Finite Group Databasebeta

← #35228615

Subgroups — #35228615

Order 20864.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782398C213yesmaximal, complemented
2Q8×Dic326C211yesmaximal, characteristic, complemented
3#23782370C213yesmaximal, complemented
4#23782334C211yesmaximal, characteristic, complemented
5#23782327C211yesmaximal, characteristic, complemented
6#23782322C211yesmaximal, characteristic, complemented
7#23782320C213yesmaximal, complemented
8#23782316C213yesmaximal, complemented
9#23782286C213yesmaximal, complemented
10#23782280C213yesmaximal, complemented
11#23782212C213yesmaximal, complemented
12#23782193C213yesmaximal, complemented
13#23782147C211yesmaximal, characteristic, complemented
14C4×Q8×D163C211yesmaximal, characteristic, complemented
15#520×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699C2216yes—
18C2×Q8×D163C2211yescharacteristic
19#19104735C2213yes—
20#19104734C2213yes—
21#19104734C2213yes—
22#19104731C2213yes—
23#19104730C2213yes—
24#19104728C2213yes—
25#19104725C2213yes—
26#19104723C2213yes—
27#19104722C2213yes—
28#19104721C2213yes—
29#19104721C2216yes—
30#19104720C2213yes—
31#19104718C2216yes—
32#19104717C2213yes—
33#19104717C2216yes—
34#19104715C2213yes—
35#19104714C2216yes—
36#19104714C2213yes—
37#19104713C2211yescharacteristic
38#19104713C2216yes—
39#19104710C2213yes—
40#19104709C2213yes—
41#19104708C2213yes—
42#19104706C2213yes—
43#19104706C2213yes—
44Q8×Dic163C2211yescharacteristic
45Q8×Dic163C2211yescharacteristic
46C4×Dic326C2213yes—
47C4×Dic326C2213yes—
48#19104700C2213yes—
49C4×D652C2213yes—
50C4×D652C2213yes—
51#19104697C2213yes—
52C4×C22⋊D163C2213yes—
53C4×C22⋊D163C2213yes—
54C4×C22⋊D163C2216yes—
55#19104692C2213yes—
56C4⋊C4×D163C2213yes—
57C4⋊C4×D163C2213yes—
58C42×D163C2213yes—
59C2×#41×C163C2211yescharacteristic
60C4×Q8×C163C2211yescharacteristic
61C4×Q8×C163C2211yescharacteristic
62C4×D4×C163C2213yes—
63C4×D4×C163C2213yes—
64#117×C163C2213yes—
65C2×C4⋊C4×C163C2213yes—
66#14579700—24no—
67#14579699—412no—
68C2×Dic326C2313yes—
69C2×Dic326C2311yescharacteristic
70C2×Dic326C2316yes—
71Q8×D163—24no—
72C2×D652C2313yes—
73C2×D652C2313yes—
74C2×C22⋊D163C2313yes—
75C2×C22⋊D163C2313yes—
76C2×C22⋊D163C2316yes—
77C2×C4×D163C2313yes—
78C2×C4×D163C2313yes—
79C2×C4×D163C2313yes—
80C2×C4×D163C2311yescharacteristic
81C2×C4×D163C2316yes—
82C4⋊Dic163C2313yes—
83C4⋊Dic163C2313yes—
84C4⋊Dic163C2313yes—
85C4⋊Dic163C2313yes—
86Dic163⋊C4C2313yes—
87Dic163⋊C4C2316yes—
88Dic163⋊C4C2311yescharacteristic
89Dic163⋊C4C2311yescharacteristic
90Dic163⋊C4C2313yes—
91Dic163⋊C4C2313yes—
92Dic163⋊C4C2313yes—
93C22⋊Dic163C2313yes—
94C22⋊Dic163C2316yes—
95C22⋊Dic163C2313yes—
96(C2×D163)⋊C4C2313yes—
97(C2×D163)⋊C4C2313yes—
98(C2×D163)⋊C4C2311yescharacteristic
99(C2×D163)⋊C4C2313yes—
100(C2×D163)⋊C4C2316yes—
101(C2×D163)⋊C4C2311yescharacteristic
102(C2×D163)⋊C4C2313yes—
103C4×Dic163C2313yes—
104C4×Dic163C2313yes—
105C4×Dic163C2313yes—
