Finite Group Databasebeta

← #36367639

Subgroups — #36367639

Order 22144.

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#26012223C213yesmaximal, complemented
2Q8×Dic346C211yesmaximal, characteristic, complemented
3#26012199C213yesmaximal, complemented
4#26012160C211yesmaximal, characteristic, complemented
5#26012152C211yesmaximal, characteristic, complemented
6#26012145C213yesmaximal, complemented
7#26012141C213yesmaximal, complemented
8#26012114C213yesmaximal, complemented
9#26012110C213yesmaximal, complemented
10#26012036C213yesmaximal, complemented
11#26012021C213yesmaximal, complemented
12#26011974C211yesmaximal, characteristic, complemented
13#26011973C211yesmaximal, characteristic, complemented
14C2×Q8×Dic173C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2216yes—
18#19149679C2213yes—
19#19149678C2213yes—
20#19149678C2213yes—
21#19149675C2213yes—
22#19149674C2213yes—
23#19149672C2213yes—
24#19149669C2213yes—
25#19149668C2213yes—
26#19149665C2213yes—
27#19149665C2216yes—
28#19149663C2216yes—
29#19149663C2213yes—
30#19149661C2216yes—
31#19149660C2213yes—
32#19149659C2216yes—
33#19149656C2216yes—
34#19149656C2211yescharacteristic
35#19149655C2213yes—
36#19149653C2213yes—
37#19149653C2213yes—
38#19149651C2213yes—
39#19149650C2213yes—
40Q8×Dic173C2211yescharacteristic
41Q8×Dic173C2213yes—
42Q8×Dic173C2211yescharacteristic
43Q8×Dic173C2213yes—
44Q8×Dic173C2211yescharacteristic
45C4×Dic346C2213yes—
46C4×D692C2213yes—
47D4×Dic173C2213yes—
48D4×Dic173C2213yes—
49#19149641C2213yes—
50C4×C22⋊D173C2213yes—
51C4×C22⋊D173C2213yes—
52#19149639C2216yes—
53#19149638C2213yes—
54#19149636C2213yes—
55C2×C4⋊Dic173C2213yes—
56C2×C4⋊Dic173C2213yes—
57C4⋊C4×D173C2213yes—
58C2×C4×Dic173C2213yes—
59C2×#41×C173C2211yescharacteristic
60C22×Q8×C173C2211yescharacteristic
61#124×C173C2213yes—
62C22⋊Q8×C173C2213yes—
63#121×C173C2213yes—
64C4×Q8×C173C2211yescharacteristic
65C4×D4×C173C2213yes—
66#14583827—24no—
67#14583825—412no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2316yes—
70C2×D692C2313yes—
71C2×C22⋊D173C2313yes—
72C2×C22⋊D173C2313yes—
73C2×C22⋊D173C2316yes—
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2313yes—
76C22×Dic173C2316yes—
77C2×C4×D173C2313yes—
78C2×C4×D173C2316yes—
79C4⋊Dic173C2313yes—
80C4⋊Dic173C2313yes—
81C4⋊Dic173C2313yes—
82C4⋊Dic173C2313yes—
83C4⋊Dic173C2316yes—
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2313yes—
86#14583805C2313yes—
87#14583805C2311yescharacteristic
88#14583805C2313yes—
89#14583805C2311yescharacteristic
90#14583805C2313yes—
91#14583805C2313yes—
92C22⋊Dic173C2313yes—
93C22⋊Dic173C2313yes—
94C22⋊Dic173C2311yescharacteristic
95C22⋊Dic173C2313yes—
96#14583803C2313yes—
97#14583803C2313yes—
98#14583803C2316yes—
99#14583803C2313yes—
100#14583803C2313yes—
101C4×Dic173C2313yes—
102C4×Dic173C2313yes—
103C4×Dic173C2313yes—
104C4×Dic173C2313yes—
105C4×Dic173C2313yes—
106#41×C173—24no—
107C2×Q8×C173—24no—
108C2×Q8×C173C2313yes—
