Finite Group Databasebeta

← #35228634

Subgroups — #35228634

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#23782401C213yesmaximal, complemented
2#23782391C211yesmaximal, characteristic, complemented
3#23782379C213yesmaximal, complemented
4#23782367C213yesmaximal, complemented
5#23782355C213yesmaximal, complemented
6#23782322C211yesmaximal, characteristic, complemented
7#23782322C211yesmaximal, characteristic, complemented
8#23782282C213yesmaximal, complemented
9#23782269C213yesmaximal, complemented
10#23782196C213yesmaximal, complemented
11#23782195C213yesmaximal, complemented
12#23782148C211yesmaximal, characteristic, complemented
13#23782141C211yesmaximal, characteristic, complemented
14#41×Dic163C211yesmaximal, characteristic, complemented
15#550×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17#19104734C2213yes—
18#19104730C2211yescharacteristic
19#19104728C2213yes—
20#19104728C2216yes—
21#19104727C2213yes—
22#19104726C2213yes—
23#19104725C2213yes—
24#19104722C2213yes—
25#19104722C2213yes—
26#19104722C2216yes—
27#19104721C2213yes—
28#19104719C2213yes—
29#19104716C2213yes—
30#19104716C2216yes—
31#19104716C2216yes—
32#19104714C2213yes—
33#19104714C2213yes—
34#19104712C2213yes—
35#19104707C2213yes—
36#19104707C2216yes—
37#19104706C2213yes—
38#19104706C2213yes—
39Q8×Dic163C2211yescharacteristic
40Q8×Dic163C2211yescharacteristic
41Q8×Dic163C2211yescharacteristic
42C4×Dic326C2213yes—
43C4×Dic326C2213yes—
44#19104700C2213yes—
45D4×Dic163C2213yes—
46D4×Dic163C2216yes—
47D4×Dic163C2213yes—
48D4×Dic163C2213yes—
49D4×Dic163C2213yes—
50C4×C22⋊D163C2213yes—
51C4×C22⋊D163C2213yes—
52#19104694C2213yes—
53#19104694C2213yes—
54#19104693C2213yes—
55#19104692C2213yes—
56#19104691C2216yes—
57C2×C4⋊Dic163C2213yes—
58C2×C4×Dic163C2213yes—
59C2×#41×C163C2211yescharacteristic
60C2×#41×C163C2211yescharacteristic
61#127×C163C2213yes—
62#121×C163C2213yes—
63#121×C163C2213yes—
64C4×Q8×C163C2211yescharacteristic
65C4×D4×C163C2213yes—
66C2×Dic326C2313yes—
67C2×Dic326C2313yes—
68Q8×D163—24no—
69C2×C22⋊D163C2313yes—
70C2×C22⋊D163C2313yes—
71C22×Dic163C2313yes—
72C22×Dic163C2313yes—
73C22×Dic163C2316yes—
74C2×C4×D163C2313yes—
75C4⋊Dic163C2313yes—
76C4⋊Dic163C2313yes—
77C4⋊Dic163C2313yes—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2313yes—
80C4⋊Dic163C2313yes—
81Dic163⋊C4C2313yes—
82Dic163⋊C4C2316yes—
83Dic163⋊C4C2313yes—
84Dic163⋊C4C2313yes—
85Dic163⋊C4C2313yes—
86Dic163⋊C4C2311yescharacteristic
87Dic163⋊C4C2313yes—
88C22⋊Dic163C2316yes—
89C22⋊Dic163C2313yes—
90C22⋊Dic163C2316yes—
91C22⋊Dic163C2313yes—
92C22⋊Dic163C2313yes—
93C22⋊Dic163C2313yes—
94C22⋊Dic163C2316yes—
95(C2×D163)⋊C4C2311yescharacteristic
96(C2×D163)⋊C4C2313yes—
97(C2×D163)⋊C4C2311yescharacteristic
98(C2×D163)⋊C4C2311yescharacteristic
99C4×Dic163C2311yescharacteristic
100C4×Dic163C2313yes—
