104666C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35#19104659C2211yescharacteristic, complemented
36#19104658C2211yescharacteristic, complemented
37#19104658C2211yescharacteristic, complemented
38#19104658C2211yescharacteristic, complemented
39#19104655C2211yescharacteristic
40#19104655C2211yescharacteristic
41#19104655C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44#19104655C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47C4×C163⋊C8C2211yescharacteristic, complemented
48#126×C163—21no—
49#125×C163—21no—
50#125×C163C2211yescharacteristic, complemented
51#123×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53C22⋊Q8×C163C2211yescharacteristic, complemented
54C4×Q8×C163C2211yescharacteristic, complemented
55C4×D4×C163C2211yescharacteristic
56#117×C163—21no—
57#117×C163C2211yescharacteristic, complemented
58C2×C4⋊C4×C163C2211yescharacteristic
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326C2311yescharacteristic
63C2×D652—22no—
64C2×D652—21no—
65C2×D652—41no—
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72#14579689—41no—
73#14579686—22no—
74#14579686—41no—
75C2×C163⋊C8—41no—
76C2×C163⋊C8—41no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C2×C163⋊C8C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88C2×Q8×C163C2311yescharacteristic
89C2×D4×C163D411yescharacteristic
90C22×C4×C163—21no—
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163—21no—
97C4⋊C4×C163—21no—
98C4⋊C4×C163—21no—
99C4⋊C4×C163D411yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163D411yescharacteristic
107C22⋊C4×C163D411yescharacteristic
108C42×C163—21no—
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116Dic326—21no—
117Dic326—21no—
118D652—21no—
119D652—21no—
120D652—21no—
121D652—42no—
122D652#4112yes—
123C22⋊D163—42no—
124C22⋊D163—81no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C4×D163—42no—
128C4×D163—81no—
129C163⋊C8—21no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163C2×D411yescharacteristic
134C23×C163C2×D411yescharacteristic
135Q8×C163—21no—
136Q8×C163—21no—
137D4×C163—22no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—42no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163#13512yes—
158C22×C163C22×D411yescharacteristic
159C2×D163—22no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—84no—
166C4×C163—21no—
167C4×C163—21no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163—41no—
173C4×C163—41no—
174C4×C163C2×#4111yescharacteristic
175C4×C163H(5,𝔽2)11yescharacteristic
176Dic163—41no—
177Dic163—41no—
178C2×C163—22no—
179C2×C163—41no—
180C2×C163#54611yescharacteristic
181C2×C163C2×#13511yescharacteristic
182C2×C163#57711yescharacteristic
183D163—42no—
184D163—81no—
185C163#329611yescharacteristic, complemented, Sylow
186#3296—1631nomaximal, Sylow
187#552D16311yescharacteristic, complemented
188#546—1631no—
189#488—1631no—
190#483—1631no—
191#482—1631no—
192#478—1631no—
193#467—1631no—
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×Q16—1631no—
205C2×SD16—1631no—
206C2×SD16—1631no—
207C22×C8—1631no—
208C22×C8—1631no—
209#126—21no—
210#125—21no—
211#125C2×D16311yescharacteristic, complemented
212#124—1631no—
213#123—21no—
214#123—3261no—
215C22⋊Q8—1631no—
216C22⋊Q8C2×D16311yescharacteristic
217C22⋊Q8C2×D16311yescharacteristic, complemented
218#121—3261no—
219#121—1631no—
220#120—3261no—
221C4×Q8C2×D16311yescharacteristic, complemented
222C4×D4—1631no—
223C4×D4C2×D16311yescharacteristic
224#117—21no—
225#117C2×D16311yescharacteristic, complemented
226C2×C4⋊C4C2×D16311yescharacteristic
227C2×C22⋊C4—3261no—
228#107—1631no—