106C4×Dic163C2313yes—
107#41×C163—24no—
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163C2313yes—
110C22×C4×C163C2313yes—
111C22×C4×C163C2316yes—
112C4⋊C4×C163C2313yes—
113C4⋊C4×C163C2313yes—
114C4⋊C4×C163C2313yes—
115C4⋊C4×C163C2311yescharacteristic
116C22⋊C4×C163C2313yes—
117C22⋊C4×C163C2313yes—
118C42×C163C2313yes—
119C42×C163C2313yes—
120C22×D163C2413yes—
121C22×D163C2411yescharacteristic
122Dic326—26no—
123Dic326—22no—
124Dic326—46no—
125D652—26no—
126D652—26no—
127C22⋊D163—26no—
128C22⋊D163—26no—
129C22⋊D163—46no—
130C2×Dic163C2411yescharacteristic
131C2×Dic163C2413yes—
132C2×Dic163C2413yes—
133C2×Dic163C2413yes—
134C2×Dic163C2411yescharacteristic
135C2×Dic163C2411yescharacteristic
136C4×D163—26no—
137C4×D163—26no—
138C4×D163—26no—
139C4×D163—22no—
140C4×D163—46no—
141C23×C163C2413yes—
142Q8×C163#4114yescomplemented
143D4×C163—26no—
144C2×C4×C163—26no—
145C2×C4×C163—46no—
146C2×C4×C163C2411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C2×C4×C163C2413yes—
149C2×C4×C163C2413yes—
150C2×C4×C163C2411yescharacteristic
151C2×C4×C163C2413yes—
152C22×C163—23no—
153C22×C163—23no—
154C22×C163—23no—
155C22×C163C2511yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—23no—
159C2×D163—23no—
160C2×D163—23no—
161C2×D163H(5,𝔽2)12yes—
162C4×C163—21no—
163C4×C163—23no—
164C4×C163—23no—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163C2×#4116yes—
168Dic163—21no—
169Dic163—23no—
170Dic163—23no—
171Dic163—23no—
172Dic163—21no—
173Dic1632−1+412yes—
174C2×C163—43no—
175C2×C163C22×#4111yescharacteristic
176C2×C163C2×H(5,𝔽2)11yescharacteristic
177C2×C163C2×2−1+411yescharacteristic
178D163—22no—
179D163—43no—
180C163#352211yescharacteristic, complemented, Sylow
181#3522—1631nomaximal, Sylow
182#559—1633no—
183Q82—1631no—
184#554—1633no—
185#550—1631no—
186#548—1631no—
187#548—1631no—
188#547—1633no—
189#547—1633no—
190#542—1633no—
191#542—1633no—
192#532—1633no—
193#529—1633no—
194#520—1631no—
195#520D16311yescharacteristic, complemented
196C2×C4×Q8—1631no—
197C2×#41—1631no—
198C2×#41—1636no—
199C2×#41C2×D16311yescharacteristic
200C22×Q8—1631no—
201C4⋊Q8—1633no—
202C4⋊Q8—1633no—
203C4⋊Q8—1633no—
204#127—1633no—
205#126—1633no—
206#126—1633no—
207#125—1633no—
208#124—1633no—
209#124—1633no—
210#124—1636no—
211#124—1633no—
212#123—1636no—
213#123—1633no—
214#123—1633no—
215#123—1636no—
216C22⋊Q8—1636no—
217C22⋊Q8—1633no—
218C22⋊Q8—1631no—
219C22⋊Q8—1636no—
220C22⋊Q8—1633no—
221#121—1633no—
222#121—1633no—
223#121—1633no—
224#121—1633no—
225#121—1633no—
226C4×Q8—1631no—
227C4×Q8—1631no—
228C4×Q8—1633no—
229C4×Q8—1633no—
230C4×Q8—1633no—
231C4×Q8C2×D16311yescharacteristic
232C4×Q8C2×D16311yescharacteristic
233C4×D4—1633no—
234C4×D4—1633no—
235C4×D4—1636no—
236C4×D4—1633no—
237C4×D4—1633no—
238C4×D4—1633no—
239C4×D4C2×D16313yes—
240C4×D4C2×D16313yes—
241#117—1633no—
242#117C2×D16313yes—
243C2×C4⋊C4—1633no—
244C2×C4⋊C4—1633no—
245C2×C4⋊C4C2×D16313yes—
246C2×C42—1633no—
