Finite Group Databasebeta

← #36367650

Subgroups — #36367650

Order 22144.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012217C211yesmaximal, characteristic, complemented
2#26012216C213yesmaximal, complemented
3#26012212C213yesmaximal, complemented
4#26012183C213yesmaximal, complemented
5#26012181C213yesmaximal, complemented
6Q8×C22⋊D173C211yesmaximal, characteristic, complemented
7Q8×C22⋊D173C211yesmaximal, characteristic, complemented
8#26012102C213yesmaximal, complemented
9#26012101C213yesmaximal, complemented
10#26012043C213yesmaximal, complemented
11#26012021C213yesmaximal
12#26011969C211yesmaximal, characteristic, complemented
13#26011967C211yesmaximal, characteristic, complemented
14C2×Q8×Dic173C211yesmaximal, characteristic, complemented
15D4×Q8×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17#19149678C2213yes—
18#19149677C2213yes—
19#19149677C2213yes—
20#19149674C2216yes—
21#19149674C2213yes—
22#19149672C2216yes—
23#19149672C2213yes—
24#19149671C2213yes—
25#19149670C2216yes—
26#19149670C2213yes—
27#19149669C2216yes—
28#19149669C2213yes—
29#19149669C2216yes—
30#19149665C2213yes—
31#19149663C2213yes—
32#19149660C2213yes—
33#19149658C2213yes—
34#19149658C2213yes—
35#19149656C2213yes—
36#19149655C2213yes—
37#19149655C2213yes—
38#19149651C2211yescharacteristic
39Q8×Dic173C2213yes—
40Q8×Dic173C2213yes—
41Q8×Dic173C2211yescharacteristic
42Q8×Dic173C2211yescharacteristic
43Q8×Dic173C2213yes—
44Q8×Dic173C2211yescharacteristic
45Q8×Dic173C2213yes—
46C4×Dic346C2213yes—
47C4×Dic346C2213yes—
48#19149644C2216yes—
49#19149644C2213yes—
50C4×C22⋊D173C2213yes—
51C4×C22⋊D173C2213yes—
52#19149638C2213yes—
53#19149638C2213yes—
54#19149637C2213yes—
55#19149636C2213yes—
56#19149635C2216yes—
57C2×C4⋊Dic173C2213yes—
58C2×C4×Dic173C2213yes—
59C22×Q8×C173C2211yescharacteristic
60C22×Q8×C173C2211yescharacteristic
61C4⋊Q8×C173C2213yes—
62C22⋊Q8×C173C2213yes—
63C22⋊Q8×C173C2213yes—
64C4×Q8×C173C2211yescharacteristic
65C4×D4×C173C2213yes—
66C2×Dic346C2313yes—
67C2×Dic346C2313yes—
68Q8×D173—24no—
69C2×C22⋊D173C2311yescharacteristic
70C2×C22⋊D173C2311yescharacteristic
71C22×Dic173C2311yescharacteristic
72C22×Dic173C2313yes—
73C2×C4×D173C2313yes—
74C4⋊Dic173C2313yes—
75C4⋊Dic173C2313yes—
76C4⋊Dic173C2316yes—
77C4⋊Dic173C2313yes—
78C4⋊Dic173C2313yes—
79C4⋊Dic173C2313yes—
80C4⋊Dic173C2313yes—
81#14583805C2316yes—
82#14583805C2316yes—
83#14583805C2313yes—
84#14583805C2313yes—
85#14583805C2313yes—
86#14583805C2313yes—
87#14583805C2316yes—
88#14583805C2311yescharacteristic
89#14583805C2313yes—
90C22⋊Dic173C2313yes—
91C22⋊Dic173C2311yescharacteristic
92C22⋊Dic173C2313yes—
93C22⋊Dic173C2313yes—
94#14583803C2311yescharacteristic
95#14583803C2313yes—
96#14583803C2313yes—
97#14583803C2313yes—
98C4×Dic173C2313yes—
99C4×Dic173C2313yes—
100C4×Dic173C2313yes—
101C4×Dic173C2313yes—
102C4×Dic173C2316yes—
