Finite Group Databasebeta

← #35226553

Subgroups — #35226553

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
2D4×C22⋊D163C211yesmaximal, characteristic
3#23782028C211yesmaximal, characteristic, complemented
4#23781988C211yesmaximal, characteristic, complemented
5#23781978C211yesmaximal, characteristic, complemented
6#23781967C211yesmaximal, characteristic, complemented
7#23781910C211yesmaximal, characteristic, complemented
8#23781894C211yesmaximal, characteristic, complemented
9#23781864C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781816C211yesmaximal, characteristic, complemented
12#23781762C211yesmaximal, characteristic, complemented
13#23781736C211yesmaximal, characteristic, complemented
14#23781668C211yesmaximal, characteristic, complemented
15C2×#102×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163—21no—
17C2×D4×D163C2211yescharacteristic
18C2×#14579688C2211yescharacteristic, complemented
19C2×#14579687C2211yescharacteristic
20C2×#14579687C2211yescharacteristic, complemented
21C2×#14579682C2211yescharacteristic, complemented
22#19104731C2211yescharacteristic
23#19104730—21no—
24#19104726C2211yescharacteristic, complemented
25#19104725—21no—
26#19104718—21no—
27#19104715C2211yescharacteristic, complemented
28#19104714C2211yescharacteristic, complemented
29#19104709C2211yescharacteristic
30#19104707—21no—
31#19104706C2211yescharacteristic
32#19104704—21no—
33#19104703—21no—
34#19104700C2211yescharacteristic, complemented
35D4×Dic163C2211yescharacteristic
36C4×C22⋊D163C2211yescharacteristic
37#19104694—21no—
38#19104693C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40#19104668C2211yescharacteristic, complemented
41#19104667C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic, complemented
48#19104655C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic, complemented
54C2×C4⋊C4×C163C2211yescharacteristic, complemented
55#102×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163—44no—
68C2×C22⋊D163D411yescharacteristic
69D4×D163—22no—
70D4×D163—41no—
71C22×Dic163—21no—
72C2×C4×D163C2311yescharacteristic
73#14579688—41no—
74#14579687—22no—
75#14579687—41no—
76#14579682—41no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
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
98C24×C163—21no—
99C2×D4×C163—22no—
100C2×D4×C163—21no—
101C2×D4×C163—41no—
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C22×C4×C163—21no—
105C22×C4×C163C2311yescharacteristic
106C2×C8×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163—21no—
110C4⋊C4×C163C2311yescharacteristic
111C4⋊C4×C163C2311yescharacteristic
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117Dic326—21no—
118Dic326—21no—
119D652—21no—
120D652—21no—
121C22⋊D163—22no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C22⋊D163—42no—
125C22⋊D163—42no—
126C22⋊D163—81no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—42no—
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C2×Dic163C2×D411yescharacteristic
134C4×D163—21no—
135C4×D163—21no—
136C163⋊C8—21no—
137C163⋊C8—21no—
138C23×C163—21no—
139C23×C163—21no—
140C23×C163—42no—
141C23×C163—42no—
142C23×C163—81no—
143C23×C163C2×D411yescharacteristic
144D4×C163—21no—
145D4×C163—21no—
146D4×C163—21no—
147D4×C163—42no—
148D4×C163#4112yes—
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—22no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—84no—
167C22×C163AGL(1,ℤ/8)12yes—
168C22×C163C22×D411yescharacteristic
169C2×D163—21no—
170C2×D163—21no—
171C2×D163—21no—
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—84no—
176C4×C163—21no—
177C4×C163—41no—
178C4×C163—41no—
179C4×C163C2×#4111yescharacteristic
180C4×C163H(5,𝔽2)11yescharacteristic
181Dic163—21no—
182Dic163—21no—
183Dic163—21no—
184Dic163—41no—
185Dic163—41no—
186Dic163—41no—
187C2×C163—22no—
188C2×C163—42no—
189C2×C163—81no—
190C2×C163#54611yescharacteristic
191C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C163#57711yescharacteristic
193D163—41no—
194D163—81no—
195C163#331011yescharacteristic, complemented, Sylow
