Finite Group Databasebeta

← #35226366

Subgroups — #35226366

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#23782349C211yesmaximal, characteristic, complemented
2#23782312C211yesmaximal, characteristic
3#23782018C211yesmaximal, characteristic, complemented
4#23781997C211yesmaximal, characteristic, complemented
5#23781986C211yesmaximal, characteristic, complemented
6#23781964C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781885C211yesmaximal, characteristic, complemented
9#23781825C211yesmaximal, characteristic, complemented
10#23781814C211yesmaximal, characteristic, complemented
11#23781750C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic, complemented
13#23781662C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×#14579689C2211yescharacteristic, complemented
19C2×#14579686C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104728—21no—
22#19104726C2211yescharacteristic, complemented
23#19104725—21no—
24#19104722C2211yescharacteristic
25#19104717—21no—
26#19104717—21no—
27#19104715C2211yescharacteristic
28#19104711C2211yescharacteristic
29#19104706—21no—
30#19104703—21no—
31Q8×Dic163C2211yescharacteristic, complemented
32#19104697C2211yescharacteristic
33C4×C22⋊D163C2211yescharacteristic
34#19104694C2211yescharacteristic, complemented
35#19104693—21no—
36C4⋊C4×D163C2211yescharacteristic
37C22⋊C4×D163—21no—
38#19104668C2211yescharacteristic, complemented
39#19104666C2211yescharacteristic, complemented
40Dic163⋊C8C2211yescharacteristic, complemented
41#19104660C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic, complemented
54C22⋊Q8×C163C2211yescharacteristic, complemented
55#121×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326C2311yescharacteristic
63C2×D652C2311yescharacteristic
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163D411yescharacteristic
67D4×D163—22no—
68D4×D163—41no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163C2311yescharacteristic
73#14579689—41no—
74#14579686—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C8×D163—41no—
78C4⋊Dic163—21no—
79C4⋊Dic163—21no—
80C4⋊Dic163—21no—
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163D411yescharacteristic
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×D4×C163—21no—
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102SD16×C163—41no—
103C2×C8×C163—41no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163—81no—
115C22×D163C2×D411yescharacteristic
116Dic326—21no—
117Dic326—21no—
118D652—21no—
119D652#4112yes—
120C22⋊D163—22no—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—42no—
132C4×D163—42no—
133C4×D163—81no—
134C163⋊C8—21no—
135C163⋊C8—21no—
136C23×C163—21no—
137C23×C163C2×D411yescharacteristic
138Q8×C163—21no—
139Q8×C163—21no—
140D4×C163—21no—
141D4×C163—21no—
142D4×C163—81no—
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—21no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163C22×D411yescharacteristic
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—22no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—84no—
165C2×D163#13512yes—
166C4×C163—21no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163C2×#4111yescharacteristic
171C4×C163H(5,𝔽2)11yescharacteristic
172Dic163—21no—
173Dic163—21no—
174Dic163—21no—
175Dic163—41no—
176Dic163—41no—
177Dic163—41no—
178C2×C163—41no—
179C2×C163—81no—
180C2×C163#54611yescharacteristic
181C2×C163C2×#13511yescharacteristic
182C2×C163#57711yescharacteristic
183D163—22no—
184D163—42no—
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#467—1631no—
194#465D16311yescharacteristic, complemented
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×SD16C2×D16311yescharacteristic
207C22×C8—1631no—
208C22×C8C2×D16311yescharacteristic, complemented
209#126—3261no—
210#125—3261no—
211#125—1631no—
212#124—1631no—
213#123—3261no—
214#123—3261no—
215C22⋊Q8—1631no—
216C22⋊Q8—1631no—
217C22⋊Q8C2×D16311yescharacteristic, complemented
218#121—3261no—
219#121C2×D16311yescharacteristic
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#107—1631no—
229#106—1631no—
230#106C2×D16311yescharacteristic, complemented
231C4⋊C8—1631no—
232#103—1631no—
233#103—1631no—
234#103—1631no—
235#103C2×D16311yescharacteristic, complemented
236#102—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102C2×D16311yescharacteristic
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—21no—
251C2×D4—3262no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—6521no—
255C2×D4—1631no—
256C2×D4—1631no—
257C2×D4C22×D16311yescharacteristic
258C22×C4—3261no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—1631no—
262C22×C4C22×D16311yescharacteristic
263Q16—6521no—
264SD16—41no—
265SD16—3262no—
266C2×C8—41no—
267C2×C8—6521no—
268C2×C8—1631no—
269C2×C8—1631no—
270C2×C8C22×D16311yescharacteristic
271C2×C8C22×D16311yescharacteristic
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—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—21no—
298C23—13041no—
299C23—3261no—
300C23—6522no—
301C23—6522no—
302C23—1631no—
303C23D4×D16311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3261no—
307Q8—3261no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—3262no—
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—13041no—
322C2×C4—3261no—
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—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—13044no—
345C22—3261no—
346C22—3261no—
347C22—3262no—
348C22—6521no—
349C22—6521no—
350C22—1632no—
351C22C2×D4×D16311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3261no—
357C4—3261no—
358C4—3261no—
359C4—6521no—
360C4—6521no—
361C4—6521no—
362C4#41×D16311yescharacteristic
363C4#1910478011yescharacteristic
364C2—41no—
365C2—81no—
366C2—3262no—
367C2—6522no—
368C2#2378230411yescharacteristic
369C2#135×D16311yescharacteristic
370C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
prime radical#23782312C2
hypercenter#465D163
hypocenterC163#3296
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #23782349C2semidirect
3. #23782018C2semidirect
4. #23781997C2semidirect
5. #23781986C2semidirect
6. #23781964C2semidirect
7. #23781905C2semidirect
8. #23781885C2semidirect
9. #23781825C2semidirect
10. #23781814C2semidirect
11. #23781750C2semidirect
12. C8×C22⋊D163C2semidirect
13. #23781662C2semidirect
14. #102×D163C2semidirect
15. #465×C163C2semidirect
18. C2×#14579689C22semidirect
22. #19104726C22semidirect
31. Q8×Dic163C22semidirect
34. #19104694C22semidirect
38. #19104668C22semidirect
39. #19104666C22semidirect
40. Dic163⋊C8C22semidirect
41. #19104660C22semidirect
42. #19104660C22semidirect
43. #19104658C22semidirect
49. C22⋊(C163⋊C8)C22semidirect
51. C8×Dic163C22semidirect
53. C22×C8×C163C22semidirect
54. C22⋊Q8×C163C22semidirect
56. #106×C163C22semidirect
57. #103×C163C22semidirect
185. C163#3296semidirect
194. #465D163semidirect
208. C22×C8C2×D163semidirect
217. C22⋊Q8C2×D163semidirect
230. #106C2×D163semidirect
235. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823492
#237823122
#237820182
#237819972
#237819862
#237819642
#237819052
#237818852
#237818252
#237818142
#237817502
C8×C22⋊D1632
#237816622
#102×D1632
#465×C1632
#3296163