Finite Group Databasebeta

← #35228638

Subgroups — #35228638

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#23782397C213yesmaximal, complemented
2#23782391C211yesmaximal, characteristic, complemented
3#23782379C213yesmaximal, complemented
4#23782361C213yesmaximal, complemented
5#23782357C213yesmaximal, complemented
6#23782334C211yesmaximal, characteristic, complemented
7#23782326C211yesmaximal, characteristic, complemented
8#23782288C213yesmaximal, complemented
9#23782268C213yesmaximal, complemented
10#23782197C213yesmaximal, complemented
11#23782194C213yesmaximal, complemented
12#23782148C211yesmaximal, characteristic, complemented
13#23782143C211yesmaximal, characteristic, complemented
14C4×#14579700C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17#19104734C2213yes—
18#19104731C2213yes—
19#19104730C2216yes—
20#19104728C2213yes—
21#19104728C2211yescharacteristic
22#19104727C2213yes—
23#19104726C2213yes—
24#19104726C2213yes—
25#19104724C2213yes—
26#19104722C2213yes—
27#19104721C2216yes—
28#19104719C2216yes—
29#19104717C2216yes—
30#19104717C2213yes—
31#19104717C2213yes—
32#19104715C2213yes—
33#19104711C2213yes—
34#19104709C2213yes—
35#19104709C2213yes—
36#19104707C2216yes—
37#19104706C2213yes—
38#19104706C2213yes—
39Q8×Dic163C2211yescharacteristic
40Q8×Dic163C2211yescharacteristic
41Q8×Dic163C2213yes—
42Q8×Dic163C2211yescharacteristic
43Q8×Dic163C2213yes—
44C4×Dic326C2213yes—
45C4×D652C2213yes—
46D4×Dic163C2213yes—
47D4×Dic163C2213yes—
48#19104697C2213yes—
49C4×C22⋊D163C2213yes—
50C4×C22⋊D163C2213yes—
51(C22⋊D163)⋊C4C2216yes—
52#19104694C2213yes—
53#19104694C2213yes—
54#19104694C2213yes—
55#19104693C2213yes—
56#19104691C2216yes—
57C2×C4⋊Dic163C2213yes—
58C42×D163C2213yes—
59C2×#41×C163C2211yescharacteristic
60C22×Q8×C163C2211yescharacteristic
61#124×C163C2213yes—
62C22⋊Q8×C163C2213yes—
63#121×C163C2213yes—
64C4×Q8×C163C2211yescharacteristic
65C4×D4×C163C2213yes—
66#14579700—24no—
67C2×Dic326C2313yes—
68C2×D652C2313yes—
69C2×C22⋊D163C2316yes—
70C2×C22⋊D163C2313yes—
71C2×C22⋊D163C2313yes—
72C22×Dic163C2316yes—
73C2×C4×D163C2313yes—
74C2×C4×D163C2313yes—
75C2×C4×D163C2313yes—
76C4⋊Dic163C2313yes—
77C4⋊Dic163C2313yes—
78C4⋊Dic163C2313yes—
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2313yes—
81C4⋊Dic163C2313yes—
82C4⋊Dic163C2313yes—
83C4⋊Dic163C2313yes—
84Dic163⋊C4C2313yes—
85Dic163⋊C4C2316yes—
86Dic163⋊C4C2313yes—
87Dic163⋊C4C2311yescharacteristic
88Dic163⋊C4C2311yescharacteristic
89C22⋊Dic163C2313yes—
90C22⋊Dic163C2311yescharacteristic
91C22⋊Dic163C2313yes—
92C22⋊Dic163C2313yes—
93C22⋊Dic163C2313yes—
94C22⋊Dic163C2311yescharacteristic
95(C2×D163)⋊C4C2316yes—
96(C2×D163)⋊C4C2316yes—
97(C2×D163)⋊C4C2313yes—
98(C2×D163)⋊C4C2313yes—
