Finite Group Databasebeta

← #35226497

Subgroups — #35226497

Order 20864.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782319C211yesmaximal, characteristic, complemented
2#23782024C211yesmaximal, characteristic, complemented
3#23781993C211yesmaximal, characteristic, complemented
4#23781981C211yesmaximal, characteristic, complemented
5#23781955C211yesmaximal, characteristic, complemented
6#23781916C211yesmaximal, characteristic, complemented
7#23781896C211yesmaximal, characteristic, complemented
8#23781877C211yesmaximal, characteristic, complemented
9#23781810C211yesmaximal, characteristic, complemented
10#23781799C211yesmaximal, characteristic, complemented
11#23781775C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic, complemented
13#23781666C211yesmaximal, characteristic, complemented
14C2×#19104656C211yesmaximal, characteristic, complemented
15#546×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×#14579686C2211yescharacteristic, complemented
18C2×#14579684C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×#14579683C2211yescharacteristic, complemented
21C22×C163⋊C8C2211yescharacteristic
22#19104735C2211yescharacteristic, complemented
23#19104719—21no—
24#19104719—21no—
25#19104715—21no—
26#19104709C2211yescharacteristic
27#19104709C2211yescharacteristic, complemented
28C4×D652C2211yescharacteristic, complemented
29C2×C4⋊Dic163C2211yescharacteristic
30C2×C4⋊Dic163C2211yescharacteristic, complemented
31#19104666C2211yescharacteristic, complemented
32#19104666C2211yescharacteristic, complemented
33#19104666C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35#19104659C2211yescharacteristic, complemented
36#19104659C2211yescharacteristic, complemented
37#19104659C2211yescharacteristic, complemented
38#19104656C2211yescharacteristic
39#19104656C2211yescharacteristic
40#19104656C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104655C2211yescharacteristic, complemented
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47#19104646C2211yescharacteristic, complemented
48C2×#41×C163C2211yescharacteristic
49C22×D4×C163C2211yescharacteristic
50#124×C163C2211yescharacteristic
51#123×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53#121×C163—21no—
54#121×C163C2211yescharacteristic
55#120×C163—21no—
56C4×D4×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic
58C2×C22⋊C4×C163—21no—
59#14579698—44no—
60C2×Dic326—21no—
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C22×Dic163—21no—
65C22×Dic163—21no—
66C2×C4×D163—21no—
67#14579686—41no—
68#14579684—22no—
69#14579684—41no—
70#14579683—41no—
71C2×C163⋊C8—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163—21no—
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84(C2×D163)⋊C4—21no—
85C24×C163—21no—
86#41×C163—22no—
87#41×C163—41no—
88C2×Q8×C163C2311yescharacteristic
89C2×D4×C163—22no—
90C2×D4×C163—21no—
91C2×D4×C163—21no—
92C2×D4×C163—21no—
93C2×D4×C163—41no—
94C2×D4×C163D411yescharacteristic
95C2×D4×C163C2311yescharacteristic
96C2×D4×C163C2311yescharacteristic
97C22×C4×C163—21no—
98C22×C4×C163—21no—
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C4⋊C4×C163—21no—
102C4⋊C4×C163D411yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C22⋊C4×C163—21no—
108C22⋊C4×C163—21no—
109C22⋊C4×C163D411yescharacteristic
110C22⋊C4×C163D411yescharacteristic
111C42×C163C2311yescharacteristic
112C22×D163—21no—
113Dic326—42no—
114D652—21no—
115D652—21no—
116C22⋊D163—42no—
117C22⋊D163—42no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—42no—
122C2×Dic163—42no—
123C4×D163—42no—
124C163⋊C8—21no—
125C163⋊C8—21no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C23×C163—21no—
129C23×C163—21no—
130C23×C163—42no—
131C23×C163—42no—
132C23×C163—81no—
133C23×C163C2×D411yescharacteristic
134C23×C163C2×D411yescharacteristic
135Q8×C163—21no—
136Q8×C163—21no—
137D4×C163—22no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—21no—
141D4×C163—42no—
142D4×C163—42no—
143D4×C163—81no—
144D4×C163#4112yescomplemented
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—42no—
152C2×C4×C163—81no—
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—84no—
170C22×C163AGL(1,ℤ/8)12yes—
171C22×C163C22×D411yescharacteristic
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—41no—
175C4×C163—21no—
176C4×C163—21no—
177C4×C163—21no—
178C4×C163—21no—
179C4×C163—41no—
180C4×C163—41no—
181C4×C163C2×#4111yescharacteristic
182C4×C163H(5,𝔽2)11yescharacteristic
183Dic163—41no—
184Dic163—41no—
185Dic163—41no—
186C2×C163—22no—
187C2×C163—41no—
188C2×C163—42no—
189C2×C163—81no—
190C2×C163#54611yescharacteristic
