Finite Group Databasebeta

← #35226611

Subgroups — #35226611

Order 20864.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782311C211yesmaximal, characteristic
2#23782027C211yesmaximal, characteristic, complemented
3#23782000C211yesmaximal, characteristic, complemented
4#23781978C211yesmaximal, characteristic, complemented
5#23781942C211yesmaximal, characteristic, complemented
6#23781913C211yesmaximal, characteristic, complemented
7#23781904C211yesmaximal, characteristic, complemented
8#23781887C211yesmaximal, characteristic, complemented
9#23781825C211yesmaximal, characteristic, complemented
10#23781815C211yesmaximal, characteristic, complemented
11#23781763C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic, complemented
13#23781664C211yesmaximal, characteristic, complemented
14C2×#19104655C211yesmaximal, characteristic, complemented
15#554×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579689C2211yescharacteristic, complemented
19C2×#14579686C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21C2×#14579683C2211yescharacteristic, complemented
22C22×C163⋊C8C2211yescharacteristic
23#19104723C2211yescharacteristic
24#19104720—21no—
25#19104714C2211yescharacteristic
26#19104709C2211yescharacteristic
27#19104708—21no—
28#19104703—21no—
29C4×D652C2211yescharacteristic
30C22⋊C4×D163—21no—
31#19104668C2211yescharacteristic, complemented
32#19104668C2211yescharacteristic, complemented
33#19104668C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35#19104659C2211yescharacteristic, complemented
36#19104658C2211yescharacteristic, complemented
37#19104658C2211yescharacteristic, complemented
38#19104658C2211yescharacteristic, complemented
39#19104655C2211yescharacteristic
40#19104655C2211yescharacteristic
41#19104655C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44#19104655C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47#19104646C2211yescharacteristic, complemented
48C4⋊Q8×C163C2211yescharacteristic, complemented
49#125×C163—21no—
50C22⋊Q8×C163—21no—
51C22⋊Q8×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53C22⋊Q8×C163C2211yescharacteristic, complemented
54C4×Q8×C163C2211yescharacteristic, complemented
55C4×D4×C163—21no—
56C4×D4×C163C2211yescharacteristic
57C2×C4⋊C4×C163C2211yescharacteristic
58C2×C4⋊C4×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326C2311yescharacteristic
63C2×D652—22no—
64C2×D652—21no—
65C2×D652—41no—
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72#14579689—41no—
73#14579686—22no—
74#14579686—41no—
75#14579683—41no—
76C2×C163⋊C8—41no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C2×C163⋊C8C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88C2×Q8×C163—21no—
89C2×Q8×C163C2311yescharacteristic
90C2×D4×C163D411yescharacteristic
91C22×C4×C163—21no—
92C22×C4×C163—21no—
93C22×C4×C163C2311yescharacteristic
94C22×C4×C163C2311yescharacteristic
95C4⋊C4×C163—21no—
96C4⋊C4×C163—21no—
97C4⋊C4×C163—21no—
98C4⋊C4×C163—21no—
99C4⋊C4×C163—21no—
100C4⋊C4×C163—21no—
101C4⋊C4×C163D411yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C22⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116Dic326—21no—
117Dic326—21no—
118D652—21no—
119D652—21no—
120D652—21no—
121D652—42no—
122D652#4112yes—
123C22⋊D163—42no—
124C22⋊D163—81no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C4×D163—42no—
128C4×D163—81no—
129C163⋊C8—21no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163C2×D411yescharacteristic
134C23×C163C2×D411yescharacteristic
135Q8×C163—21no—
136Q8×C163—21no—
137Q8×C163—81no—
138D4×C163—22no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—81no—
148C2×C4×C163—81no—
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2×D411yescharacteristic
153C2×C4×C163C2411yescharacteristic
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—21no—
159C22×C163#13712yes—
160C22×C163C22×D411yescharacteristic
161C2×D163—22no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—84no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—21no—
171C4×C163—21no—
172C4×C163—41no—
173C4×C163—41no—
174C4×C163—41no—
175C4×C163—41no—
176C4×C163C2×#4111yescharacteristic
177C4×C163H(5,𝔽2)11yescharacteristic
178Dic163—41no—
179Dic163—41no—
180C2×C163—22no—
181C2×C163—41no—
182C2×C163#54611yescharacteristic
183C2×C163C2×#13711yescharacteristic
184C2×C163#57611yescharacteristic
185D163—42no—
186D163—81no—
187C163#331411yescharacteristic, complemented, Sylow
188#3314—1631nomaximal, Sylow
