Finite Group Databasebeta

← #35228544

Subgroups — #35228544

Order 20864.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic326C211yesmaximal, characteristic, complemented
2#23782386C213yesmaximal, complemented
3#23782385C213yesmaximal, complemented
4#23782384C213yesmaximal, complemented
5#23782379C213yesmaximal, complemented
6Q8×C22⋊D163C211yesmaximal, characteristic, complemented
7#23782322C211yesmaximal, characteristic, complemented
8#23782284C213yesmaximal, complemented
9#23782282C213yesmaximal, complemented
10#23782217C213yesmaximal, complemented
11#23782195C213yesmaximal, complemented
12#23782143C211yesmaximal, characteristic, complemented
13#23782141C211yesmaximal, characteristic, complemented
14#41×Dic163C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17#19104735C2213yes—
18#19104734C2213yes—
19#19104734C2213yes—
20#19104733C2213yes—
21#19104730C2213yes—
22#19104728C2213yes—
23#19104728C2216yes—
24#19104728C2216yes—
25#19104725C2213yes—
26#19104722C2213yes—
27#19104722C2216yes—
28#19104721C2211yescharacteristic
29#19104721C2216yes—
30#19104719C2213yes—
31#19104716C2213yes—
32#19104714C2213yes—
33#19104714C2213yes—
34#19104712C2213yes—
35#19104712C2216yes—
36#19104711C2213yes—
37#19104707C2213yes—
38#19104706C2213yes—
39Q8×Dic163C2213yes—
40Q8×Dic163C2211yescharacteristic
41Q8×Dic163C2211yescharacteristic
42Q8×Dic163C2213yes—
43Q8×Dic163C2211yescharacteristic
44C4×Dic326C2213yes—
45C4×Dic326C2213yes—
46#19104700C2213yes—
47#19104700C2216yes—
48D4×Dic163C2213yes—
49D4×Dic163C2213yes—
50C4×C22⋊D163C2213yes—
51C4×C22⋊D163C2213yes—
52#19104694C2213yes—
53#19104694C2213yes—
54#19104694C2213yes—
55#19104693C2213yes—
56#19104692C2213yes—
57#19104691C2216yes—
58C2×C4×Dic163C2213yes—
59C2×#41×C163C2211yescharacteristic
60C22×Q8×C163C2211yescharacteristic
61#124×C163C2213yes—
62C22⋊Q8×C163C2213yes—
63#121×C163C2213yes—
64C4×Q8×C163C2211yescharacteristic
65C4×D4×C163C2213yes—
66C2×Dic326C2313yes—
67C2×Dic326C2311yescharacteristic
68C2×Dic326C2316yes—
69Q8×D163—24no—
70C2×C22⋊D163C2311yescharacteristic
71C2×C22⋊D163C2313yes—
72C22×Dic163C2313yes—
73C22×Dic163C2313yes—
74C2×C4×D163C2313yes—
75C4⋊Dic163C2313yes—
76C4⋊Dic163C2313yes—
77C4⋊Dic163C2313yes—
78C4⋊Dic163C2313yes—
79C4⋊Dic163C2313yes—
80C4⋊Dic163C2313yes—
81Dic163⋊C4C2313yes—
82Dic163⋊C4C2313yes—
83Dic163⋊C4C2313yes—
84Dic163⋊C4C2313yes—
85Dic163⋊C4C2313yes—
86Dic163⋊C4C2316yes—
87Dic163⋊C4C2313yes—
88C22⋊Dic163C2316yes—
89C22⋊Dic163C2313yes—
90C22⋊Dic163C2311yescharacteristic
91C22⋊Dic163C2316yes—
92C22⋊Dic163C2313yes—
93C22⋊Dic163C2313yes—
94(C2×D163)⋊C4C2311yescharacteristic
95(C2×D163)⋊C4C2313yes—
96(C2×D163)⋊C4C2311yescharacteristic
97(C2×D163)⋊C4C2313yes—
98C4×Dic163C2311yescharacteristic
99C4×Dic163C2313yes—