99C4×Dic163C2311yescharacteristic
100C4×Dic163C2313yes—
101C4×Dic163C2313yes—
102C4×Dic163C2316yes—
103C4×Dic163C2313yes—
104C4×Dic163C2313yes—
105#41×C163—24no—
106C2×Q8×C163—24no—
107C2×Q8×C163C2313yes—
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163C2313yes—
110C2×D4×C163C2313yes—
111C22×C4×C163C2313yes—
112C22×C4×C163C2313yes—
113C4⋊C4×C163C2313yes—
114C4⋊C4×C163C2313yes—
115C22⋊C4×C163C2311yescharacteristic
116C22⋊C4×C163C2313yes—
117C22⋊C4×C163C2316yes—
118C42×C163C2313yes—
119C22×D163C2413yes—
120Dic326—43no—
121D652—26no—
122C22⋊D163—43no—
123C22⋊D163—43no—
124C22⋊D163—46no—
125C2×Dic163—46no—
126C2×Dic163C2413yes—
127C2×Dic163C2411yescharacteristic
128C2×Dic163C2411yescharacteristic
129C2×Dic163C2411yescharacteristic
130C2×Dic163C2413yes—
131C2×Dic163C2411yescharacteristic
132C2×Dic163C2413yes—
133C4×D163—26no—
134C4×D163—43no—
135C4×D163—43no—
136C23×C163C2413yes—
137C23×C163C2411yescharacteristic
138Q8×C163—26no—
139Q8×C163#4114yes—
140D4×C163—26no—
141D4×C163—43no—
142C2×C4×C163—26no—
143C2×C4×C163—26no—
144C2×C4×C163C2411yescharacteristic
145C2×C4×C163C2413yes—
146C2×C4×C163C2413yes—
147C2×C4×C163C2411yescharacteristic
148C2×C4×C163C2413yes—
149C22×C163—21no—
150C22×C163—23no—
151C22×C163—21no—
152C22×C163—23no—
153C22×C163—23no—
154C22×C163—21no—
155C22×C163C2511yescharacteristic
156C2×D163—23no—
157C2×D163—23no—
158C2×D163—23no—
159C4×C163—21no—
160C4×C163—23no—
161C4×C163—23no—
162C4×C163—21no—
163C4×C163C2×#4116yes—
164Dic163—21no—
165Dic163—21no—
166Dic163—23no—
167Dic163—23no—
168Dic163—21no—
169Dic163—23no—
170Dic163—21no—
171C2×C163—41no—
172C2×C163—43no—
173C2×C163C22×#4111yescharacteristic
174C2×C163#58511yescharacteristic
175C2×C163#58511yescharacteristic
176D163—43no—
177C163#352411yescharacteristic, complemented, Sylow
178#3524—1631nomaximal, Sylow
179#559—1633no—
180#557—1631no—
181#555—1633no—
182#553—1633no—
183#552—1633no—
184#550—1631no—
185#548—1631no—
186#548D16311yescharacteristic, complemented
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×#41C2×D16311yescharacteristic
196C22×Q8C2×D16311yescharacteristic
197C4⋊Q8—1633no—
198#127—1633no—
199#126—1636no—
200#126—1631no—
201#126—1633no—
202#125—1633no—
203#125—1633no—
204#125—1633no—
205#124—1633no—
206#124—1633no—
207#124—1636no—
208#124C2×D16313yes—
209#123—1636no—
210#123—1633no—
211#123—1633no—
212#123—1636no—
213C22⋊Q8—1633no—
214C22⋊Q8—1633no—
215C22⋊Q8C2×D16313yes—
216#121—1633no—
217#121—1633no—
218#121—1636no—
219#121—1633no—
220#121—1633no—
221#121C2×D16313yes—
222C4×Q8—1631no—
223C4×Q8—1633no—
224C4×Q8—1631no—
225C4×Q8—1633no—
226C4×Q8—1631no—
227C4×Q8—1633no—
228C4×Q8C2×D16311yescharacteristic
229C4×D4—1633no—
230C4×D4—1633no—
231C4×D4—1633no—