101C4×Dic163C2313yes—
102C4×Dic163C2313yes—
103C4×Dic163C2316yes—
104C4×Dic163C2313yes—
105#41×C163—24no—
106#41×C163—24no—
107C2×Q8×C163C2311yescharacteristic
108C2×D4×C163C2313yes—
109C2×D4×C163C2313yes—
110C2×D4×C163C2313yes—
111C2×D4×C163C2316yes—
112C22×C4×C163C2313yes—
113C22×C4×C163C2313yes—
114C4⋊C4×C163C2311yescharacteristic
115C4⋊C4×C163C2313yes—
116C22⋊C4×C163C2313yes—
117C22⋊C4×C163C2313yes—
118C42×C163C2313yes—
119C22×D163C2411yescharacteristic
120Dic326—26no—
121Dic326—43no—
122C22⋊D163—43no—
123C22⋊D163—43no—
124C2×Dic163—43no—
125C2×Dic163—46no—
126C2×Dic163—43no—
127C2×Dic163C2411yescharacteristic
128C2×Dic163C2411yescharacteristic
129C2×Dic163C2413yes—
130C2×Dic163C2411yescharacteristic
131C2×Dic163C2413yes—
132C2×Dic163C2413yes—
133C2×Dic163C2413yes—
134C4×D163—26no—
135C23×C163C2413yes—
136C23×C163C2413yes—
137Q8×C163#4114yes—
138D4×C163—26no—
139D4×C163—26no—
140D4×C163—43no—
141D4×C163—46no—
142C2×C4×C163—26no—
143C2×C4×C163—26no—
144C2×C4×C163C2411yescharacteristic
145C2×C4×C163C2413yes—
146C2×C4×C163C2411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C2×C4×C163C2413yes—
149C22×C163—23no—
150C22×C163—23no—
151C22×C163—23no—
152C22×C163—23no—
153C22×C163—23no—
154C22×C163—23no—
155C22×C163C2511yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—21no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—23no—
163C4×C163C2×#4116yes—
164Dic163—23no—
165Dic163—21no—
166Dic163—21no—
167Dic163—23no—
168Dic163—23no—
169Dic163—23no—
170Dic163—21no—
171C2×C163—43no—
172C2×C163—43no—
173C2×C163C22×#4111yescharacteristic
174C2×C163#58511yescharacteristic
175C2×C163#58511yescharacteristic
176D163—41no—
177C163#352411yescharacteristic, complemented, Sylow
178#3524—1631nomaximal, Sylow
179#559—1633no—
180#557—1631no—
181#555—1633no—
182#553—1633no—
183#552—1633no—
184#550D16311yescharacteristic, complemented
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×#41C2×D16311yescharacteristic
195C2×#41C2×D16311yescharacteristic
196C22×Q8—1631no—
197C4⋊Q8—1633no—
198#127C2×D16313yes—
199#126—1631no—
200#126—1633no—
201#126—1636no—
202#125—1633no—
203#125—1633no—
204#125—1633no—
205#124—1633no—
206#124—1633no—
207#124—1636no—
208#124—1633no—
209#123—1636no—
210#123—1633no—
211#123—1636no—
212#123—1633no—
213C22⋊Q8—1633no—
214C22⋊Q8—1633no—
215C22⋊Q8—1633no—
216#121—1633no—
217#121—1633no—
218#121—1633no—
219#121—1636no—
220#121C2×D16313yes—
221#121C2×D16313yes—
222C4×Q8—1631no—
223C4×Q8—1631no—
224C4×Q8—1631no—
225C4×Q8—1633no—
226C4×Q8—1633no—
227C4×Q8—1633no—
228C4×Q8C2×D16311yescharacteristic
229C4×D4—1633no—
230C4×D4—1633no—
231C4×D4—1633no—
232C4×D4—1633no—
233C4×D4—1636no—
234C4×D4—1633no—
235C4×D4—1633no—
236C4×D4C2×D16313yes—
237#117—1633no—
238#117—1633no—
239#117—1633no—
240#117—1636no—
241#117—1633no—
242C2×C4⋊C4—1633no—
243C2×C42—1633no—
