Finite Group Databasebeta

← #35228612

Subgroups — #35228612

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#23782397C213yesmaximal, complemented
2Q8×Dic326C211yesmaximal, characteristic, complemented
3#23782373C213yesmaximal, complemented
4#23782334C211yesmaximal, characteristic, complemented
5#23782326C211yesmaximal, characteristic, complemented
6#23782319C213yesmaximal, complemented
7#23782315C213yesmaximal, complemented
8#23782288C213yesmaximal, complemented
9#23782284C213yesmaximal, complemented
10#23782210C213yesmaximal, complemented
11#23782195C213yesmaximal, complemented
12#23782148C211yesmaximal, characteristic, complemented
13#23782147C211yesmaximal, characteristic, complemented
14C2×Q8×Dic163C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2216yes—
18#19104735C2213yes—
19#19104734C2213yes—
20#19104734C2213yes—
21#19104731C2213yes—
22#19104730C2213yes—
23#19104728C2213yes—
24#19104725C2213yes—
25#19104724C2213yes—
26#19104721C2213yes—
27#19104721C2216yes—
28#19104719C2213yes—
29#19104719C2216yes—
30#19104717C2216yes—
31#19104716C2213yes—
32#19104715C2216yes—
33#19104712C2211yescharacteristic
34#19104712C2216yes—
35#19104711C2213yes—
36#19104709C2213yes—
37#19104709C2213yes—
38#19104707C2213yes—
39#19104706C2213yes—
40Q8×Dic163C2211yescharacteristic
41Q8×Dic163C2211yescharacteristic
42Q8×Dic163C2213yes—
43Q8×Dic163C2211yescharacteristic
44Q8×Dic163C2213yes—
45C4×Dic326C2213yes—
46C4×D652C2213yes—
47D4×Dic163C2213yes—
48D4×Dic163C2213yes—
49#19104697C2213yes—
50C4×C22⋊D163C2213yes—
51C4×C22⋊D163C2213yes—
52(C22⋊D163)⋊C4C2216yes—
53#19104694C2213yes—
54#19104692C2213yes—
55C2×C4⋊Dic163C2213yes—
56C2×C4⋊Dic163C2213yes—
57C4⋊C4×D163C2213yes—
58C2×C4×Dic163C2213yes—
59C2×#41×C163C2211yescharacteristic
60C22×Q8×C163C2211yescharacteristic
61#124×C163C2213yes—
62C22⋊Q8×C163C2213yes—
63#121×C163C2213yes—
64C4×Q8×C163C2211yescharacteristic
65C4×D4×C163C2213yes—
66#14579700—24no—
67#14579698—412no—
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2316yes—
70C2×D652C2313yes—
71C2×C22⋊D163C2313yes—
72C2×C22⋊D163C2313yes—
73C2×C22⋊D163C2316yes—
74C22×Dic163C2313yes—
75C22×Dic163C2311yescharacteristic
76C22×Dic163C2316yes—
77C2×C4×D163C2313yes—
78C2×C4×D163C2316yes—
79C4⋊Dic163C2313yes—
80C4⋊Dic163C2313yes—
81C4⋊Dic163C2313yes—
82C4⋊Dic163C2313yes—
83C4⋊Dic163C2316yes—
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2313yes—
86Dic163⋊C4C2313yes—
87Dic163⋊C4C2311yescharacteristic
88Dic163⋊C4C2313yes—
89Dic163⋊C4C2311yescharacteristic
90Dic163⋊C4C2313yes—
91Dic163⋊C4C2313yes—
92C22⋊Dic163C2313yes—
93C22⋊Dic163C2313yes—
94C22⋊Dic163C2311yescharacteristic
95C22⋊Dic163C2313yes—
96(C2×D163)⋊C4C2313yes—
97(C2×D163)⋊C4C2313yes—
98(C2×D163)⋊C4C2316yes—
99(C2×D163)⋊C4C2313yes—
100(C2×D163)⋊C4C2313yes—
101C4×Dic163C2313yes—
102C4×Dic163C2313yes—
103C4×Dic163C2313yes—