189#554D16311yescharacteristic, complemented
190#546—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#474—1631no—
195#468—1631no—
196#467—1631no—
197#464—1631no—
198#452—1631no—
199#450—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#102—1631no—
204C2×#41—1631no—
205C22×D4—1631no—
206C2×Q16—1631no—
207C2×SD16—1631no—
208C2×SD16—1631no—
209C2×M16—1631no—
210C22×C8—1631no—
211C4⋊Q8C2×D16311yescharacteristic, complemented
212#125—21no—
213#124—1631no—
214#123—3261no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—1631no—
218C22⋊Q8C2×D16311yescharacteristic
219C22⋊Q8C2×D16311yescharacteristic, complemented
220#121—3261no—
221#121—1631no—
222#120—3261no—
223C4×Q8C2×D16311yescharacteristic, complemented
224C4×D4—21no—
225C4×D4—1631no—
226C4×D4C2×D16311yescharacteristic
227C2×C4⋊C4C2×D16311yescharacteristic
228C2×C4⋊C4C2×D16311yescharacteristic, complemented
229C2×C22⋊C4—3261no—
230#107—1631no—
231#107—1631no—
232#107—1631no—
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97—1631no—
247C24—3261no—
248#41—3262no—
249#41—6521no—
250C2×Q8—21no—
251C2×Q8—1631no—
252C2×Q8C22×D16311yescharacteristic
253C2×D4—3262no—
254C2×D4—3261no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—6521no—
258C2×D4—1631no—
259C2×D4—1631no—
260C2×D4C22⋊D16311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4C22×D16311yescharacteristic
266C22×C4C22×D16311yescharacteristic
267Q16—6521no—
268SD16—3262no—
269SD16—6521no—
270M16—6521no—
271C2×C8—6521no—
272C2×C8—1631no—
273C2×C8—1631no—
274C2×C8—1631no—
275C2×C8—1631no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—3261no—
283C4⋊C4—1631no—
284C4⋊C4C22⋊D16311yescharacteristic
285C4⋊C4C22×D16311yescharacteristic
286C4⋊C4C22×D16311yescharacteristic
287C4⋊C4C22×D16311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4C22⋊D16311yescharacteristic
296C22⋊C4C22⋊D16311yescharacteristic
297C42—21no—
298C42C22×D16311yescharacteristic
299C23—13041no—
300C23—3261no—
301C23—3261no—
302C23—6522no—
303C23—6522no—
304C23C2×C22⋊D16311yescharacteristic
305C23C2×C22⋊D16311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—3261no—
310Q8—3261no—
311D4—22no—
312D4—13041no—
313D4—3261no—
314D4—3261no—
315D4—3261no—
316D4—6522no—
317D4—6522no—
318D4—1632no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—13041no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—6522no—
333C2×C4C2×C22⋊D16311yescharacteristic
334C2×C4C2×C22⋊D16311yescharacteristic
335C2×C4C2×C22⋊D16311yescharacteristic
336C2×C4C2×C22⋊D16311yescharacteristic
337C2×C4C23×D16311yescharacteristic
338C8—3261no—
339C8—3261no—
340C8—3261no—
341C8—3261no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—13044no—
348C22—3262no—
349C22—6521no—
350C22—6521no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22#1910476612yes—
355C22C22×C22⋊D16311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—6521no—
365C4—6521no—
366C4#41×D16311yescharacteristic
367C4#1910478111yescharacteristic
368C2—22no—
369C2—41no—
370C2—13041no—
371C2—6522no—
372C2#2378230311yescharacteristic
373C2C2×#1910476611yescharacteristic
374C2#2378246811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#554×C163C2
socleC22×C163C22×D4
prime radical#23782311C2
hypercenter#554D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #23782027C2semidirect
3. #23782000C2semidirect
4. #23781978C2semidirect
5. #23781942C2semidirect
6. #23781913C2semidirect
7. #23781904C2semidirect
8. #23781887C2semidirect
9. #23781825C2semidirect
10. #23781815C2semidirect
11. #23781763C2semidirect
12. #23781737C2semidirect
13. #23781664C2semidirect
14. C2×#19104655C2semidirect
15. #554×C163C2semidirect
18. C2×#14579689C22semidirect
21. C2×#14579683C22semidirect
31. #19104668C22semidirect
32. #19104668C22semidirect
33. #19104668C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. #19104659C22semidirect
36. #19104658C22semidirect
37. #19104658C22semidirect
38. #19104658C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
47. #19104646C22semidirect
48. C4⋊Q8×C163C22semidirect
53. C22⋊Q8×C163C22semidirect
54. C4×Q8×C163C22semidirect
58. C2×C4⋊C4×C163C22semidirect
187. C163#3314semidirect
189. #554D163semidirect
211. C4⋊Q8C2×D163semidirect
219. C22⋊Q8C2×D163semidirect
223. C4×Q8C2×D163semidirect
228. C2×C4⋊C4C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823112
#237820272
#237820002
#237819782
#237819422
#237819132
#237819042
#237818872
#237818252
#237818152
#237817632
#237817372
#237816642
C2×#191046552
#554×C1632
#3314163