244#41—24no—
245#41—24no—
246C2×Q8—3264no—
247C2×Q8—1633no—
248C2×Q8—1633no—
249C2×Q8C22×D16311yescharacteristic
250C2×D4—1633no—
251C2×D4—1633no—
252C2×D4C22×D16313yes—
253C2×D4C22×D16313yes—
254C2×D4C22×D16316yes—
255C2×D4C22×D16313yes—
256C22×C4—1633no—
257C22×C4—1633no—
258C22×C4—1633no—
259C22×C4—1636no—
260C22×C4C22×D16313yes—
261C22×C4C22×D16313yes—
262C4⋊C4—1633no—
263C4⋊C4—1633no—
264C4⋊C4—1633no—
265C4⋊C4—1631no—
266C4⋊C4—1633no—
267C4⋊C4—1633no—
268C4⋊C4—1633no—
269C4⋊C4—1633no—
270C4⋊C4—1636no—
271C4⋊C4—1633no—
272C4⋊C4—1633no—
273C4⋊C4—1631no—
274C4⋊C4—1633no—
275C4⋊C4C22×D16313yes—
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—1633no—
278C22⋊C4—1636no—
279C22⋊C4—1633no—
280C22⋊C4—1636no—
281C22⋊C4—1633no—
282C22⋊C4—1636no—
283C22⋊C4—1633no—
284C22⋊C4—1631no—
285C22⋊C4—1633no—
286C22⋊C4—1631no—
287C22⋊C4—1631no—
288C22⋊C4C22×D16313yes—
289C22⋊C4C22×D16313yes—
290C42—1631no—
291C42—1633no—
292C42—1633no—
293C42—1633no—
294C42—1636no—
295C42—1633no—
296C42C22×D16313yes—
297C23—1631no—
298C23C23×D16313yes—
299C23C23×D16313yes—
300Q8—3266no—
301Q8—6523no—
302Q8#1457969814yes—
303D4—26no—
304D4—26no—
305D4—46no—
306D4—43no—
307D4—6523no—
308D4—6523no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—3266no—
312C2×C4—6523no—
313C2×C4—6523no—
314C2×C4—6526no—
315C2×C4—1631no—
316C2×C4—1631no—
317C2×C4—1633no—
318C2×C4—1631no—
319C2×C4—1633no—
320C2×C4—1633no—
321C2×C4—1633no—
322C2×C4C23×D16311yescharacteristic
323C2×C4C23×D16313yes—
324C2×C4C23×D16311yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C2×C4C23×D16313yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—3261no—
334C22—3261no—
335C22—3261no—
336C22C24×D16311yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—3263no—
342C4—3261no—
343C4—3263no—
344C4—3261no—
345C4—3261no—
346C4—3263no—
347C4—3263no—
348C4C2×#1457969816yes—
349C2—43no—
350C2—43no—
351C2—6521no—
352C2C22×#1457969811yescharacteristic
353C2#2378250011yescharacteristic
354C2#2378250111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#550×C163C2
hypercenter#550D163
hypocenterC163#3524
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #23782401C2semidirect
2. #23782391C2semidirect
3. #23782379C2semidirect
4. #23782367C2semidirect
5. #23782355C2semidirect
6. #23782322C2semidirect
7. #23782322C2semidirect
8. #23782282C2semidirect
9. #23782269C2semidirect
10. #23782196C2semidirect
11. #23782195C2semidirect
12. #23782148C2semidirect
13. #23782141C2semidirect
14. #41×Dic163C2semidirect
15. #550×C163C2semidirect
177. C163#3524semidirect
184. #550D163semidirect

Maximal subgroups

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

subgroupindex
#237824012
#237823912
#237823792
#237823672
#237823552
#237823222
#237822822
#237822692
#237821962
#237821952
#237821482
#237821412
#41×Dic1632
#550×C1632
#3524163