247#41—24no—
248#41—3264no—
249#41—65212no—
250C2×Q8—3264no—
251C2×Q8—1631no—
252C2×Q8—1633no—
253C2×Q8—1636no—
254C2×Q8C22×D16311yescharacteristic
255C2×D4—1633no—
256C2×D4—1633no—
257C2×D4—1636no—
258C2×D4—1633no—
259C2×D4—1633no—
260C2×D4C22×D16313yes—
261C22×C4—1631no—
262C22×C4—1633no—
263C22×C4—1633no—
264C22×C4—1633no—
265C22×C4—1636no—
266C22×C4C22×D16313yes—
267C22×C4C22×D16316yes—
268C4⋊C4—1633no—
269C4⋊C4—1633no—
270C4⋊C4—1633no—
271C4⋊C4—1636no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4—1633no—
275C4⋊C4—1633no—
276C4⋊C4—1633no—
277C4⋊C4—1633no—
278C4⋊C4—1633no—
279C4⋊C4C22×D16313yes—
280C4⋊C4C22×D16313yes—
281C4⋊C4C22×D16313yes—
282C4⋊C4C22×D16311yescharacteristic
283C22⋊C4—1633no—
284C22⋊C4—1636no—
285C22⋊C4—1633no—
286C22⋊C4—1633no—
287C22⋊C4—1633no—
288C22⋊C4—1631no—
289C22⋊C4—1631no—
290C22⋊C4—1633no—
291C22⋊C4—1636no—
292C22⋊C4—1633no—
293C22⋊C4C22×D16313yes—
294C22⋊C4C22×D16313yes—
295C42—1633no—
296C42—1633no—
297C42—1633no—
298C42—1633no—
299C42C22×D16313yes—
300C42C22×D16313yes—
301C23—1633no—
302C23—1631no—
303C23C23×D16313yes—
304Q8—3262no—
305Q8—3266no—
306Q8—6526no—
307Q8#1457969914yescomplemented
308D4—26no—
309D4—3266no—
310D4—3266no—
311D4—3266no—
312D4—3266no—
313D4—6526no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—3266no—
317C2×C4—3262no—
318C2×C4—3266no—
319C2×C4—3266no—
320C2×C4—6526no—
321C2×C4—1631no—
322C2×C4—1633no—
323C2×C4—1633no—
324C2×C4—1633no—
325C2×C4—1631no—
326C2×C4—1631no—
327C2×C4C23×D16311yescharacteristic
328C2×C4C23×D16313yes—
329C2×C4C23×D16311yescharacteristic
330C2×C4C23×D16313yes—
331C2×C4C23×D16311yescharacteristic
332C2×C4C23×D16313yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—3261no—
337C22—3261no—
338C22—3263no—
339C22—3263no—
340C22—3263no—
341C22—1632no—
342C22C24×D16311yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—3261no—
349C4—3263no—
350C4—3263no—
351C4—3263no—
352C4—3261no—
353C4—1632no—
354C4C2×#1457969916yes—
355C2—43no—
356C2—3262no—
357C2—6523no—
358C2C22×#1457969911yescharacteristic
359C2H(5,𝔽2)×D16311yescharacteristic
360C22−1+4×D16311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#520×C163C2
hypercenter#520D163
hypocenterC163#3522
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23782398C2semidirect
2. Q8×Dic326C2semidirect
3. #23782370C2semidirect
4. #23782334C2semidirect
5. #23782327C2semidirect
6. #23782322C2semidirect
7. #23782320C2semidirect
8. #23782316C2semidirect
9. #23782286C2semidirect
10. #23782280C2semidirect
11. #23782212C2semidirect
12. #23782193C2semidirect
13. #23782147C2semidirect
14. C4×Q8×D163C2semidirect
15. #520×C163C2semidirect
142. Q8×C163#41semidirect
180. C163#3522semidirect
195. #520D163semidirect
307. Q8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823982
Q8×Dic3262
#237823702
#237823342
#237823272
#237823222
#237823202
#237823162
#237822862
#237822802
#237822122
#237821932
#237821472
C4×Q8×D1632
#520×C1632
#3522163