Finite Group Databasebeta

← #35226557

Subgroups — #35226557

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#23782355C211yesmaximal, characteristic, complemented
2#23782029C211yesmaximal, characteristic, complemented
3#23781986C211yesmaximal, characteristic, complemented
4#23781976C211yesmaximal, characteristic, complemented
5#23781965C211yesmaximal, characteristic, complemented
6#23781909C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781862C211yesmaximal, characteristic, complemented
9#23781816C211yesmaximal, characteristic, complemented
10#23781816C211yesmaximal, characteristic, complemented
11#23781765C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic, complemented
13#23781670C211yesmaximal, characteristic, complemented
14C2×#19104656C211yesmaximal, characteristic, complemented
15D42×C163C211yesmaximal, characteristic
16C2×#14579687C2211yescharacteristic
17C2×#14579687C2211yescharacteristic, complemented
18C2×#14579687C2211yescharacteristic, complemented
19C2×#14579683C2211yescharacteristic, complemented
20C22×C163⋊C8C2211yescharacteristic, complemented
21#19104728—21no—
22#19104727C2211yescharacteristic, complemented
23#19104725—21no—
24#19104716—21no—
25#19104714C2211yescharacteristic, complemented
26#19104714C2211yescharacteristic, complemented
27C4×Dic326C2211yescharacteristic, complemented
28#19104694C2211yescharacteristic, complemented
29#19104691—21no—
30C2×C4⋊Dic163C2211yescharacteristic, complemented
31#19104668C2211yescharacteristic, complemented
32#19104668C2211yescharacteristic, complemented
33#19104666C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35#19104658C2211yescharacteristic, complemented
36#19104656C2211yescharacteristic
37#19104656C2211yescharacteristic
38#19104656C2211yescharacteristic
39#19104656C2211yescharacteristic, complemented
40#19104656C2211yescharacteristic, complemented
41#19104656C2211yescharacteristic, complemented
42#19104656C2211yescharacteristic, complemented
43#19104656C2211yescharacteristic, complemented
44#19104656C2211yescharacteristic, complemented
45C22⋊(C163⋊C8)C2211yescharacteristic, complemented
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47#19104646C2211yescharacteristic, complemented
48C22×D4×C163—21no—
49C22×D4×C163C2211yescharacteristic
50C22×D4×C163C2211yescharacteristic
51#127×C163C2211yescharacteristic
52#121×C163—21no—
53#121×C163C2211yescharacteristic
54#121×C163C2211yescharacteristic
55#120×C163—21no—
56#120×C163—21no—
57C4×D4×C163C2211yescharacteristic
58C4×D4×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C22×Dic163—21no—
61#14579687—22no—
62#14579687—41no—
63#14579687—41no—
64#14579683—41no—
65C2×C163⋊C8—41no—
66C2×C163⋊C8C2311yescharacteristic
67C2×C163⋊C8C2311yescharacteristic
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163—21no—
71C4⋊Dic163C2311yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74Dic163⋊C4—21no—
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C22⋊Dic163—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C4×Dic163—21no—
83C4×Dic163—21no—
84C24×C163—21no—
85C24×C163—21no—
86C2×D4×C163—22no—
87C2×D4×C163—22no—
88C2×D4×C163—21no—
89C2×D4×C163—21no—
90C2×D4×C163—21no—
91C2×D4×C163—21no—
92C2×D4×C163—21no—
93C2×D4×C163—21no—
94C2×D4×C163—41no—
95C2×D4×C163—41no—
96C2×D4×C163—44no—
97C2×D4×C163D411yescharacteristic
98C2×D4×C163C2311yescharacteristic
99C2×D4×C163C2311yescharacteristic
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163—21no—
102C22×C4×C163C2311yescharacteristic
103C22×C4×C163C2311yescharacteristic
104C4⋊C4×C163D411yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C22⋊C4×C163—21no—
107C22⋊C4×C163—21no—
108C22⋊C4×C163—21no—
109C22⋊C4×C163D411yescharacteristic
110C22⋊C4×C163D411yescharacteristic
111C42×C163C2311yescharacteristic
112Dic326—21no—
113Dic326—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—42no—
119C163⋊C8—21no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163—21no—
124C23×C163—21no—
125C23×C163—21no—
126C23×C163—42no—
127C23×C163—42no—
128C23×C163—42no—
129C23×C163—42no—
130C23×C163—81no—
131C23×C163—81no—
132C23×C163C2×D411yescharacteristic
133C23×C163C2×D411yescharacteristic
134D4×C163—22no—
135D4×C163—21no—
136D4×C163—21no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—42no—
141D4×C163—42no—
142D4×C163—42no—
143D4×C163—42no—
144D4×C163—42no—
145D4×C163—81no—
146D4×C163#4112yes—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—42no—
153C2×C4×C163C2×D411yescharacteristic
154C2×C4×C163C2×D411yescharacteristic
155C2×C4×C163C2×D411yescharacteristic
156C2×C4×C163C2×D411yescharacteristic
157C2×C4×C163C2411yescharacteristic
158C22×C163—21no—
159C22×C163—21no—
160C22×C163—21no—
161C22×C163—22no—
