Finite Group Databasebeta

← #35226371

Subgroups — #35226371

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#23782355C211yesmaximal, characteristic, complemented
2#23782019C211yesmaximal, characteristic, complemented
3#23781993C211yesmaximal, characteristic, complemented
4#23781986C211yesmaximal, characteristic, complemented
5#23781965C211yesmaximal, characteristic, complemented
6#23781904C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781884C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781816C211yesmaximal, characteristic, complemented
11C4×#14579687C211yesmaximal, characteristic, complemented
12D4×C163⋊C8C211yesmaximal, characteristic, complemented
13#23781661C211yesmaximal, characteristic, complemented
14C2×#19104656C211yesmaximal, characteristic, complemented
15#546×C163C211yesmaximal, characteristic
16C2×#14579689C2211yescharacteristic, complemented
17C2×#14579687C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C22×C163⋊C8C2211yescharacteristic
20C22×C163⋊C8C2211yescharacteristic, complemented
21#19104728—21no—
22#19104727C2211yescharacteristic, complemented
23#19104725—21no—
24#19104716—21no—
25#19104714C2211yescharacteristic
26#19104714C2211yescharacteristic, complemented
27C4×Dic326C2211yescharacteristic, complemented
28#19104694C2211yescharacteristic, complemented
29#19104691—21no—
30C2×C4⋊Dic163C2211yescharacteristic
31#19104668C2211yescharacteristic, complemented
32#19104666C2211yescharacteristic, complemented
33#19104666C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35#19104659C2211yescharacteristic, complemented
36#19104659C2211yescharacteristic, complemented
37#19104659C2211yescharacteristic, complemented
38#19104658C2211yescharacteristic, complemented
39#19104656C2211yescharacteristic
40#19104656C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104656C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47C4×C163⋊C8C2211yescharacteristic, complemented
48C2×#41×C163C2211yescharacteristic
49C22×D4×C163C2211yescharacteristic
50#124×C163C2211yescharacteristic
51#123×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53#121×C163—21no—
54#121×C163C2211yescharacteristic
55#120×C163—21no—
56C4×D4×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic
58C2×C22⋊C4×C163—21no—
59C2×Dic326C2311yescharacteristic
60C22×Dic163—21no—
61#14579689—41no—
62#14579687—22no—
63#14579687—41no—
64C2×C163⋊C8—41no—
65C2×C163⋊C8—41no—
66C2×C163⋊C8C2311yescharacteristic
67C2×C163⋊C8C2311yescharacteristic
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163—21no—
71C4⋊Dic163C2311yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74Dic163⋊C4—21no—
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C22⋊Dic163—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C4×Dic163—21no—
83C4×Dic163—21no—
84C24×C163—21no—
85#41×C163—22no—
86#41×C163—41no—
87C2×Q8×C163C2311yescharacteristic
88C2×D4×C163—22no—
89C2×D4×C163—21no—
90C2×D4×C163—21no—
91C2×D4×C163—21no—
92C2×D4×C163—41no—
93C2×D4×C163D411yescharacteristic
94C2×D4×C163C2311yescharacteristic
95C2×D4×C163C2311yescharacteristic
96C22×C4×C163—21no—
97C22×C4×C163—21no—
98C22×C4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163D411yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C22⋊C4×C163—21no—
108C22⋊C4×C163D411yescharacteristic
109C22⋊C4×C163D411yescharacteristic
110C42×C163C2311yescharacteristic
111Dic326—21no—
112Dic326—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—42no—
118C163⋊C8—21no—
119C163⋊C8—21no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C23×C163—21no—
123C23×C163—21no—
124C23×C163—42no—
125C23×C163—42no—
126C23×C163—81no—
127C23×C163C2×D411yescharacteristic
128C23×C163C2×D411yescharacteristic
129Q8×C163—21no—
130Q8×C163—21no—
131D4×C163—22no—
132D4×C163—21no—
133D4×C163—21no—
134D4×C163—21no—
135D4×C163—42no—
136D4×C163—42no—
137D4×C163—81no—
138D4×C163#4112yes—
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—42no—