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173C2313yes—
111C2×D4×C173C2313yes—
112C22×C4×C173C2313yes—
113C22×C4×C173C2313yes—
114C4⋊C4×C173C2313yes—
115C4⋊C4×C173C2313yes—
116C22⋊C4×C173C2316yes—
117C22⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173C2313yes—
119C42×C173C2313yes—
120C22×D173C2413yes—
121Dic346—22no—
122Dic346—46no—
123D692—26no—
124C22⋊D173—26no—
125C22⋊D173—26no—
126C22⋊D173—46no—
127C2×Dic173—26no—
128C2×Dic173—22no—
129C2×Dic173—46no—
130C2×Dic173C2411yescharacteristic
131C2×Dic173C2413yes—
132C2×Dic173C2413yes—
133C2×Dic173C2413yes—
134C2×Dic173C2411yescharacteristic
135C2×Dic173C2411yescharacteristic
136C2×Dic173C2411yescharacteristic
137C4×D173—26no—
138C4×D173—46no—
139C23×C173C2413yes—
140C23×C173C2411yescharacteristic
141Q8×C173—26no—
142Q8×C173#4114yescomplemented
143D4×C173—26no—
144D4×C173—26no—
145C2×C4×C173—26no—
146C2×C4×C173—26no—
147C2×C4×C173C2413yes—
148C2×C4×C173C2411yescharacteristic
149C2×C4×C173C2413yes—
150C2×C4×C173C2411yescharacteristic
151C2×C4×C173C2413yes—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—23no—
155C22×C173—23no—
156C22×C173—23no—
157C22×C173H(5,𝔽2)12yes—
158C22×C173C2511yescharacteristic
159C2×D173—23no—
160C2×D173—23no—
161C2×D173—23no—
162C4×C173—21no—
163C4×C173—23no—
164C4×C173—23no—
165C4×C173—21no—
166C4×C173C2×#4116yes—
167Dic173—21no—
168Dic173—23no—
169Dic173—23no—
170Dic173—23no—
171Dic173—21no—
172Dic173—21no—
173Dic1732−1+412yes—
174C2×C173—22no—
175C2×C173—43no—
176C2×C173C22×#4111yescharacteristic
177C2×C173C2×H(5,𝔽2)11yescharacteristic
178C2×C173C2×2−1+411yescharacteristic
179D173—43no—
180C173#352211yescharacteristic, complemented, Sylow
181#3522—1731nomaximal, Sylow
182#559—1733no—
183Q82—1731no—
184#554—1733no—
185#550—1731no—
186#548—1731no—
187#548D17311yescharacteristic, complemented
188#547—1733no—
189#547—1733no—
190#542—1733no—
191#542—1733no—
192#532—1733no—
193#529—1733no—
194#520—1731no—
195#520—1731no—
196C2×C4×Q8—1731no—
197C2×#41—1731no—
198C2×#41—1736no—
199C2×#41C2×D17311yescharacteristic
200C22×Q8C2×D17311yescharacteristic
201C4⋊Q8—1733no—
202C4⋊Q8—1733no—
203C4⋊Q8—1733no—
204#127—1733no—
205#126—1733no—
206#126—1733no—
207#125—1733no—
208#124—1733no—
209#124—1733no—
210#124—1736no—
211#124C2×D17313yes—
212#123—1733no—
213#123—1733no—
214#123—1736no—
215#123—1736no—
216C22⋊Q8—1733no—
217C22⋊Q8—1736no—
218C22⋊Q8—1731no—
219C22⋊Q8—1736no—
220C22⋊Q8C2×D17313yes—
221#121—1733no—
222#121—1733no—
223#121—1733no—
224#121—1733no—
225#121C2×D17313yes—
226C4×Q8—1731no—
227C4×Q8—1733no—
228C4×Q8—1731no—
229C4×Q8—1733no—
230C4×Q8—1731no—
231C4×Q8—1733no—
232C4×Q8C2×D17311yescharacteristic
233C4×D4—1733no—
234C4×D4—1733no—
235C4×D4—1733no—
236C4×D4—1733no—
237C4×D4—1733no—
238C4×D4—1736no—
239C4×D4—1733no—
240C4×D4C2×D17313yes—
241#117—1733no—
242#117—1733no—
243C2×C4⋊C4—1733no—
244C2×C4⋊C4—1733no—
245C2×C4⋊C4—1733no—
246C2×C42—1733no—
247#41—24no—
248#41—3464no—