162C22×C163—21no—
163C22×C163—21no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—41no—
170C22×C163—41no—
171C22×C163—41no—
172C22×C163—84no—
173C22×C163—84no—
174C22×C163AGL(1,ℤ/8)12yes—
175C22×C163C22×D411yescharacteristic
176C4×C163—21no—
177C4×C163—21no—
178C4×C163—21no—
179C4×C163—21no—
180C4×C163—41no—
181C4×C163C2×#4111yescharacteristic
182C4×C163H(5,𝔽2)11yescharacteristic
183Dic163—41no—
184Dic163—41no—
185Dic163—41no—
186Dic163—41no—
187C2×C163—22no—
188C2×C163—41no—
189C2×C163—42no—
190C2×C163—81no—
191C2×C163—81no—
192C2×C163#54611yescharacteristic
193C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
194C2×C163#57711yescharacteristic
195C163#331011yescharacteristic, complemented, Sylow
196#3310—1631nomaximal, Sylow
197#552—1631no—
198D42D16311yescharacteristic
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×#102—1631no—
212C22×D4—21no—
213C22×D4C2×D16311yescharacteristic
214C22×D4C2×D16311yescharacteristic
215C2×SD16—1631no—
216C2×SD16—1631no—
217C2×SD16—1631no—
218C2×M16—1631no—
219C22×C8—1631no—
220#127C2×D16311yescharacteristic
221#126—3261no—
222#125—3261no—
223#125—1631no—
224#123—3261no—
225C22⋊Q8—1631no—
226C22⋊Q8—1631no—
227#121—21no—
228#121C2×D16311yescharacteristic
229#121C2×D16311yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8—1631no—
233C4×D4C2×D16311yescharacteristic
234C4×D4C2×D16311yescharacteristic
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#102—1631no—
251#102—1631no—
252C22⋊C8—1631no—
253C22⋊C8—1631no—
254#97—1631no—
255C24—21no—
256C24—21no—
257C2×Q8—1631no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D16311yescharacteristic
270C2×D4C22×D16311yescharacteristic
271C2×D4C22×D16311yescharacteristic
272C2×D4C22×D16311yescharacteristic
273C22×C4—21no—
274C22×C4—3261no—
275C22×C4C22×D16311yescharacteristic
276C22×C4C22×D16311yescharacteristic
277SD16—3262no—
278SD16—6521no—
279SD16—6521no—
280M16—6521no—
281C2×C8—6521no—
282C2×C8—1631no—
283C2×C8—1631no—
284C2×C8—1631no—
285C2×C8—1631no—
286C4⋊C4—3261no—
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—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3261no—
301C22⋊C4—3261no—
302C22⋊C4—3261no—
303C22⋊C4C22⋊D16311yescharacteristic
304C22⋊C4C22⋊D16311yescharacteristic
305C42—3261no—
306C42—3261no—
307C42C22×D16311yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D16311yescharacteristic
318C23C2×C22⋊D16311yescharacteristic
319Q8—3261no—
320Q8—3261no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#1457969812yes—
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×C4C2×C22⋊D16311yescharacteristic
346C2×C4C2×C22⋊D16311yescharacteristic
347C2×C4C2×C22⋊D16311yescharacteristic
348C2×C4C2×C22⋊D16311yescharacteristic
349C2×C4C23×D16311yescharacteristic
350C8—3261no—
351C8—3261no—
352C8—3261no—
353C8—3261no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#1910476112yes—
371C22C22×C22⋊D16311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—6521no—
378C4—6521no—
379C4—6521no—
380C4—6521no—
381C4C2×#1457969811yescharacteristic
382C4#1910478011yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#2378230911yescharacteristic
389C2C2×#1910476111yescharacteristic
390C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting, prime radicalD42×C163C2
socleC22×C163C22×D4
hypercenterD42D163
hypocenterC163#3310
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #23782355C2semidirect
2. #23782029C2semidirect
3. #23781986C2semidirect
4. #23781976C2semidirect
5. #23781965C2semidirect
6. #23781909C2semidirect
7. #23781892C2semidirect
8. #23781862C2semidirect
9. #23781816C2semidirect
10. #23781816C2semidirect
11. #23781765C2semidirect
12. #23781737C2semidirect
13. #23781670C2semidirect
14. C2×#19104656C2semidirect
17. C2×#14579687C22semidirect
18. C2×#14579687C22semidirect
19. C2×#14579683C22semidirect
20. C22×C163⋊C8C22semidirect
22. #19104727C22semidirect
25. #19104714C22semidirect
26. #19104714C22semidirect
27. C4×Dic326C22semidirect
28. #19104694C22semidirect
30. C2×C4⋊Dic163C22semidirect
31. #19104668C22semidirect
32. #19104668C22semidirect
33. #19104666C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. #19104658C22semidirect
39. #19104656C22semidirect
40. #19104656C22semidirect
41. #19104656C22semidirect
42. #19104656C22semidirect
43. #19104656C22semidirect
44. #19104656C22semidirect
45. C22⋊(C163⋊C8)C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
47. #19104646C22semidirect
195. C163#3310semidirect

Maximal subgroups

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

subgroupindex
#237823552
#237820292
#237819862
#237819762
#237819652
#237819092
#237818922
#237818622
#237818162
#237817652
#237817372
#237816702
C2×#191046562
D42×C1632
#3310163