Finite Group Databasebeta

← #35228648

Subgroups — #35228648

Order 20864.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782400C213yesmaximal, complemented
2#23782381C213yesmaximal, complemented
3#23782368C213yesmaximal, complemented
4#23782353C213yesmaximal, complemented
5#23782337C211yesmaximal, characteristic, complemented
6#23782327C211yesmaximal, characteristic, complemented
7#23782327C211yesmaximal, characteristic, complemented
8#23782289C213yesmaximal, complemented
9#23782271C213yesmaximal, complemented
10#23782202C213yesmaximal, complemented
11#23782200C213yesmaximal, complemented
12#23782147C211yesmaximal, characteristic, complemented
13#23782141C211yesmaximal, characteristic, complemented
14C4×#14579700C211yesmaximal, characteristic, complemented
15#557×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579700C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic
19#19104732C2213yes—
20#19104730C2213yes—
21#19104730C2211yescharacteristic
22#19104730C2216yes—
23#19104726C2213yes—
24#19104724C2216yes—
25#19104724C2213yes—
26#19104723C2213yes—
27#19104723C2213yes—
28#19104720C2213yes—
29#19104720C2216yes—
30#19104720C2216yes—
31#19104720C2213yes—
32#19104715C2213yes—
33#19104715C2213yes—
34#19104711C2213yes—
35#19104710C2213yes—
36#19104710C2213yes—
37#19104710C2213yes—
38#19104710C2213yes—
39#19104710C2216yes—
40#19104710C2213yes—
41Q8×Dic163C2211yescharacteristic
42C4×Dic326C2213yes—
43#19104700C2213yes—
44C4×D652C2213yes—
45C4×D652C2213yes—
46C4×D652C2213yes—
47C4×D652C2213yes—
48C4×D652C2213yes—
49#19104697C2216yes—
50#19104697C2213yes—
51#19104697C2213yes—
52#19104693C2216yes—
53#19104693C2213yes—
54#19104693C2213yes—
55#19104692C2213yes—
56#19104692C2213yes—
57C4⋊C4×D163C2213yes—
58C42×D163C2213yes—
59C4⋊Q8×C163C2213yes—
60#125×C163C2213yes—
61#125×C163C2213yes—
62C4×Q8×C163C2213yes—
63C4×Q8×C163C2211yescharacteristic
64C4×Q8×C163C2211yescharacteristic
65C4×Q8×C163C2211yescharacteristic
66#14579700—24no—
67#14579700—24no—
68C2×Dic326C2313yes—
69Q8×D163—24no—
70C2×D652C2313yes—
71C2×D652C2313yes—
72C2×D652C2316yes—
73C2×D652C2313yes—
74C2×D652C2313yes—
75C2×D652C2313yes—
76C2×C4×D163C2313yes—
77C2×C4×D163C2313yes—
78C2×C4×D163C2313yes—
79C2×C4×D163C2313yes—
80C2×C4×D163C2313yes—
81C2×C4×D163C2316yes—
82C4⋊Dic163C2313yes—
83C4⋊Dic163C2313yes—
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2313yes—
86Dic163⋊C4C2313yes—
87Dic163⋊C4C2311yescharacteristic
88Dic163⋊C4C2313yes—
89Dic163⋊C4C2313yes—
90(C2×D163)⋊C4C2311yescharacteristic
91(C2×D163)⋊C4C2313yes—
92(C2×D163)⋊C4C2313yes—
93(C2×D163)⋊C4C2313yes—
94(C2×D163)⋊C4C2313yes—
95(C2×D163)⋊C4C2316yes—
96(C2×D163)⋊C4C2316yes—
97(C2×D163)⋊C4C2313yes—
98(C2×D163)⋊C4C2316yes—
99(C2×D163)⋊C4C2311yescharacteristic
100(C2×D163)⋊C4C2311yescharacteristic
101(C2×D163)⋊C4C2313yes—
102(C2×D163)⋊C4C2313yes—