191C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C163#57711yescharacteristic
193D163—81no—
194C163#330611yescharacteristic, complemented, Sylow
195#3306—1631nomaximal, Sylow
196#547—1631no—
197#546D16311yescharacteristic, complemented
198#489—1631no—
199#483—1631no—
200#481—1631no—
201#476—1631no—
202#469—1631no—
203#465—1631no—
204#463—1631no—
205#449—1631no—
206#447—1631no—
207#442—1631no—
208#435—1631no—
209#417—1631no—
210C2×#102—1631no—
211C2×#41—3261no—
212C2×#41C2×D16311yescharacteristic
213C22×D4C2×D16311yescharacteristic
214C2×SD16—1631no—
215C2×D8—1631no—
216C2×D8—1631no—
217C2×M16—1631no—
218C22×C8—1631no—
219C4⋊Q8—1631no—
220#124C2×D16311yescharacteristic
221#123—21no—
222#123—3261no—
223#123—3261no—
224C22⋊Q8—3261no—
225C22⋊Q8C2×D16311yescharacteristic
226#121—21no—
227#121—1631no—
228#121—1631no—
229#121C2×D16311yescharacteristic
230#120—21no—
231C4×D4—1631no—
232C4×D4C2×D16311yescharacteristic
233C4×D4C2×D16311yescharacteristic
234C2×C4⋊C4—1631no—
235C2×C4⋊C4—1631no—
236C2×C22⋊C4—21no—
237#106—1631no—
238#106—1631no—
239#106—1631no—
240C4⋊C8—1631no—
241#103—1631no—
242#103—1631no—
243#103—1631no—
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102—1631no—
251C22⋊C8—1631no—
252C22⋊C8—1631no—
253#97—1631no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—6524no—
258C2×Q8—3261no—
259C2×Q8C22×D16311yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—3261no—
266C2×D4—3261no—
267C2×D4—1631no—
268C2×D4C22⋊D16311yescharacteristic
269C2×D4C22×D16311yescharacteristic
270C2×D4C22×D16311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3261no—
274C22×C4—3261no—
275C22×C4—3261no—
276C22×C4C22×D16311yescharacteristic
277C22×C4C22×D16311yescharacteristic
278SD16—6521no—
279D8—3262no—
280D8—6521no—
281M16—6521no—
282C2×C8—6521no—
283C2×C8—1631no—
284C2×C8—1631no—
285C2×C8—1631no—
286C2×C8—1631no—
287C4⋊C4—21no—
288C4⋊C4—3261no—
289C4⋊C4—3261no—
290C4⋊C4—3261no—
291C4⋊C4—1631no—
292C4⋊C4—1631no—
293C4⋊C4—1631no—
294C4⋊C4C22⋊D16311yescharacteristic
295C4⋊C4C22×D16311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—3261no—
302C22⋊C4—3261no—
303C22⋊C4—3261no—
304C22⋊C4C22⋊D16311yescharacteristic
305C22⋊C4C22⋊D16311yescharacteristic
306C42C22×D16311yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—3261no—
313C23C2×C22⋊D16311yescharacteristic
314C23C2×C22⋊D16311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6522no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—3261no—
326D4—3261no—
327D4—6522no—
328D4—6522no—
329D4#1457969812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—3261no—
341C2×C4—6522no—
342C2×C4—6522no—
343C2×C4—6522no—
344C2×C4C2×C22⋊D16311yescharacteristic
345C2×C4C2×C22⋊D16311yescharacteristic
346C2×C4C2×C22⋊D16311yescharacteristic
347C2×C4C2×C22⋊D16311yescharacteristic
348C2×C4C23×D16311yescharacteristic
349C8—3261no—
350C8—3261no—
351C8—3261no—
352C8—3261no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—6521no—
366C22—6521no—
367C22—6521no—
368C22#1910476012yes—
369C22C22×C22⋊D16311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—6521no—
377C4—6521no—
378C4—6521no—
379C4C2×#1457969811yescharacteristic
380C4#1910478011yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—13041no—
386C2#2378230911yescharacteristic
387C2C2×#1910476011yescharacteristic
388C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#546×C163C2
socleC22×C163C22×D4
hypercenter#546D163
hypocenterC163#3306
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #23782319C2semidirect
2. #23782024C2semidirect
3. #23781993C2semidirect
4. #23781981C2semidirect
5. #23781955C2semidirect
6. #23781916C2semidirect
7. #23781896C2semidirect
8. #23781877C2semidirect
9. #23781810C2semidirect
10. #23781799C2semidirect
11. #23781775C2semidirect
12. #23781737C2semidirect
13. #23781666C2semidirect
14. C2×#19104656C2semidirect
15. #546×C163C2semidirect
17. C2×#14579686C22semidirect
20. C2×#14579683C22semidirect
22. #19104735C22semidirect
27. #19104709C22semidirect
28. C4×D652C22semidirect
30. C2×C4⋊Dic163C22semidirect
31. #19104666C22semidirect
32. #19104666C22semidirect
33. #19104666C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. #19104659C22semidirect
36. #19104659C22semidirect
37. #19104659C22semidirect
44. #19104655C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
47. #19104646C22semidirect
144. D4×C163#41semidirect
194. C163#3306semidirect
197. #546D163semidirect
329. D4#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823192
#237820242
#237819932
#237819812
#237819552
#237819162
#237818962
#237818772
#237818102
#237817992
#237817752
#237817372
#237816662
C2×#191046562
#546×C1632
#3306163