100C4×Dic163C2313yes—
101C4×Dic163C2313yes—
102C4×Dic163C2316yes—
103C4×Dic163C2316yes—
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×D163C2411yescharacteristic
120Dic326—26no—
121Dic326—22no—
122Dic326—46no—
123C22⋊D163—22no—
124C22⋊D163—43no—
125C2×Dic163—26no—
126C2×Dic163—43no—
127C2×Dic163C2413yes—
128C2×Dic163C2411yescharacteristic
129C2×Dic163C2411yescharacteristic
130C2×Dic163C2413yes—
131C2×Dic163C2413yes—
132C2×Dic163C2413yes—
133C2×Dic163C2411yescharacteristic
134C4×D163—26no—
135C23×C163C2413yes—
136C23×C163C2411yescharacteristic
137Q8×C163—26no—
138Q8×C163#4114yes—
139D4×C163—26no—
140D4×C163—43no—
141C2×C4×C163—26no—
142C2×C4×C163—26no—
143C2×C4×C163C2411yescharacteristic
144C2×C4×C163C2413yes—
145C2×C4×C163C2413yes—
146C2×C4×C163C2411yescharacteristic
147C2×C4×C163C2413yes—
148C22×C163—21no—
149C22×C163—23no—
150C22×C163—23no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163—23no—
154C22×C163C2511yescharacteristic
155C2×D163—21no—
156C2×D163—21no—
157C2×D163—21no—
158C4×C163—21no—
159C4×C163—23no—
160C4×C163—21no—
161C4×C163—23no—
162C4×C163C2×#4116yes—
163Dic163—23no—
164Dic163—23no—
165Dic163—21no—
166Dic163—21no—
167Dic163—23no—
168Dic163—23no—
169Dic1632−1+412yes—
170C2×C163—41no—
171C2×C163—43no—
172C2×C163C22×#4111yescharacteristic
173C2×C163C2×2−1+411yescharacteristic
174C2×C163#58511yescharacteristic
175D163—41no—
176C163#351811yescharacteristic, complemented, Sylow
177#3518—1631nomaximal, Sylow
178Q82—1631no—
179#556—1633no—
180#556—1633no—
181#556—1633no—
182#555—1633no—
183D4×Q8—1631no—
184#548—1631no—
185#548D16311yescharacteristic, complemented
186#542—1633no—
187#542—1633no—
188#533—1633no—
189#529—1633no—
190#519—1631no—
191#519—1631no—
192C4×#41—1631no—
193C2×#41C2×D16311yescharacteristic
194C22×Q8—1631no—
195C22×Q8C2×D16311yescharacteristic
196C4⋊Q8—1633no—
197C4⋊Q8—1633no—
198C4⋊Q8—1633no—
199C4⋊Q8—1633no—
200#126—1633no—
201#126—1633no—
202#126—1636no—
203#126—1636no—
204#125—1633no—
205#124—1633no—
206#124—1636no—
207#124—1631no—
208#124—1636no—
209#124C2×D16313yes—
210#123—1633no—
211#123—1633no—
212C22⋊Q8—1633no—
213C22⋊Q8—1633no—
214C22⋊Q8—1633no—
215C22⋊Q8—1633no—
216C22⋊Q8—1636no—
217C22⋊Q8C2×D16313yes—
218#121—1633no—
219#121—1633no—
220#121C2×D16313yes—
221C4×Q8—1631no—
222C4×Q8—1631no—
223C4×Q8—1633no—
224C4×Q8—1631no—
225C4×Q8—1633no—
226C4×Q8—1633no—
227C4×Q8—1636no—
228C4×Q8—1633no—
229C4×Q8—1633no—
230C4×Q8C2×D16311yescharacteristic
231C4×D4—1633no—
232C4×D4—1633no—
233C4×D4—1633no—
234C4×D4—1633no—
235C4×D4C2×D16313yes—
236#117—1633no—
237#117—1633no—
238#117—1633no—
239#117—1633no—
240#117—1636no—
241#117—1633no—
242C2×C42—1633no—
243#41—24no—
244C2×Q8—24no—
245C2×Q8—3264no—
246C2×Q8—1631no—
