Finite Group Databasebeta

← #35226550

Subgroups — #35226550

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#23782349C211yesmaximal, characteristic, complemented
2D4×C22⋊D163C211yesmaximal, characteristic
3#23782028C211yesmaximal, characteristic, complemented
4#23781990C211yesmaximal, characteristic, complemented
5#23781977C211yesmaximal, characteristic, complemented
6#23781967C211yesmaximal, characteristic, complemented
7#23781913C211yesmaximal, characteristic, complemented
8#23781891C211yesmaximal, characteristic, complemented
9#23781863C211yesmaximal, characteristic, complemented
10#23781814C211yesmaximal, characteristic, complemented
11#23781766C211yesmaximal, characteristic, complemented
12#23781739C211yesmaximal, characteristic, complemented
13#23781670C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#450×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163—21no—
17C2×D4×D163C2211yescharacteristic
18C2×#14579686C2211yescharacteristic, complemented
19C2×#14579683C2211yescharacteristic, complemented
20C2×C8×D163C2211yescharacteristic, complemented
21#19104731C2211yescharacteristic
22#19104728—21no—
23#19104726C2211yescharacteristic, complemented
24#19104725—21no—
25#19104717—21no—
26#19104711C2211yescharacteristic, complemented
27#19104709C2211yescharacteristic
28#19104707—21no—
29#19104706C2211yescharacteristic
30#19104704—21no—
31#19104703—21no—
32Q8×Dic163C2211yescharacteristic, complemented
33D4×Dic163C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35#19104694C2211yescharacteristic, complemented
36#19104693—21no—
37C4⋊C4×D163C2211yescharacteristic, complemented
38#19104669C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40#19104667C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic, complemented
55C22⋊Q8×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×D652C2311yescharacteristic
61C2×C22⋊D163—21no—
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—44no—
67C2×C22⋊D163D411yescharacteristic
68D4×D163—22no—
69D4×D163—41no—
70C22×Dic163—21no—
71C2×C4×D163—21no—
72C2×C4×D163C2311yescharacteristic
73#14579686—41no—
74#14579683—41no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C8×D163—41no—
78C4⋊Dic163—21no—
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
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)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C24×C163—21no—
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—22no—
100C2×D4×C163—21no—
101C2×D4×C163—41no—
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105SD16×C163—22no—
106SD16×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163—21no—
110C4⋊C4×C163C2311yescharacteristic
111C22⋊C4×C163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117D652—21no—
118D652—21no—
119C22⋊D163—22no—
120C22⋊D163—42no—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C22⋊D163—81no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—42no—
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C2×Dic163C2×D411yescharacteristic
132C4×D163—21no—
133C4×D163—21no—
134C4×D163—42no—
135C163⋊C8—21no—
136C163⋊C8—21no—
137C23×C163—21no—
138C23×C163—21no—
139C23×C163—42no—
140C23×C163—42no—
141C23×C163—81no—
142C23×C163C2×D411yescharacteristic
143Q8×C163—21no—
144Q8×C163—21no—
145D4×C163—21no—
146D4×C163—21no—
147D4×C163—21no—
148D4×C163—42no—
149D4×C163#4112yes—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163C2×D411yescharacteristic
155C2×C4×C163C2411yescharacteristic
156C8×C163—21no—
157C8×C163—21no—
158C22×C163—21no—
159C22×C163—22no—
160C22×C163—21no—
161C22×C163—21no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163—84no—
168C22×C163C22×D411yescharacteristic
169C2×D163—21no—
170C2×D163—21no—
171C2×D163—41no—
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—84no—
175C2×D163AGL(1,ℤ/8)12yes—
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—42no—
188C2×C163—41no—
189C2×C163—81no—
190C2×C163#54611yescharacteristic
191C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C163#57711yescharacteristic
193D163—22no—
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#450D16311yescharacteristic, complemented
208#440—1631no—
209#435—1631no—
210#418—1631no—
211C2×#102—1631no—
212C22×D4—3261no—
213C22×D4—1631no—
214C22×D4C2×D16311yescharacteristic
215C2×SD16—1631no—
216C2×SD16C2×D16311yescharacteristic
217C2×SD16C2×D16311yescharacteristic, complemented
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⋊Q8C2×D16311yescharacteristic, complemented
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⋊C4—1631no—
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#102—1631no—
249#102—1631no—
250#102C2×D16311yescharacteristic
251#102C2×D16311yescharacteristic, complemented
252C22⋊C8—1631no—
253C22⋊C8C2×D16311yescharacteristic, complemented
254#97—1631no—
255C24—21no—
256C24—3261no—
257C2×Q8C22×D16311yescharacteristic
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—3261no—
274C22×C4—3261no—
275C22×C4—1631no—
276C22×C4C22×D16311yescharacteristic
277SD16—22no—
278SD16—41no—
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⋊C4—1631no—
295C4⋊C4—1631no—
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—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—13041no—
314C23—3261no—
315C23—6522no—
316C23—6522no—
317C23—1631no—
318C23D4×D16311yescharacteristic
319Q8—21no—
320Q8—21no—
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#1457969812yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—3261no—
341C2×C4—3261no—
342C2×C4—3261no—
343C2×C4—6522no—
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—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—13044no—
365C22—3261no—
366C22—3261no—
367C22—6521no—
368C22—6521no—
369C22—6521no—
370C22—1632no—
371C22C2×D4×D16311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3261no—
376C4—3261no—
377C4—3261no—
378C4—6521no—
379C4—6521no—
380C4—6521no—
381C4C2×#1457969811yescharacteristic
382C4#1910478011yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—13041no—
387C2—3262no—
388C2#2378230211yescharacteristic
389C2AGL(1,ℤ/8)×D16311yescharacteristic
390C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#450×C163C2
socleC22×C163C22×D4
prime radicalD4×C22⋊D163C2
hypercenter#450D163
hypocenterC163#3310
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23782349C2semidirect
3. #23782028C2semidirect
4. #23781990C2semidirect
5. #23781977C2semidirect
6. #23781967C2semidirect
7. #23781913C2semidirect
8. #23781891C2semidirect
9. #23781863C2semidirect
10. #23781814C2semidirect
11. #23781766C2semidirect
12. #23781739C2semidirect
13. #23781670C2semidirect
14. #102×D163C2semidirect
15. #450×C163C2semidirect
18. C2×#14579686C22semidirect
19. C2×#14579683C22semidirect
20. C2×C8×D163C22semidirect
23. #19104726C22semidirect
26. #19104711C22semidirect
32. Q8×Dic163C22semidirect
35. #19104694C22semidirect
37. C4⋊C4×D163C22semidirect
38. #19104669C22semidirect
39. #19104668C22semidirect
40. #19104667C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104659C22semidirect
43. #19104657C22semidirect
44. #19104657C22semidirect
47. #19104656C22semidirect
48. #19104656C22semidirect
49. #19104655C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
54. C2×SD16×C163C22semidirect
55. C22⋊Q8×C163C22semidirect
57. #102×C163C22semidirect
58. C22⋊C8×C163C22semidirect
195. C163#3310semidirect
207. #450D163semidirect
217. C2×SD16C2×D163semidirect
226. C22⋊Q8C2×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
#237823492
D4×C22⋊D1632
#237820282
#237819902
#237819772
#237819672
#237819132
#237818912
#237818632
#237818142
#237817662
#237817392
#237816702
#102×D1632
#450×C1632
#3310163