146C2×C4×C163—81no—
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—22no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163—84no—
164C22×C163#13512yes—
165C22×C163C22×D411yescharacteristic
166C4×C163—21no—
167C4×C163—21no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177Dic163—41no—
178C2×C163—22no—
179C2×C163—41no—
180C2×C163—42no—
181C2×C163—81no—
182C2×C163#54611yescharacteristic
183C2×C163C2×#13511yescharacteristic
184C2×C163#57711yescharacteristic
185C163#329611yescharacteristic, complemented, Sylow
186#3296—1631nomaximal, Sylow
187#552—1631no—
188#546D16311yescharacteristic
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×#41C2×D16311yescharacteristic
203C22×D4C2×D16311yescharacteristic
204C2×Q16—1631no—
205C2×SD16—1631no—
206C2×SD16—1631no—
207C22×C8—1631no—
208C22×C8—1631no—
209#126—3261no—
210#125—3261no—
211#125—1631no—
212#124C2×D16311yescharacteristic
213#123—21no—
214#123—3261no—
215C22⋊Q8—1631no—
216C22⋊Q8—1631no—
217C22⋊Q8C2×D16311yescharacteristic
218#121—21no—
219#121C2×D16311yescharacteristic
220#120—21no—
221C4×Q8—1631no—
222C4×D4C2×D16311yescharacteristic
223C4×D4C2×D16311yescharacteristic
224#117—3261no—
225#117—1631no—
226C2×C4⋊C4—1631no—
227C2×C22⋊C4—21no—
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—21no—
246#41—22no—
247#41—41no—
248C2×Q8—1631no—
249C2×Q8C22×D16311yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D16311yescharacteristic
256C2×D4C22×D16311yescharacteristic
257C2×D4C22×D16311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
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—3261no—
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—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4C22⋊D16311yescharacteristic
293C22⋊C4C22⋊D16311yescharacteristic
294C42—3261no—
295C42—3261no—
296C42C22×D16311yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D16311yescharacteristic
303C23C2×C22⋊D16311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3261no—
307Q8—3261no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1457969812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—3261no—
325C2×C4—3261no—
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—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1910475512yes—
351C22C22×C22⋊D16311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—6521no—
359C4—6521no—
360C4—6521no—
361C4—6521no—
362C4C2×#1457969811yescharacteristic
363C4#1910478011yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2378230911yescharacteristic
369C2C2×#1910475511yescharacteristic
370C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting, prime radical#546×C163C2
socleC22×C163C22×D4
hypercenter#546D163
hypocenterC163#3296
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #23782355C2semidirect
2. #23782019C2semidirect
3. #23781993C2semidirect
4. #23781986C2semidirect
5. #23781965C2semidirect
6. #23781904C2semidirect
7. #23781892C2semidirect
8. #23781884C2semidirect
9. #23781827C2semidirect
10. #23781816C2semidirect
11. C4×#14579687C2semidirect
12. D4×C163⋊C8C2semidirect
13. #23781661C2semidirect
14. C2×#19104656C2semidirect
16. C2×#14579689C22semidirect
20. C22×C163⋊C8C22semidirect
22. #19104727C22semidirect
26. #19104714C22semidirect
27. C4×Dic326C22semidirect
28. #19104694C22semidirect
31. #19104668C22semidirect
32. #19104666C22semidirect
33. #19104666C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. #19104659C22semidirect
36. #19104659C22semidirect
37. #19104659C22semidirect
38. #19104658C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
47. C4×C163⋊C8C22semidirect
185. C163#3296semidirect

Maximal subgroups

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

subgroupindex
#237823552
#237820192
#237819932
#237819862
#237819652
#237819042
#237818922
#237818842
#237818272
#237818162
C4×#145796872
D4×C163⋊C82
#237816612
C2×#191046562
#546×C1632
#3296163