Finite Group Databasebeta

← #35226555

Subgroups — #35226555

Order 20864.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782350C211yesmaximal, characteristic, complemented
2#23782299C211yesmaximal, characteristic
3#23782028C211yesmaximal, characteristic, complemented
4#23781992C211yesmaximal, characteristic, complemented
5#23781968C211yesmaximal, characteristic, complemented
6#23781912C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781864C211yesmaximal, characteristic, complemented
9#23781818C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781764C211yesmaximal, characteristic, complemented
12#23781733C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic, complemented
14C2×#19104655C211yesmaximal, characteristic, complemented
15#480×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C2×D4×D163—21no—
18C2×D4×D163C2211yescharacteristic
19C2×#14579688C2211yescharacteristic
20C2×#14579688C2211yescharacteristic, complemented
21C2×#14579687C2211yescharacteristic, complemented
22C2×#14579682C2211yescharacteristic, complemented
23C22×C163⋊C8C2211yescharacteristic, complemented
24#19104731C2211yescharacteristic
25#19104730—21no—
26#19104726C2211yescharacteristic, complemented
27#19104725—21no—
28#19104718—21no—
29#19104715C2211yescharacteristic, complemented
30#19104714C2211yescharacteristic, complemented
31#19104710C2211yescharacteristic
32#19104708—21no—
33#19104705—21no—
34#19104703—21no—
35#19104700C2211yescharacteristic, complemented
36#19104697C2211yescharacteristic
37C4×C22⋊D163C2211yescharacteristic
38#19104694—21no—
39#19104693C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic, complemented
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52#121×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic, complemented
54#107×C163C2211yescharacteristic, complemented
55#107×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C23×D163—21no—
61C2×Dic326C2311yescharacteristic
62C2×D652—22no—
63C2×D652—21no—
64C2×D652—21no—
65C2×D652—41no—
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163D411yescharacteristic
72D4×D163—22no—
73D4×D163—41no—
74D4×D163—44no—
75C2×C4×D163—21no—
76C2×C4×D163C2311yescharacteristic
77#14579688—22no—
78#14579688—41no—
79#14579687—41no—
80#14579682—41no—
81C2×C163⋊C8—41no—
82C2×C163⋊C8C2311yescharacteristic
83C2×C163⋊C8C2311yescharacteristic
84C4⋊Dic163—21no—
85C4⋊Dic163—21no—
86C4⋊Dic163C2311yescharacteristic
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4D411yescharacteristic
91C22⋊Dic163—21no—
92C22⋊Dic163D411yescharacteristic
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4D411yescharacteristic
98C4×Dic163—21no—
99C4×Dic163—21no—
100C4×Dic163C2311yescharacteristic
101C2×D4×C163—21no—
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163—21no—
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22⋊C4×C163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—42no—
115C22×D163—42no—
116C22×D163—42no—
117C22×D163—42no—
118C22×D163—81no—
119C22×D163—81no—
120C22×D163C2×D411yescharacteristic
121Dic326—21no—
122Dic326—21no—
123D652—21no—
124D652—21no—
125D652—21no—
126D652—42no—
127D652—42no—
128D652#4112yes—
129C22⋊D163—22no—
130C22⋊D163—42no—
131C22⋊D163—42no—
132C22⋊D163—81no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163C2×D411yescharacteristic
136C2×Dic163C2×D411yescharacteristic
137C2×Dic163C2×D411yescharacteristic
138C4×D163—21no—
139C4×D163—21no—
140C4×D163—42no—
141C163⋊C8—21no—
142C163⋊C8—21no—
143C23×C163—21no—
144C23×C163C2×D411yescharacteristic
145D4×C163—21no—
146D4×C163—21no—
147D4×C163—42no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—42no—
154C2×C4×C163C2×D411yescharacteristic
155C2×C4×C163C2411yescharacteristic
156C8×C163—21no—
157C8×C163—21no—
158C22×C163—21no—
159C22×C163—21no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163AGL(1,ℤ/8)12yes—
164C22×C163C22×D411yescharacteristic
165C2×D163—21no—
166C2×D163—22no—
167C2×D163—21no—
168C2×D163—21no—
169C2×D163—41no—
170C2×D163—41no—
171C2×D163—41no—
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—84no—
175C2×D163—84no—
176C4×C163—21no—
177C4×C163—41no—
178C4×C163—41no—
179C4×C163—41no—
180C4×C163C2×#4111yescharacteristic
181C4×C163H(5,𝔽2)11yescharacteristic
182Dic163—21no—
183Dic163—21no—
184Dic163—21no—
185Dic163—41no—
186Dic163—41no—
187C2×C163—22no—
188C2×C163—81no—
189C2×C163#54611yescharacteristic
190C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C163#57711yescharacteristic
192D163—41no—
193D163—42no—
194D163—81no—
195C163#331011yescharacteristic, complemented, Sylow
196#3310—1631nomaximal, Sylow
197#552—1631no—
198D42—1631no—
199#490—1631no—
200#482—1631no—
201#480D16311yescharacteristic, complemented
202#478—1631no—
203#468—1631no—
204#465—1631no—
205#460—1631no—
206#450—1631no—
207#450—1631no—
208#440—1631no—
209#435—1631no—
210#418—1631no—
211C2×#102—1631no—