232C4×D4—1633no—
233C4×D4—1633no—
234C4×D4—1636no—
235C4×D4—1633no—
236C4×D4C2×D16313yes—
237#117—1633no—
238#117—1633no—
239#117—1633no—
240#117—1633no—
241#117—1636no—
242C2×C4⋊C4—1633no—
243C2×C42—1633no—
244#41—24no—
245#41—3264no—
246C2×Q8—24no—
247C2×Q8—1633no—
248C2×Q8C22×D16313yes—
249C2×Q8C22×D16311yescharacteristic
250C2×D4—1633no—
251C2×D4—1636no—
252C2×D4—1633no—
253C2×D4—1633no—
254C2×D4C22×D16313yes—
255C2×D4C22×D16313yes—
256C22×C4—1633no—
257C22×C4—1636no—
258C22×C4—1633no—
259C22×C4—1633no—
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—1631no—
272C4⋊C4—1633no—
273C4⋊C4—1631no—
274C4⋊C4—1633no—
275C4⋊C4C22×D16313yes—
276C4⋊C4C22×D16313yes—
277C22⋊C4—1631no—
278C22⋊C4—1633no—
279C22⋊C4—1633no—
280C22⋊C4—1633no—
281C22⋊C4—1633no—
282C22⋊C4—1631no—
283C22⋊C4—1636no—
284C22⋊C4—1633no—
285C22⋊C4—1636no—
286C22⋊C4—1633no—
287C22⋊C4C22×D16316yes—
288C22⋊C4C22×D16313yes—
289C22⋊C4C22×D16311yescharacteristic
290C42—1631no—
291C42—1633no—
292C42—1633no—
293C42—1636no—
294C42—1633no—
295C42—1633no—
296C42C22×D16313yes—
297C23—1633no—
298C23C23×D16311yescharacteristic
299C23C23×D16313yes—
300Q8—26no—
301Q8—6523no—
302Q8#1457969814yes—
303D4—26no—
304D4—43no—
305D4—3266no—
306D4—6526no—
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—1631no—
321C2×C4—1633no—
322C2×C4C23×D16313yes—
323C2×C4C23×D16311yescharacteristic
324C2×C4C23×D16313yes—
325C2×C4C23×D16311yescharacteristic
326C2×C4C23×D16313yes—
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C22—3263no—
334C22—3263no—
335C22—3263no—
336C22C24×D16311yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—3261no—
342C4—3261no—
343C4—3263no—
344C4—3261no—
345C4—3263no—
346C4—3263no—
347C4—3261no—
348C4C2×#1457969816yes—
349C2—41no—
350C2—43no—
351C2—6523no—
352C2C22×#1457969811yescharacteristic
353C2#2378250111yescharacteristic
354C2#2378250211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#548×C163C2
hypercenter#548D163
hypocenterC163#3524
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23782397C2semidirect
2. #23782391C2semidirect
3. #23782379C2semidirect
4. #23782361C2semidirect
5. #23782357C2semidirect
6. #23782334C2semidirect
7. #23782326C2semidirect
8. #23782288C2semidirect
9. #23782268C2semidirect
10. #23782197C2semidirect
11. #23782194C2semidirect
12. #23782148C2semidirect
13. #23782143C2semidirect
14. C4×#14579700C2semidirect
15. #548×C163C2semidirect
177. C163#3524semidirect
186. #548D163semidirect

Maximal subgroups

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

subgroupindex
#237823972
#237823912
#237823792
#237823612
#237823572
#237823342
#237823262
#237822882
#237822682
#237821972
#237821942
#237821482
#237821432
C4×#145797002
#548×C1632
#3524163