249#41—69212no—
250C2×Q8—24no—
251C2×Q8—1731no—
252C2×Q8—1736no—
253C2×Q8C22×D17313yes—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—1733no—
256C2×D4—1733no—
257C2×D4—1736no—
258C2×D4—1733no—
259C2×D4C22×D17313yes—
260C2×D4C22×D17313yes—
261C22×C4—1731no—
262C22×C4—1733no—
263C22×C4—1733no—
264C22×C4—1736no—
265C22×C4—1736no—
266C22×C4C22×D17313yes—
267C22×C4C22×D17313yes—
268C4⋊C4—1733no—
269C4⋊C4—1733no—
270C4⋊C4—1733no—
271C4⋊C4—1733no—
272C4⋊C4—1733no—
273C4⋊C4—1736no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4—1733no—
278C4⋊C4—1733no—
279C4⋊C4—1733no—
280C4⋊C4—1733no—
281C4⋊C4C22×D17313yes—
282C4⋊C4C22×D17313yes—
283C22⋊C4—1733no—
284C22⋊C4—1733no—
285C22⋊C4—1731no—
286C22⋊C4—1733no—
287C22⋊C4—1733no—
288C22⋊C4—1733no—
289C22⋊C4—1733no—
290C22⋊C4—1736no—
291C22⋊C4—1733no—
292C22⋊C4C22×D17316yes—
293C22⋊C4C22×D17311yescharacteristic
294C22⋊C4C22×D17313yes—
295C42—1733no—
296C42—1733no—
297C42—1733no—
298C42—1733no—
299C42—1733no—
300C42C22×D17313yes—
301C23—1733no—
302C23C23×D17313yes—
303C23C23×D17311yescharacteristic
304Q8—26no—
305Q8—3462no—
306Q8—6926no—
307Q8#1458382514yescomplemented
308D4—26no—
309D4—26no—
310D4—3466no—
311D4—3466no—
312D4—3466no—
313D4—6926no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—3466no—
317C2×C4—3462no—
318C2×C4—3466no—
319C2×C4—1731no—
320C2×C4—1731no—
321C2×C4—1733no—
322C2×C4—1733no—
323C2×C4—1733no—
324C2×C4—1731no—
325C2×C4—1731no—
326C2×C4—6926no—
327C2×C4—6926no—
328C2×C4C23×D17313yes—
329C2×C4C23×D17311yescharacteristic
330C2×C4C23×D17313yes—
331C2×C4C23×D17311yescharacteristic
332C2×C4C23×D17313yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3463no—
339C22—3463no—
340C22—3463no—
341C22#1914972512yes—
342C22C24×D17311yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—3461no—
348C4—3463no—
349C4—3463no—
350C4—3463no—
351C4—3461no—
352C4—3461no—
353C4—1732no—
354C4C2×#1458382516yes—
355C2—22no—
356C2—43no—
357C2—6923no—
358C2C22×#1458382511yescharacteristic
359C2C2×#1914972511yescharacteristic
360C22−1+4×D17311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#548×C173C2
hypercenter#548D173
hypocenterC173#3522
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #26012223C2semidirect
2. Q8×Dic346C2semidirect
3. #26012199C2semidirect
4. #26012160C2semidirect
5. #26012152C2semidirect
6. #26012145C2semidirect
7. #26012141C2semidirect
8. #26012114C2semidirect
9. #26012110C2semidirect
10. #26012036C2semidirect
11. #26012021C2semidirect
12. #26011974C2semidirect
13. #26011973C2semidirect
14. C2×Q8×Dic173C2semidirect
15. #548×C173C2semidirect
142. Q8×C173#41semidirect
180. C173#3522semidirect
187. #548D173semidirect
307. Q8#14583825semidirect

Maximal subgroups

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

subgroupindex
#260122232
Q8×Dic3462
#260121992
#260121602
#260121522
#260121452
#260121412
#260121142
#260121102
#260120362
#260120212
#260119742
#260119732
C2×Q8×Dic1732
#548×C1732
#3522173