104C4×Dic163C2313yes—
105C4×Dic163C2313yes—
106#41×C163—24no—
107C2×Q8×C163—24no—
108C2×Q8×C163C2313yes—
109C2×Q8×C163C2311yescharacteristic
110C2×D4×C163C2313yes—
111C2×D4×C163C2313yes—
112C22×C4×C163C2313yes—
113C22×C4×C163C2313yes—
114C4⋊C4×C163C2313yes—
115C4⋊C4×C163C2313yes—
116C22⋊C4×C163C2311yescharacteristic
117C22⋊C4×C163C2313yes—
118C22⋊C4×C163C2316yes—
119C42×C163C2313yes—
120C22×D163C2413yes—
121Dic326—22no—
122Dic326—46no—
123D652—26no—
124C22⋊D163—26no—
125C22⋊D163—26no—
126C22⋊D163—46no—
127C2×Dic163—26no—
128C2×Dic163—22no—
129C2×Dic163—46no—
130C2×Dic163C2411yescharacteristic
131C2×Dic163C2411yescharacteristic
132C2×Dic163C2413yes—
133C2×Dic163C2413yes—
134C2×Dic163C2413yes—
135C2×Dic163C2411yescharacteristic
136C2×Dic163C2411yescharacteristic
137C4×D163—26no—
138C4×D163—46no—
139C23×C163C2413yes—
140C23×C163C2411yescharacteristic
141Q8×C163—26no—
142Q8×C163#4114yescomplemented
143D4×C163—26no—
144D4×C163—26no—
145C2×C4×C163—26no—
146C2×C4×C163—26no—
147C2×C4×C163C2411yescharacteristic
148C2×C4×C163C2413yes—
149C2×C4×C163C2413yes—
150C2×C4×C163C2411yescharacteristic
151C2×C4×C163C2413yes—
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—23no—
155C22×C163—23no—
156C22×C163—23no—
157C22×C163H(5,𝔽2)12yes—
158C22×C163C2511yescharacteristic
159C2×D163—23no—
160C2×D163—23no—
161C2×D163—23no—
162C4×C163—21no—
163C4×C163—23no—
164C4×C163—23no—
165C4×C163—21no—
166C4×C163C2×#4116yes—
167Dic163—21no—
168Dic163—23no—
169Dic163—23no—
170Dic163—23no—
171Dic163—21no—
172Dic163—21no—
173Dic1632−1+412yes—
174C2×C163—22no—
175C2×C163—43no—
176C2×C163C22×#4111yescharacteristic
177C2×C163C2×H(5,𝔽2)11yescharacteristic
178C2×C163C2×2−1+411yescharacteristic
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#548D16311yescharacteristic, complemented
188#547—1633no—
189#547—1633no—
190#542—1633no—
191#542—1633no—
192#532—1633no—
193#529—1633no—
194#520—1631no—
195#520—1631no—
196C2×C4×Q8—1631no—
197C2×#41—1631no—
198C2×#41—1636no—
199C2×#41C2×D16311yescharacteristic
200C22×Q8C2×D16311yescharacteristic
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#124C2×D16313yes—
212#123—1633no—
213#123—1633no—
214#123—1636no—
215#123—1636no—
216C22⋊Q8—1633no—
217C22⋊Q8—1636no—
218C22⋊Q8—1631no—
219C22⋊Q8—1636no—
220C22⋊Q8C2×D16313yes—
221#121—1633no—
222#121—1633no—
223#121—1633no—
224#121—1633no—
225#121C2×D16313yes—
226C4×Q8—1631no—
227C4×Q8—1633no—
228C4×Q8—1631no—
229C4×Q8—1633no—
230C4×Q8—1631no—
231C4×Q8—1633no—
232C4×Q8C2×D16311yescharacteristic
233C4×D4—1633no—
234C4×D4—1633no—
235C4×D4—1633no—
236C4×D4—1633no—
237C4×D4—1633no—
238C4×D4—1636no—
239C4×D4—1633no—
240C4×D4C2×D16313yes—
241#117—1633no—
242#117—1633no—
243C2×C4⋊C4—1633no—
244C2×C4⋊C4—1633no—
245C2×C4⋊C4—1633no—
246C2×C42—1633no—
247#41—24no—
248#41—3264no—
249#41—65212no—
250C2×Q8—24no—
251C2×Q8—1631no—
252C2×Q8—1636no—