103C4×Dic173C2316yes—
104C4×Dic173C2313yes—
105C2×Q8×C173—24no—
106C2×Q8×C173—24no—
107C2×Q8×C173C2313yes—
108C2×Q8×C173C2313yes—
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C22×C4×C173C2313yes—
112C22×C4×C173C2313yes—
113C4⋊C4×C173C2313yes—
114C4⋊C4×C173C2316yes—
115C4⋊C4×C173C2313yes—
116C22⋊C4×C173C2313yes—
117C22⋊C4×C173C2313yes—
118C42×C173C2313yes—
119C22×D173C2411yescharacteristic
120Dic346—26no—
121Dic346—26no—
122C22⋊D173—22no—
123C22⋊D173—22no—
124C2×Dic173—26no—
125C2×Dic173—22no—
126C2×Dic173C2413yes—
127C2×Dic173C2411yescharacteristic
128C2×Dic173C2413yes—
129C2×Dic173C2413yes—
130C2×Dic173C2411yescharacteristic
131C2×Dic173C2413yes—
132C2×Dic173C2411yescharacteristic
133C4×D173—26no—
134C23×C173C2411yescharacteristic
135C23×C173C2411yescharacteristic
136Q8×C173—26no—
137Q8×C173—26no—
138Q8×C173#4114yes—
139D4×C173—22no—
140C2×C4×C173—26no—
141C2×C4×C173—26no—
142C2×C4×C173C2413yes—
143C2×C4×C173C2411yescharacteristic
144C2×C4×C173C2413yes—
145C2×C4×C173C2413yes—
146C2×C4×C173C2413yes—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C1732−1+412yes—
153C22×C173C2511yescharacteristic
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—21no—
157C4×C173—23no—
158C4×C173—23no—
159C4×C173—21no—
160C4×C173—23no—
161C4×C173C2×#4116yes—
162Dic173—23no—
163Dic173—21no—
164Dic173—23no—
165Dic173—23no—
166Dic173—21no—
167Dic173—23no—
168Dic1732−1+412yes—
169C2×C173—22no—
170C2×C173—41no—
171C2×C173C22×#4111yescharacteristic
172C2×C173C2×2−1+411yescharacteristic
173C2×C173C2×2−1+411yescharacteristic
174D173—41no—
175C173#352311yescharacteristic, complemented, Sylow
176#3523—1731nomaximal, Sylow
177#557—1733no—
178#557—1731no—
179#556—1733no—
180#552—1733no—
181#552—1733no—
182D4×Q8—1731no—
183D4×Q8—1731no—
184D4×Q8D17311yescharacteristic, complemented
185#541—1733no—
186#541—1733no—
187#533—1733no—
188#529—1733no—
189#519—1731no—
190#519—1731no—
191C2×C4×Q8—1731no—
192C22×Q8—1731no—
193C22×Q8C2×D17311yescharacteristic
194C22×Q8C2×D17311yescharacteristic
195C4⋊Q8—1733no—
196C4⋊Q8—1733no—
197C4⋊Q8—1733no—
198C4⋊Q8C2×D17313yes—
199#126—1733no—
200#126—1736no—
201#126—1733no—
202#126—1736no—
203#125—1733no—
204#125—1736no—
205#125—1736no—
206#125—1736no—
207#125—1733no—
208#125—1733no—
209#124—1733no—
210#123—1733no—
211#123—1733no—
212C22⋊Q8—1733no—
213C22⋊Q8—1733no—
214C22⋊Q8—1733no—
215C22⋊Q8—1733no—
216C22⋊Q8—1733no—
217C22⋊Q8C2×D17313yes—
218C22⋊Q8C2×D17313yes—
219#121—1731no—
220C4×Q8—1731no—
221C4×Q8—1733no—
222C4×Q8—1731no—
223C4×Q8—1733no—
224C4×Q8—1733no—
225C4×Q8—1731no—
226C4×Q8—1733no—
227C4×Q8—1733no—
228C4×Q8—1736no—
229C4×Q8—1733no—
230C4×Q8—1733no—
231C4×Q8C2×D17311yescharacteristic
232C4×D4—1733no—
233C4×D4—1733no—
234C4×D4C2×D17313yes—
235#117—1733no—
236#117—1733no—
237#117—1733no—
238#117—1736no—
239#117—1733no—
240C2×C4⋊C4—1733no—