212C22×D4—3261no—
213C22×D4—1631no—
214C22×D4—1631no—
215C2×SD16—1631no—
216C2×SD16—1631no—
217C2×SD16—1631no—
218C2×M16—1631no—
219C22×C8—1631no—
220#127—1631no—
221#126—3261no—
222#125—3261no—
223#125—1631no—
224#123—3261no—
225C22⋊Q8—1631no—
226C22⋊Q8—1631no—
227#121—3261no—
228#121—1631no—
229#121C2×D16311yescharacteristic
230#120—3261no—
231#120—3261no—
232C4×Q8—1631no—
233C4×D4—1631no—
234C4×D4—1631no—
235#117—3261no—
236#117—1631no—
237C2×C4⋊C4C2×D16311yescharacteristic, complemented
238#107C2×D16311yescharacteristic, complemented
239#107C2×D16311yescharacteristic, complemented
240#106—1631no—
241C4⋊C8—1631no—
242#103—1631no—
243#102—1631no—
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102C2×D16311yescharacteristic, complemented
251#102C2×D16311yescharacteristic, complemented
252C22⋊C8—1631no—
253C22⋊C8C2×D16311yescharacteristic, complemented
254#97—1631no—
255C24—3261no—
256C24—3261no—
257C2×Q8—1631no—
258C2×D4—21no—
259C2×D4—3262no—
260C2×D4—3261no—
261C2×D4—3262no—
262C2×D4—3261no—
263C2×D4—3261no—
264C2×D4—3261no—
265C2×D4—3261no—
266C2×D4—6521no—
267C2×D4—6521no—
268C2×D4—6524no—
269C2×D4—1631no—
270C2×D4—1631no—
271C2×D4—1631no—
272C2×D4C22×D16311yescharacteristic
273C22×C4—21no—
274C22×C4—3261no—
275C22×C4—1631no—
276C22×C4C22×D16311yescharacteristic
277SD16—3262no—
278SD16—6521no—
279SD16—6521no—
280M16—6521no—
281C2×C8—6521no—
282C2×C8—1631no—
283C2×C8—1631no—
284C2×C8C22×D16311yescharacteristic
285C2×C8C22×D16311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3261no—
288C4⋊C4—3261no—
289C4⋊C4—3261no—
290C4⋊C4—3261no—
291C4⋊C4—3261no—
292C4⋊C4—1631no—
293C4⋊C4—1631no—
294C4⋊C4C22×D16311yescharacteristic
295C4⋊C4C22×D16311yescharacteristic
296C4⋊C4C22×D16311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—3261no—
299C22⋊C4—3261no—
300C22⋊C4—3261no—
301C22⋊C4—3261no—
302C22⋊C4—3261no—
303C22⋊C4—1631no—
304C22⋊C4—1631no—
305C42—3261no—
306C42—3261no—
307C42—1631no—
308C23—21no—
309C23—13041no—
310C23—13041no—
311C23—3261no—
312C23—3261no—
313C23—6522no—
314C23—6522no—
315C23—6522no—
316C23—6522no—
317C23—1631no—
318C23D4×D16311yescharacteristic
319Q8—3261no—
320Q8—3261no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—13041no—
325D4—3262no—
326D4—3261no—
327D4—3261no—
328D4—3261no—
329D4—6522no—
330D4—6522no—
331D4—6522no—
332D4—6522no—
333D4—1632no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—3261no—
341C2×C4—3261no—
342C2×C4—3261no—
343C2×C4—3261no—
344C2×C4—6522no—
345C2×C4—1631no—
346C2×C4—1631no—
347C2×C4—1631no—
348C2×C4D4×D16311yescharacteristic
349C2×C4C23×D16311yescharacteristic
350C8—21no—
351C8—21no—
352C8—3261no—
353C8—3261no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—13044no—
360C22—13044no—
361C22—3261no—
362C22—3261no—
363C22—3261no—
364C22—3262no—
365C22—6521no—
366C22—6521no—
367C22—6521no—
368C22—6521no—
369C22—6521no—
370C22#1910475912yes—
371C22C2×D4×D16311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—3261no—
377C4—3261no—
378C4—3261no—
379C4—6521no—
380C4—6521no—
381C4#41×D16311yescharacteristic
382C4#1910478111yescharacteristic
383C2—22no—
384C2—81no—
385C2—13041no—
386C2—6521no—
387C2—6522no—
388C2#2378230711yescharacteristic
389C2C2×#1910475911yescharacteristic
390C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#480×C163C2
socleC22×C163C22×D4
prime radical#23782299C2
hypercenter#480D163
hypocenterC163#3310
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #23782350C2semidirect
3. #23782028C2semidirect
4. #23781992C2semidirect
5. #23781968C2semidirect
6. #23781912C2semidirect
7. #23781892C2semidirect
8. #23781864C2semidirect
9. #23781818C2semidirect
10. #23781817C2semidirect
11. #23781764C2semidirect
12. #23781733C2semidirect
13. #23781668C2semidirect
14. C2×#19104655C2semidirect
15. #480×C163C2semidirect
20. C2×#14579688C22semidirect
21. C2×#14579687C22semidirect
22. C2×#14579682C22semidirect
23. C22×C163⋊C8C22semidirect
26. #19104726C22semidirect
29. #19104715C22semidirect
30. #19104714C22semidirect
35. #19104700C22semidirect
39. #19104693C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104658C22semidirect
44. #19104657C22semidirect
45. #19104656C22semidirect
48. #19104655C22semidirect
49. #19104655C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
53. C2×C4⋊C4×C163C22semidirect
54. #107×C163C22semidirect
55. #107×C163C22semidirect
56. #102×C163C22semidirect
57. #102×C163C22semidirect
58. C22⋊C8×C163C22semidirect
195. C163#3310semidirect
201. #480D163semidirect
237. C2×C4⋊C4C2×D163semidirect
238. #107C2×D163semidirect
239. #107C2×D163semidirect
250. #102C2×D163semidirect
251. #102C2×D163semidirect
253. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823502
#237822992
#237820282
#237819922
#237819682
#237819122
#237818922
#237818642
#237818182
#237818172
#237817642
#237817332
#237816682
C2×#191046552
#480×C1632
#3310163