229#106—1631no—
230#106—1631no—
231C4⋊C8—1631no—
232#103—1631no—
233#103—1631no—
234#103—1631no—
235#103—1631no—
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×Q8—1631no—
249C2×Q8C22×D16311yescharacteristic
250C2×D4—3262no—
251C2×D4—3261no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—6521no—
255C2×D4—1631no—
256C2×D4—1631no—
257C2×D4C22⋊D16311yescharacteristic
258C22×C4—21no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4C22×D16311yescharacteristic
262C22×C4C22×D16311yescharacteristic
263Q16—6521no—
264SD16—3262no—
265SD16—6521no—
266C2×C8—6521no—
267C2×C8—6521no—
268C2×C8—1631no—
269C2×C8—1631no—
270C2×C8—1631no—
271C2×C8—1631no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3261no—
279C4⋊C4—1631no—
280C4⋊C4C22⋊D16311yescharacteristic
281C4⋊C4C22×D16311yescharacteristic
282C4⋊C4C22×D16311yescharacteristic
283C4⋊C4C22×D16311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4C22⋊D16311yescharacteristic
293C22⋊C4C22⋊D16311yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D16311yescharacteristic
297C23—13041no—
298C23—3261no—
299C23—3261no—
300C23—6522no—
301C23—6522no—
302C23C2×C22⋊D16311yescharacteristic
303C23C2×C22⋊D16311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3261no—
307Q8—3261no—
308D4—22no—
309D4—13041no—
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—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—13041no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—6522no—
329C2×C4C2×C22⋊D16311yescharacteristic
330C2×C4C2×C22⋊D16311yescharacteristic
331C2×C4C2×C22⋊D16311yescharacteristic
332C2×C4C2×C22⋊D16311yescharacteristic
333C2×C4C23×D16311yescharacteristic
334C8—3261no—
335C8—3261no—
336C8—3261no—
337C8—3261no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—13044no—
344C22—3262no—
345C22—6521no—
346C22—6521no—
347C22—6521no—
348C22—6521no—
349C22—6521no—
350C22#1910475512yes—
351C22C22×C22⋊D16311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—6521no—
361C4—6521no—
362C4#41×D16311yescharacteristic
363C4#1910478111yescharacteristic
364C2—22no—
365C2—41no—
366C2—13041no—
367C2—6522no—
368C2#2378230311yescharacteristic
369C2C2×#1910475511yescharacteristic
370C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#552×C163C2
socleC22×C163C22×D4
prime radical#23782311C2
hypercenter#552D163
hypocenterC163#3296
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #23782022C2semidirect
3. #23781995C2semidirect
4. #23781989C2semidirect
5. #23781962C2semidirect
6. #23781904C2semidirect
7. #23781896C2semidirect
8. #23781887C2semidirect
9. #23781825C2semidirect
10. #23781815C2semidirect
11. C4×#14579686C2semidirect
12. D4×C163⋊C8C2semidirect
13. #23781659C2semidirect
14. C2×#19104655C2semidirect
15. #552×C163C2semidirect
18. C2×#14579689C22semidirect
22. C22×C163⋊C8C22semidirect
31. #19104668C22semidirect
32. #19104666C22semidirect
33. #19104666C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. #19104659C22semidirect
36. #19104658C22semidirect
37. #19104658C22semidirect
38. #19104658C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
47. C4×C163⋊C8C22semidirect
50. #125×C163C22semidirect
53. C22⋊Q8×C163C22semidirect
54. C4×Q8×C163C22semidirect
57. #117×C163C22semidirect
185. C163#3296semidirect
187. #552D163semidirect
211. #125C2×D163semidirect
217. C22⋊Q8C2×D163semidirect
221. C4×Q8C2×D163semidirect
225. #117C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823112
#237820222
#237819952
#237819892
#237819622
#237819042
#237818962
#237818872
#237818252
#237818152
C4×#145796862
D4×C163⋊C82
#237816592
C2×#191046552
#552×C1632
#3296163