241C2×C42—1733no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—3464no—
245C2×Q8—1733no—
246C2×Q8—1733no—
247C2×Q8C22×D17313yes—
248C2×Q8C22×D17313yes—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—1731no—
251C2×D4—1731no—
252C2×D4C22×D17311yescharacteristic
253C22×C4—1731no—
254C22×C4—1733no—
255C22×C4—1733no—
256C22×C4C22×D17313yes—
257C22×C4C22×D17313yes—
258C4⋊C4—1733no—
259C4⋊C4—1733no—
260C4⋊C4—1736no—
261C4⋊C4—1733no—
262C4⋊C4—1733no—
263C4⋊C4—1733no—
264C4⋊C4—1733no—
265C4⋊C4—1733no—
266C4⋊C4—1736no—
267C4⋊C4—1736no—
268C4⋊C4—1733no—
269C4⋊C4—1733no—
270C4⋊C4—1731no—
271C4⋊C4—1736no—
272C4⋊C4—1733no—
273C4⋊C4—1733no—
274C4⋊C4C22×D17313yes—
275C4⋊C4C22×D17313yes—
276C4⋊C4C22×D17316yes—
277C22⋊C4—1731no—
278C22⋊C4—1733no—
279C22⋊C4—1733no—
280C22⋊C4—1733no—
281C22⋊C4—1731no—
282C22⋊C4—1733no—
283C22⋊C4—1733no—
284C22⋊C4—1733no—
285C22⋊C4C22×D17313yes—
286C22⋊C4C22×D17313yes—
287C42—1733no—
288C42—1733no—
289C42—1733no—
290C42—1733no—
291C42—1736no—
292C42—1736no—
293C42—1733no—
294C42C22×D17313yes—
295C23—1731no—
296C23C23×D17311yescharacteristic
297C23C23×D17311yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—3466no—
301Q8—3466no—
302Q8#1458382514yes—
303D4—22no—
304D4—3462no—
305D4—3462no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—3466no—
309C2×C4—3462no—
310C2×C4—3466no—
311C2×C4—1731no—
312C2×C4—1733no—
313C2×C4—1733no—
314C2×C4—1731no—
315C2×C4—1733no—
316C2×C4—1733no—
317C2×C4—1731no—
318C2×C4C23×D17313yes—
319C2×C4C23×D17313yes—
320C2×C4C23×D17313yes—
321C2×C4C23×D17311yescharacteristic
322C2×C4C23×D17313yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—3461no—
329C22—3461no—
330C22—3461no—
331C22#1914972712yes—
332C22C24×D17311yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—3463no—
338C4—3461no—
339C4—3463no—
340C4—3463no—
341C4—3463no—
342C4—3461no—
343C4—1732no—
344C4C2×#1458382516yes—
345C2—22no—
346C2—41no—
347C2—6921no—
348C2C22×#1458382511yescharacteristic
349C22−1+4×D17311yescharacteristic
350C2C2×#1914972711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
FittingD4×Q8×C173C2
prime radical#19149651C22
hypercenterD4×Q8D173
hypocenterC173#3523
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #26012217C2semidirect
2. #26012216C2semidirect
3. #26012212C2semidirect
4. #26012183C2semidirect
5. #26012181C2semidirect
6. Q8×C22⋊D173C2semidirect
7. Q8×C22⋊D173C2semidirect
8. #26012102C2semidirect
9. #26012101C2semidirect
10. #26012043C2semidirect
12. #26011969C2semidirect
13. #26011967C2semidirect
14. C2×Q8×Dic173C2semidirect
15. D4×Q8×C173C2semidirect
175. C173#3523semidirect
184. D4×Q8D173semidirect

Maximal subgroups

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

subgroupindex
#260122172
#260122162
#260122122
#260121832
#260121812
Q8×C22⋊D1732
#260121022
#260121012
#260120432
#260120212
#260119692
#260119672
C2×Q8×Dic1732
D4×Q8×C1732
#3523173