253C2×Q8C22×D16313yes—
254C2×Q8C22×D16311yescharacteristic
255C2×D4—1633no—
256C2×D4—1633no—
257C2×D4—1636no—
258C2×D4—1633no—
259C2×D4C22×D16313yes—
260C2×D4C22×D16313yes—
261C22×C4—1631no—
262C22×C4—1633no—
263C22×C4—1633no—
264C22×C4—1636no—
265C22×C4—1636no—
266C22×C4C22×D16313yes—
267C22×C4C22×D16313yes—
268C4⋊C4—1633no—
269C4⋊C4—1633no—
270C4⋊C4—1633no—
271C4⋊C4—1633no—
272C4⋊C4—1633no—
273C4⋊C4—1636no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4—1631no—
277C4⋊C4—1633no—
278C4⋊C4—1633no—
279C4⋊C4—1633no—
280C4⋊C4—1633no—
281C4⋊C4C22×D16313yes—
282C4⋊C4C22×D16313yes—
283C22⋊C4—1633no—
284C22⋊C4—1633no—
285C22⋊C4—1631no—
286C22⋊C4—1633no—
287C22⋊C4—1633no—
288C22⋊C4—1633no—
289C22⋊C4—1633no—
290C22⋊C4—1636no—
291C22⋊C4—1633no—
292C22⋊C4C22×D16316yes—
293C22⋊C4C22×D16311yescharacteristic
294C22⋊C4C22×D16313yes—
295C42—1633no—
296C42—1633no—
297C42—1633no—
298C42—1633no—
299C42—1633no—
300C42C22×D16313yes—
301C23—1633no—
302C23C23×D16313yes—
303C23C23×D16311yescharacteristic
304Q8—26no—
305Q8—3262no—
306Q8—6526no—
307Q8#1457969814yescomplemented
308D4—26no—
309D4—26no—
310D4—3266no—
311D4—3266no—
312D4—3266no—
313D4—6526no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—3266no—
317C2×C4—3262no—
318C2×C4—3266no—
319C2×C4—6526no—
320C2×C4—6526no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4—1633no—
324C2×C4—1633no—
325C2×C4—1633no—
326C2×C4—1631no—
327C2×C4—1631no—
328C2×C4C23×D16313yes—
329C2×C4C23×D16311yescharacteristic
330C2×C4C23×D16313yes—
331C2×C4C23×D16311yescharacteristic
332C2×C4C23×D16313yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3263no—
339C22—3263no—
340C22—3263no—
341C22#1910478112yes—
342C22C24×D16311yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—3261no—
348C4—3263no—
349C4—3263no—
350C4—3263no—
351C4—3261no—
352C4—3261no—
353C4—1632no—
354C4C2×#1457969816yes—
355C2—22no—
356C2—43no—
357C2—6523no—
358C2C22×#1457969811yescharacteristic
359C2C2×#1910478111yescharacteristic
360C22−1+4×D16311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#548×C163C2
hypercenter#548D163
hypocenterC163#3522
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23782397C2semidirect
2. Q8×Dic326C2semidirect
3. #23782373C2semidirect
4. #23782334C2semidirect
5. #23782326C2semidirect
6. #23782319C2semidirect
7. #23782315C2semidirect
8. #23782288C2semidirect
9. #23782284C2semidirect
10. #23782210C2semidirect
11. #23782195C2semidirect
12. #23782148C2semidirect
13. #23782147C2semidirect
14. C2×Q8×Dic163C2semidirect
15. #548×C163C2semidirect
142. Q8×C163#41semidirect
180. C163#3522semidirect
187. #548D163semidirect
307. Q8#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823972
Q8×Dic3262
#237823732
#237823342
#237823262
#237823192
#237823152
#237822882
#237822842
#237822102
#237821952
#237821482
#237821472
C2×Q8×Dic1632
#548×C1632
#3522163