196#3310—1631nomaximal, Sylow
197#552—1631no—
198D42—1631no—
199#490—1631no—
200#482—1631no—
201#480—1631no—
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×#102D16311yescharacteristic, complemented
212C22×D4—3261no—
213C22×D4—1631no—
214C22×D4C2×D16311yescharacteristic
215C2×SD16—1631no—
216C2×SD16—1631no—
217C2×SD16—1631no—
218C2×M16—1631no—
219C22×C8C2×D16311yescharacteristic, complemented
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#121—1631no—
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#107—1631no—
239#107—1631no—
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#102C2×D16311yescharacteristic
249#102C2×D16311yescharacteristic
250#102C2×D16311yescharacteristic, complemented
251#102C2×D16311yescharacteristic, complemented
252C22⋊C8—1631no—
253C22⋊C8—1631no—
254#97—1631no—
255C24—21no—
256C24—3261no—
257C2×Q8—1631no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—3262no—
262C2×D4—3261no—
263C2×D4—3261no—
264C2×D4—3261no—
265C2×D4—3261no—
266C2×D4—3261no—
267C2×D4—6521no—
268C2×D4—6524no—
269C2×D4—1631no—
270C2×D4—1631no—
271C2×D4C22×D16311yescharacteristic
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—41no—
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⋊C4—1631no—
295C4⋊C4C22×D16311yescharacteristic
296C4⋊C4C22×D16311yescharacteristic
297C22⋊C4—3261no—
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—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—13041no—
314C23—3261no—
315C23—6522no—
316C23—6522no—
317C23—1631no—
318C23D4×D16311yescharacteristic
319Q8—3261no—
320Q8—3261no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—13041no—
326D4—3262no—
327D4—3261no—
328D4—3261no—
329D4—6522no—
330D4—6522no—
331D4—6522no—
332D4—6522no—
333D4#1457969912yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—3261no—
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—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22—13044no—
364C22—3261no—
365C22—3261no—
366C22—3261no—
367C22—6521no—
368C22—6521no—
369C22—6521no—
370C22#1910475912yes—
371C22C2×D4×D16311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3261no—
376C4—3261no—
377C4—3261no—
378C4—6521no—
379C4—6521no—
380C4—6521no—
381C4C2×#1457969911yescharacteristic
382C4#1910478011yescharacteristic
383C2—22no—
384C2—42no—
385C2—81no—
386C2—13041no—
387C2—6521no—
388C2#2378230511yescharacteristic
389C2C2×#1910475911yescharacteristic
390C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#102×C163C2
socleC22×C163C22×D4
prime radicalD4×C22⋊D163C2
hypercenterC2×#102D163
hypocenterC163#3310
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. #23782028C2semidirect
4. #23781988C2semidirect
5. #23781978C2semidirect
6. #23781967C2semidirect
7. #23781910C2semidirect
8. #23781894C2semidirect
9. #23781864C2semidirect
10. #23781817C2semidirect
11. #23781816C2semidirect
12. #23781762C2semidirect
13. #23781736C2semidirect
14. #23781668C2semidirect
15. C2×#102×C163C2semidirect
18. C2×#14579688C22semidirect
20. C2×#14579687C22semidirect
21. C2×#14579682C22semidirect
24. #19104726C22semidirect
27. #19104715C22semidirect
28. #19104714C22semidirect
34. #19104700C22semidirect
38. #19104693C22semidirect
39. #19104668C22semidirect
40. #19104668C22semidirect
41. #19104667C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104660C22semidirect
44. #19104657C22semidirect
46. #19104656C22semidirect
47. #19104655C22semidirect
48. #19104655C22semidirect
49. C22⋊(C163⋊C8)C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
53. C22×C8×C163C22semidirect
54. C2×C4⋊C4×C163C22semidirect
57. #102×C163C22semidirect
58. #102×C163C22semidirect
195. C163#3310semidirect
211. C2×#102D163semidirect
219. C22×C8C2×D163semidirect
237. C2×C4⋊C4C2×D163semidirect
250. #102C2×D163semidirect
251. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823502
D4×C22⋊D1632
#237820282
#237819882
#237819782
#237819672
#237819102
#237818942
#237818642
#237818172
#237818162
#237817622
#237817362
#237816682
C2×#102×C1632
#3310163