Finite Group Databasebeta

← #35224679

Subgroups — #35224679

Order 20864.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782200C211yesmaximal, characteristic, complemented
2C4×D4×D163C211yesmaximal, characteristic, complemented
3#23781997C211yesmaximal, characteristic, complemented
4#23781980C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781881C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781817C211yesmaximal, characteristic, complemented
9#23781800C211yesmaximal, characteristic
10#23781776C211yesmaximal, characteristic, complemented
11#23781774C211yesmaximal, characteristic, complemented
12#23781764C211yesmaximal, characteristic, complemented
13#23781762C211yesmaximal, characteristic, complemented
14#23781721C211yesmaximal, characteristic, complemented
15#478×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C22×C4×D163—21no—
18C2×#14579688C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×D1304C2211yescharacteristic
21C2×#14579684C2211yescharacteristic
22#19104730—21no—
23#19104723C2211yescharacteristic, complemented
24#19104720—21no—
25#19104715C2211yescharacteristic
26#19104710C2211yescharacteristic
27#19104700C2211yescharacteristic, complemented
28C4×D652C2211yescharacteristic
29D4×Dic163C2211yescharacteristic
30#19104697—21no—
31#19104697C2211yescharacteristic
32#19104697C2211yescharacteristic, complemented
33C4×C22⋊D163—21no—
34(C22⋊D163)⋊C4—21no—
35#19104693—21no—
36C4⋊C4×D163C2211yescharacteristic
37C22⋊C4×D163—21no—
38C42×D163C2211yescharacteristic
39#19104668C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41C4⋊(C163⋊C8)C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104655C2211yescharacteristic
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic, complemented
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50#19104645C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52#125×C163C2211yescharacteristic, complemented
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56C4⋊C8×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652—21no—
62C2×D652C2311yescharacteristic
63C2×D652C2311yescharacteristic
64C2×C22⋊D163—21no—
65D4×D163—22no—
66D4×D163—22no—
67C22×Dic163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—44no—
73C2×C4×D163D411yescharacteristic
74C2×C4×D163D411yescharacteristic
75C2×C4×D163C2311yescharacteristic
76#14579688—22no—
77#14579687—22no—
78D1304—22no—
79#14579684—22no—
80C2×C163⋊C8C2311yescharacteristic
81C2×C163⋊C8C2311yescharacteristic
82C4⋊Dic163—21no—
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
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—
94C4×Dic163—21no—
95C4×Dic163D411yescharacteristic
96C4×Dic163D411yescharacteristic
97C4×Dic163C2311yescharacteristic
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163—21no—
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163—21no—
103C4⋊C4×C163—21no—
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C42×C163C2311yescharacteristic
109C22×D163—24no—
110C22×D163—21no—
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163C2×D411yescharacteristic
115Dic326—21no—
116Dic326—21no—
117D652—21no—
118D652—21no—
119D652—21no—
120D652—42no—
121D652#4112yes—
122C22⋊D163—24no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—42no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—22no—
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—41no—
133C4×D163—42no—
134C4×D163—42no—
135C4×D163—42no—
136C4×D163—42no—
137C163⋊C8—21no—
138C163⋊C8—21no—
139C23×C163—21no—
140D4×C163—21no—
141D4×C163#4112yes—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—42no—
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C8×C163—21no—
150C8×C163—21no—
151C22×C163—22no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163C22×D411yescharacteristic
155C2×D163—21no—
156C2×D163—21no—
157C2×D163—22no—
158C2×D163—48no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163AGL(1,ℤ/8)12yes—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163C2×#4111yescharacteristic
171C4×C163C2×#4111yescharacteristic
172Dic163—21no—
173Dic163—21no—
174Dic163—21no—
175Dic163—41no—
176Dic163—41no—
177C2×C163—42no—
178C2×C163#52511yescharacteristic
179C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C163#57511yescharacteristic
181D163—22no—
182D163—42no—
183D163—81no—
184C163#311411yescharacteristic, complemented, Sylow
185#3114—1631nomaximal, Sylow
186#529—1631no—
187C2×C4×D4—1631no—
188#483—1631no—
189#480—1631no—
190#478—1631no—