103C4×Dic163C2311yescharacteristic
104C4×Dic163C2313yes—
105C4×Dic163C2313yes—
106C4×Dic163C2316yes—
107C2×Q8×C163C2313yes—
108C2×Q8×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2313yes—
111C4⋊C4×C163C2313yes—
112C4⋊C4×C163C2313yes—
113C4⋊C4×C163C2313yes—
114C4⋊C4×C163C2313yes—
115C4⋊C4×C163C2316yes—
116C42×C163C2313yes—
117C42×C163C2313yes—
118C42×C163C2313yes—
119C22×D163C2411yescharacteristic
120C22×D163C2413yes—
121C22×D163C2413yes—
122Dic326—26no—
123D652—26no—
124D652—26no—
125D652—43no—
126D652—46no—
127D652—43no—
128D652—43no—
129C2×Dic163C2411yescharacteristic
130C2×Dic163C2413yes—
131C2×Dic163C2411yescharacteristic
132C2×Dic163C2413yes—
133C2×Dic163C2411yescharacteristic
134C4×D163—26no—
135C4×D163—26no—
136C4×D163—26no—
137C4×D163—43no—
138C4×D163—43no—
139C4×D163—46no—
140Q8×C163—43no—
141Q8×C163#4114yes—
142C2×C4×C163C2411yescharacteristic
143C2×C4×C163C2411yescharacteristic
144C2×C4×C163C2411yescharacteristic
145C2×C4×C163C2413yes—
146C2×C4×C163C2413yes—
147C2×C4×C163C2413yes—
148C2×C4×C163C2413yes—
149C22×C163C2511yescharacteristic
150C2×D163—21no—
151C2×D163—23no—
152C2×D163—23no—
153C2×D163—23no—
154C2×D163—23no—
155C2×D163—23no—
156C2×D163—21no—
157C2×D163—23no—
158C2×D163—21no—
159C4×C163—21no—
160C4×C163—23no—
161C4×C163—23no—
162C4×C163—21no—
163C4×C163—23no—
164C4×C163—21no—
165C4×C163C2×#4116yes—
166Dic163—21no—
167Dic163—21no—
168Dic163—23no—
169Dic163—21no—
170Dic163—23no—
171C2×C163C22×#4111yescharacteristic
172C2×C163#58511yescharacteristic
173C2×C163#58511yescharacteristic
174D163—41no—
175D163—43no—
176D163—43no—
177C163#352411yescharacteristic, complemented, Sylow
178#3524—1631nomaximal, Sylow
179#559—1633no—
180#557D16311yescharacteristic, complemented
181#555—1633no—
182#553—1633no—
183#552—1633no—
184#550—1631no—
185#548—1631no—
186#548—1631no—
187#542—1633no—
188#540—1633no—
189#529—1633no—
190#529—1633no—
191#520—1631no—
192#519—1631no—
193C4×#41—1631no—
194C2×#41—1631no—
195C2×#41—1631no—
196C22×Q8—1631no—
197C4⋊Q8C2×D16313yes—
198#127—1633no—
199#126—1631no—
200#126—1633no—
201#126—1636no—
202#125—1633no—
203#125C2×D16313yes—
204#125C2×D16313yes—
205#124—1633no—
206#124—1636no—
207#124—1633no—
208#124—1633no—
209#123—1636no—
210#123—1633no—
211#123—1633no—
212#123—1636no—
213C22⋊Q8—1633no—
214C22⋊Q8—1633no—
215C22⋊Q8—1633no—
216#121—1633no—
217#121—1633no—
218#121—1633no—
219#121—1633no—
220#121—1636no—
221#121—1633no—
222C4×Q8—1631no—
223C4×Q8—1633no—
224C4×Q8—1633no—
225C4×Q8C2×D16311yescharacteristic
226C4×Q8C2×D16311yescharacteristic
227C4×Q8C2×D16311yescharacteristic
228C4×Q8C2×D16313yes—
229C4×D4—1633no—
230C4×D4—1633no—
231C4×D4—1636no—
232C4×D4—1633no—
233C4×D4—1633no—
234C4×D4—1633no—
235C4×D4—1633no—
236C4×D4—1633no—
237#117—1633no—
238#117—1633no—
239#117—1636no—
240#117—1633no—
241#117—1633no—
242C2×C4⋊C4—1633no—
243C2×C42—1633no—
