Finite Group Databasebeta

← #35226367

Subgroups — #35226367

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#23782350C211yesmaximal, characteristic, complemented
2#23782309C211yesmaximal, characteristic
3#23782018C211yesmaximal, characteristic, complemented
4#23781995C211yesmaximal, characteristic, complemented
5#23781987C211yesmaximal, characteristic, complemented
6#23781963C211yesmaximal, characteristic, complemented
7#23781906C211yesmaximal, characteristic, complemented
8#23781893C211yesmaximal, characteristic, complemented
9#23781886C211yesmaximal, characteristic, complemented
10#23781828C211yesmaximal, characteristic, complemented
11#23781816C211yesmaximal, characteristic, complemented
12#23781756C211yesmaximal, characteristic, complemented
13C8×C22⋊D163C211yesmaximal, characteristic, complemented
14#23781658C211yesmaximal, characteristic, complemented
15C2×#102×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×Dic652C2211yescharacteristic, complemented
18C2×#14579687C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic, complemented
21#19104730—21no—
22#19104726C2211yescharacteristic, complemented
23#19104725—21no—
24#19104722C2211yescharacteristic
25#19104718—21no—
26#19104716—21no—
27#19104715C2211yescharacteristic, complemented
28#19104714C2211yescharacteristic
29#19104714C2211yescharacteristic
30#19104707—21no—
31#19104706C2211yescharacteristic
32#19104704—21no—
33#19104700C2211yescharacteristic, complemented
34D4×Dic163C2211yescharacteristic
35C4×C22⋊D163C2211yescharacteristic
36#19104694—21no—
37#19104693C2211yescharacteristic, complemented
38C2×C22⋊Dic163—21no—
39#19104669C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41#19104666C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104658C2211yescharacteristic, complemented
46#19104658C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#102×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—41no—
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163D411yescharacteristic
66C22×Dic163—21no—
67C22×Dic163—21no—
68C2×C4×D163C2311yescharacteristic
69Dic652—41no—
70#14579687—22no—
71#14579687—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C8×D163—41no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163D411yescharacteristic
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C24×C163—21no—
98C2×D4×C163—22no—
99C2×D4×C163—21no—
100C2×D4×C163—41no—
101C2×D4×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105C2×C8×C163—41no—
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163—21no—
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116C22⋊D163—22no—
117C22⋊D163—42no—
118C22⋊D163—81no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—42no—
124C2×Dic163—81no—
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C4×D163—21no—
129C4×D163—21no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163—21no—
133C23×C163—21no—
134C23×C163—42no—
135C23×C163—42no—
136C23×C163—81no—
137C23×C163C2×D411yescharacteristic
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—21no—
141D4×C163—42no—
142D4×C163#4112yes—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—42no—
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2411yescharacteristic
150C8×C163—21no—
151C8×C163—21no—
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—22no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—84no—
161C22×C163#13512yes—
162C22×C163C22×D411yescharacteristic
163C2×D163—21no—
164C2×D163—21no—
165C2×D163—21no—
166C4×C163—21no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163C2×#4111yescharacteristic
170C4×C163H(5,𝔽2)11yescharacteristic
171Dic163—21no—
172Dic163—21no—
173Dic163—21no—
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177Dic163—41no—
178C2×C163—22no—
179C2×C163—42no—
180C2×C163—81no—
181C2×C163#54611yescharacteristic
182C2×C163C2×#13511yescharacteristic
183C2×C163#57711yescharacteristic
184D163—41no—
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#465—1631no—
195#464—1631no—
196#452—1631no—
197#450—1631no—
198C4×SD16—1631no—
199C8×D4—1631no—
200#416—1631no—
201C2×#102D16311yescharacteristic, complemented
202C2×#41—1631no—
203C22×D4C2×D16311yescharacteristic
204C2×Q16—1631no—
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⋊Q8—1631no—
217C22⋊Q8—1631no—
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⋊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#102C2×D16311yescharacteristic
239#102C2×D16311yescharacteristic
240#102C2×D16311yescharacteristic
241#102C2×D16311yescharacteristic
242C22⋊C8—1631no—
243C22⋊C8—1631no—
244C4×C8—1631no—
245C24—21no—
246#41—3262no—
247#41—6521no—
248C2×Q8—1631no—
249C2×Q8—1631no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—3261no—
254C2×D4—3261no—
255C2×D4—1631no—
256C2×D4C22×D16311yescharacteristic
257C2×D4C22×D16311yescharacteristic
258C22×C4—21no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—1631no—
262C22×C4C22×D16311yescharacteristic
263Q16—6521no—
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—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—3261no—
285C22⋊C4—3261no—
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—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1631no—
303C23D4×D16311yescharacteristic
304Q8—3261no—
305Q8—3261no—
306Q8—3261no—
307Q8—3261no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—13041no—
313D4—3262no—
314D4—6522no—
315D4#1457969912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—13041no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
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—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—3261no—
348C22—3261no—
349C22—3261no—
350C22#1910475412yes—
351C22C2×D4×D16311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—3261no—
356C4—3261no—
357C4—3261no—
358C4—6521no—
359C4—6521no—
360C4—6521no—
361C4—6521no—
362C4C2×#1457969911yescharacteristic
363C4#1910478011yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—6521no—
368C2#2378230511yescharacteristic
369C2C2×#1910475411yescharacteristic
370C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#102×C163C2
socleC22×C163C22×D4
prime radical#23782309C2
hypercenterC2×#102D163
hypocenterC163#3296
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #23782350C2semidirect
3. #23782018C2semidirect
4. #23781995C2semidirect
5. #23781987C2semidirect
6. #23781963C2semidirect
7. #23781906C2semidirect
8. #23781893C2semidirect
9. #23781886C2semidirect
10. #23781828C2semidirect
11. #23781816C2semidirect
12. #23781756C2semidirect
13. C8×C22⋊D163C2semidirect
14. #23781658C2semidirect
15. C2×#102×C163C2semidirect
17. C2×Dic652C22semidirect
20. C2×C8×D163C22semidirect
22. #19104726C22semidirect
27. #19104715C22semidirect
33. #19104700C22semidirect
37. #19104693C22semidirect
39. #19104669C22semidirect
40. #19104666C22semidirect
41. #19104666C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
45. #19104658C22semidirect
46. #19104658C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
185. C163#3296semidirect
201. C2×#102D163semidirect

Maximal subgroups

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

subgroupindex
#237823502
#237823092
#237820182
#237819952
#237819872
#237819632
#237819062
#237818932
#237818862
#237818282
#237818162
#237817562
C8×C22⋊D1632
#237816582
C2×#102×C1632
#3296163