191#478D16311yescharacteristic, complemented
192#463—1631no—
193#463—1631no—
194#450—1631no—
195#447—1631no—
196#442—1631no—
197#442—1631no—
198#440—1631no—
199#440—1631no—
200#432—1631no—
201C22×D4—1631no—
202C23×C4—3261no—
203C2×SD16—1631no—
204C2×SD16—1631no—
205C2×D8—1631no—
206C2×D8—1631no—
207#126—3261no—
208#125C2×D16311yescharacteristic, complemented
209#124—1631no—
210#123—3261no—
211C22⋊Q8—1631no—
212#121—1631no—
213C4×Q8—1631no—
214C4×D4—3261no—
215C4×D4—3261no—
216C4×D4—3261no—
217C4×D4—1631no—
218C4×D4—1631no—
219C4×D4—1631no—
220C4×D4—1631no—
221C4×D4C2×D16311yescharacteristic
222#117—3261no—
223C2×C4⋊C4—1631no—
224C2×C22⋊C4—3261no—
225C2×C42—1631no—
226#107—1631no—
227#107C2×D16311yescharacteristic, complemented
228#106—1631no—
229#106C2×D16311yescharacteristic, complemented
230C4⋊C8—1631no—
231C4⋊C8C2×D16311yescharacteristic, complemented
232#103—1631no—
233#103—1631no—
234#102—1631no—
235#102—1631no—
236#102—1631no—
237#102—1631no—
238#102C2×D16311yescharacteristic
239#102C2×D16311yescharacteristic
240C22⋊C8—1631no—
241C22⋊C8—1631no—
242#97—1631no—
243#97—1631no—
244C24—3261no—
245C2×Q8—1631no—
246C2×D4—3261no—
247C2×D4—3262no—
248C2×D4—3262no—
249C2×D4—3261no—
250C2×D4—1631no—
251C2×D4—1631no—
252C2×D4C22×D16311yescharacteristic
253C22×C4—21no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4—3261no—
257C22×C4—3261no—
258C22×C4—3261no—
259C22×C4—6524no—
260C22×C4—1631no—
261C22×C4—1631no—
262C22×C4—1631no—
263SD16—3262no—
264SD16—3262no—
265D8—3262no—
266D8—3262no—
267C2×C8—1631no—
268C2×C8—1631no—
269C2×C8C22×D16311yescharacteristic
270C2×C8C22×D16311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—1631no—
277C4⋊C4C22×D16311yescharacteristic
278C4⋊C4C22×D16311yescharacteristic
279C4⋊C4C22×D16311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C42—3261no—
290C42—1631no—
291C42—1631no—
292C42—1631no—
293C42C22×D16311yescharacteristic
294C23—21no—
295C23—3264no—
296C23—3261no—
297C23—3261no—
298C23—6522no—
299C23—6522no—
300C23—1631no—
301Q8—3261no—
302Q8—3261no—
303D4—21no—
304D4—3264no—
305D4—3261no—
306D4—3261no—
307D4—3261no—
308D4—6522no—
309D4—1632no—
310D4#1457970012yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—3262no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—6521no—
321C2×C4—6522no—
322C2×C4—6522no—
323C2×C4—6522no—
324C2×C4—6522no—
325C2×C4—6522no—
326C2×C4—1631no—
327C2×C4—1631no—
328C2×C4—1631no—
329C2×C4D4×D16311yescharacteristic
330C2×C4D4×D16311yescharacteristic
331C2×C4C23×D16311yescharacteristic
332C8—21no—
333C8—21no—
334C8—3261no—
335C8—3261no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—3261no—
340C22—3261no—
341C22—3262no—
342C22—6528no—
343C22—6521no—
344C22—6521no—
345C22—6521no—
346C22—6521no—
347C22—6521no—
348C22—1632no—
349C22C2×D4×D16311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3261no—
356C4—3261no—
357C4—3261no—
358C4—6521no—
359C4—6521no—
360C4#41×D16311yescharacteristic
361C4C2×#1457970011yescharacteristic
362C2—42no—
363C2—13041no—
364C2—3262no—
365C2—6522no—
366C2#2378217811yescharacteristic
367C2AGL(1,ℤ/8)×D16311yescharacteristic
368C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#478×C163C2
socleC22×C163C22×D4
prime radical#23781800C2
hypercenter#478D163
hypocenterC163#3114
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #23782200C2semidirect
2. C4×D4×D163C2semidirect
3. #23781997C2semidirect
4. #23781980C2semidirect
5. #23781962C2semidirect
6. #23781881C2semidirect
7. #23781880C2semidirect
8. #23781817C2semidirect
10. #23781776C2semidirect
11. #23781774C2semidirect
12. #23781764C2semidirect
13. #23781762C2semidirect
14. #23781721C2semidirect
15. #478×C163C2semidirect
23. #19104723C22semidirect
27. #19104700C22semidirect
32. #19104697C22semidirect
39. #19104668C22semidirect
40. #19104666C22semidirect
41. C4⋊(C163⋊C8)C22semidirect
42. #19104660C22semidirect
43. #19104658C22semidirect
44. #19104657C22semidirect
47. #19104655C22semidirect
50. #19104645C22semidirect
51. #19104645C22semidirect
52. #125×C163C22semidirect
54. #107×C163C22semidirect
55. #106×C163C22semidirect
56. C4⋊C8×C163C22semidirect
184. C163#3114semidirect
191. #478D163semidirect
208. #125C2×D163semidirect
227. #107C2×D163semidirect
229. #106C2×D163semidirect
231. C4⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822002
C4×D4×D1632
#237819972
#237819802
#237819622
#237818812
#237818802
#237818172
#237818002
#237817762
#237817742
#237817642
#237817622
#237817212
#478×C1632
#3114163