244#41—3264no—
245#41—3264no—
246C2×Q8—3264no—
247C2×Q8—1633no—
248C2×Q8C22×D16313yes—
249C2×Q8C22×D16311yescharacteristic
250C2×D4—1633no—
251C2×D4—1633no—
252C2×D4—1633no—
253C2×D4—1633no—
254C2×D4—1636no—
255C2×D4—1633no—
256C22×C4—1633no—
257C22×C4—1633no—
258C22×C4—1633no—
259C22×C4—1633no—
260C22×C4—1633no—
261C22×C4—1636no—
262C4⋊C4—1633no—
263C4⋊C4—1633no—
264C4⋊C4—1631no—
265C4⋊C4—1633no—
266C4⋊C4—1633no—
267C4⋊C4—1633no—
268C4⋊C4—1631no—
269C4⋊C4—1633no—
270C4⋊C4C22×D16313yes—
271C4⋊C4C22×D16313yes—
272C4⋊C4C22×D16313yes—
273C4⋊C4C22×D16316yes—
274C4⋊C4C22×D16313yes—
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16313yes—
277C22⋊C4—1631no—
278C22⋊C4—1633no—
279C22⋊C4—1633no—
280C22⋊C4—1633no—
281C22⋊C4—1636no—
282C22⋊C4—1633no—
283C22⋊C4—1636no—
284C22⋊C4—1631no—
285C22⋊C4—1633no—
286C22⋊C4—1631no—
287C22⋊C4—1633no—
288C22⋊C4—1636no—
289C22⋊C4—1633no—
290C42—1631no—
291C42—1633no—
292C42—1633no—
293C42—1636no—
294C42C22×D16313yes—
295C42C22×D16313yes—
296C42C22×D16313yes—
297C23—1631no—
298C23—1633no—
299C23—1633no—
300Q8—43no—
301Q8—3266no—
302Q8#1457970014yes—
303D4—3266no—
304D4—3266no—
305D4—6523no—
306D4—6526no—
307D4—6523no—
308D4—6523no—
309C2×C4—3266no—
310C2×C4—3266no—
311C2×C4—3266no—
312C2×C4—6523no—
313C2×C4—6523no—
314C2×C4—6526no—
315C2×C4—1631no—
316C2×C4—1631no—
317C2×C4—1633no—
318C2×C4—1633no—
319C2×C4—1631no—
320C2×C4C23×D16313yes—
321C2×C4C23×D16311yescharacteristic
322C2×C4C23×D16313yes—
323C2×C4C23×D16311yescharacteristic
324C2×C4C23×D16311yescharacteristic
325C2×C4C23×D16313yes—
326C2×C4C23×D16313yes—
327C22—3261no—
328C22—3263no—
329C22—3263no—
330C22—3263no—
331C22—3261no—
332C22—3263no—
333C22—3263no—
334C22—3263no—
335C22—3261no—
336C22C24×D16311yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—3261no—
344C4—3261no—
345C4—3263no—
346C4—3261no—
347C4—3263no—
348C4C2×#1457970016yes—
349C2—6521no—
350C2—6523no—
351C2—6523no—
352C2C22×#1457970011yescharacteristic
353C2#2378250111yescharacteristic
354C2#2378250211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#557×C163C2
hypercenter#557D163
hypocenterC163#3524
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #23782400C2semidirect
2. #23782381C2semidirect
3. #23782368C2semidirect
4. #23782353C2semidirect
5. #23782337C2semidirect
6. #23782327C2semidirect
7. #23782327C2semidirect
8. #23782289C2semidirect
9. #23782271C2semidirect
10. #23782202C2semidirect
11. #23782200C2semidirect
12. #23782147C2semidirect
13. #23782141C2semidirect
14. C4×#14579700C2semidirect
15. #557×C163C2semidirect
177. C163#3524semidirect
180. #557D163semidirect

Maximal subgroups

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

subgroupindex
#237824002
#237823812
#237823682
#237823532
#237823372
#237823272
#237822892
#237822712
#237822022
#237822002
#237821472
#237821412
C4×#145797002
#557×C1632
#3524163