247C2×Q8—1633no—
248C2×Q8—1636no—
249C2×Q8C22×D16313yes—
250C2×Q8C22×D16311yescharacteristic
251C2×D4—1631no—
252C2×D4—1633no—
253C2×D4C22×D16313yes—
254C2×D4C22×D16313yes—
255C22×C4—1633no—
256C22×C4—1633no—
257C22×C4—1633no—
258C22×C4C22×D16313yes—
259C22×C4C22×D16313yes—
260C4⋊C4—1633no—
261C4⋊C4—1633no—
262C4⋊C4—1633no—
263C4⋊C4—1633no—
264C4⋊C4—1633no—
265C4⋊C4—1633no—
266C4⋊C4—1633no—
267C4⋊C4—1633no—
268C4⋊C4—1633no—
269C4⋊C4—1633no—
270C4⋊C4—1636no—
271C4⋊C4—1633no—
272C4⋊C4—1633no—
273C4⋊C4C22×D16313yes—
274C4⋊C4C22×D16313yes—
275C22⋊C4—1633no—
276C22⋊C4—1631no—
277C22⋊C4—1636no—
278C22⋊C4—1636no—
279C22⋊C4—1633no—
280C22⋊C4—1633no—
281C22⋊C4—1633no—
282C22⋊C4—1631no—
283C22⋊C4—1631no—
284C22⋊C4—1633no—
285C22⋊C4C22×D16316yes—
286C22⋊C4C22×D16311yescharacteristic
287C22⋊C4C22×D16313yes—
288C42—1631no—
289C42—1633no—
290C42—1633no—
291C42—1633no—
292C42—1636no—
293C42—1636no—
294C42—1633no—
295C42C22×D16313yes—
296C23—1631no—
297C23C23×D16311yescharacteristic
298C23C23×D16313yes—
299Q8—26no—
300Q8—3262no—
301Q8—3266no—
302Q8—6526no—
303Q8#1457969814yes—
304D4—26no—
305D4—43no—
306D4—3262no—
307D4—6523no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—3266no—
311C2×C4—3266no—
312C2×C4—6523no—
313C2×C4—1633no—
314C2×C4—1631no—
315C2×C4—1631no—
316C2×C4—1633no—
317C2×C4—1633no—
318C2×C4—1633no—
319C2×C4—1631no—
320C2×C4C23×D16311yescharacteristic
321C2×C4C23×D16313yes—
322C2×C4C23×D16313yes—
323C2×C4C23×D16311yescharacteristic
324C2×C4C23×D16313yes—
325C22—21no—
326C22—23no—
327C22—23no—
328C22—21no—
329C22—23no—
330C22—21no—
331C22—3261no—
332C22—3261no—
333C22—3261no—
334C22C24×D16311yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—3263no—
340C4—3263no—
341C4—3261no—
342C4—3261no—
343C4—3263no—
344C4—3263no—
345C4—1632no—
346C4C2×#1457969816yes—
347C2—41no—
348C2—43no—
349C2—6521no—
350C2C22×#1457969811yescharacteristic
351C22−1+4×D16311yescharacteristic
352C2#2378250111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#548×C163C2
hypercenter#548D163
hypocenterC163#3518
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. Q8×Dic326C2semidirect
2. #23782386C2semidirect
3. #23782385C2semidirect
4. #23782384C2semidirect
5. #23782379C2semidirect
6. Q8×C22⋊D163C2semidirect
7. #23782322C2semidirect
8. #23782284C2semidirect
9. #23782282C2semidirect
10. #23782217C2semidirect
11. #23782195C2semidirect
12. #23782143C2semidirect
13. #23782141C2semidirect
14. #41×Dic163C2semidirect
15. #548×C163C2semidirect
176. C163#3518semidirect
185. #548D163semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3262
#237823862
#237823852
#237823842
#237823792
Q8×C22⋊D1632
#237823222
#237822842
#237822822
#237822172
#237821952
#237821432
#237821412
#41×Dic1632
#548×C1632
#3518163