Finite Group Databasebeta

← #36335812

Subgroups — #36335812

Order 21376.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D1336C211yesmaximal, characteristic, complemented
2C2×D8×D167C211yesmaximal, characteristic, complemented
3#24923903C212yesmaximal, complemented
4#24923644C211yesmaximal, characteristic, complemented
5#24923506C211yesmaximal, characteristic, complemented
6#24923463C212yesmaximal, complemented
7#24923404C212yesmaximal, complemented
8#24923402C212yesmaximal, complemented
9#24923351C211yesmaximal, characteristic, complemented
10C8×C22⋊D167C211yesmaximal, characteristic, complemented
11#466×C167C211yesmaximal, characteristic, complemented
12C22×D668C2212yescomplemented
13C2×D4×D167—22no—
14C2×D4×D167C2212yescomplemented
15C2×D1336—24no—
16C2×D1336C2211yescharacteristic
17C2×D1336C2212yescomplemented
18C2×D1336C2211yescharacteristic, complemented
19C2×#14581935C2212yescomplemented
20C2×#14581935C2212yescomplemented
21D8×D167—24no—
22C2×C8×D167C2211yescharacteristic, complemented
23#19128828C2212yescomplemented
24#19128807C2212yescomplemented
25#19128805—22no—
26#19128802—22no—
27#19128800—22no—
28#19128794C2212yescomplemented
29C4×C22⋊D167C2211yescharacteristic
30Dic167⋊C8C2211yescharacteristic, complemented
31#19128752C2212yescomplemented
32#19128752C2212yescomplemented
33C22⋊(C167⋊C8)C2211yescharacteristic, complemented
34(C2×D167)⋊C8C2211yescharacteristic, complemented
35C8×Dic167C2211yescharacteristic, complemented
36C2×D8×C167C2211yescharacteristic, complemented
37C22×C8×C167C2211yescharacteristic, complemented
38#121×C167C2212yescomplemented
39#107×C167C2211yescharacteristic, complemented
40#102×C167C2212yescomplemented
41C23×D167—22no—
42C23×D167—22no—
43C2×D668—24no—
44C2×D668—22no—
45C2×D668—22no—
46C2×D668—42no—
47C2×D668C2312yes—
48C2×D668C2312yes—
49C2×C22⋊D167—22no—
50C2×C22⋊D167—22no—
51C2×C22⋊D167—22no—
52C2×C22⋊D167D411yescharacteristic
53D4×D167—24no—
54D4×D167—42no—
55D4×D167—48no—
56C2×C4×D167—22no—
57C2×C4×D167C2311yescharacteristic
58D1336—24no—
59D1336—22no—
60D1336D414yescomplemented
61#14581935—24no—
62#14581935—42no—
63C2×C167⋊C8C2311yescharacteristic
64C2×C167⋊C8C2311yescharacteristic
65C8×D167—22no—
66Dic167⋊C4D411yescharacteristic
67C22⋊Dic167D411yescharacteristic
68(C2×D167)⋊C4—22no—
69(C2×D167)⋊C4—22no—
70(C2×D167)⋊C4D411yescharacteristic
71C4×Dic167C2311yescharacteristic
72C2×D4×C167—22no—
73C2×D4×C167C2312yes—
74C22×C4×C167C2311yescharacteristic
75D8×C167—22no—
76C2×C8×C167—22no—
77C2×C8×C167C2311yescharacteristic
78C2×C8×C167C2311yescharacteristic
79C4⋊C4×C167C2312yes—
80C22⋊C4×C167—22no—
81C22×D167—22no—
82C22×D167—22no—
83C22×D167—44no—
84C22×D167—44no—
85C22×D167—44no—
86C22×D167—44no—
87C22×D167—82no—
88C22×D167—82no—
89C22×D167C2×D411yescharacteristic
90D668—22no—
91D668—22no—
92D668—22no—
93D668—44no—
94D668—44no—
95D668C2×D414yescomplemented
96C22⋊D167—44no—
97C22⋊D167—44no—
98C22⋊D167—82no—
99C22⋊D167D814yescomplemented
100C2×Dic167C2×D411yescharacteristic
101C2×Dic167C2×D411yescharacteristic
102C2×Dic167C2×D411yescharacteristic
103C4×D167—21no—
104C4×D167—21no—
105C4×D167—44no—
106C167⋊C8—21no—
107C167⋊C8—21no—
108C23×C167—22no—
109C23×C167C2×D411yescharacteristic
110D4×C167—22no—
111D4×C167—22no—
112D4×C167—44no—
113C2×C4×C167—21no—
114C2×C4×C167—21no—
115C2×C4×C167—22no—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2411yescharacteristic
118C8×C167—21no—
119C8×C167C2×D412yescomplemented
120C22×C167—21no—
121C22×C167—21no—
122C22×C167—42no—
123C22×C167—42no—
124C22×C167—42no—
125C22×C167C2×D812yescomplemented
126C22×C167C22×D411yescharacteristic
127C2×D167—21no—
128C2×D167—21no—
129C2×D167—24no—
130C2×D167—42no—
131C2×D167—42no—
132C2×D167—42no—
133C2×D167—42no—
134C2×D167—42no—
135C2×D167—88no—
136C2×D167—88no—
137C2×D167C2×D812yescomplemented
138C4×C167—21no—
139C4×C167—42no—
140C4×C167C22×D411yescharacteristic
141C4×C167H(5,𝔽2)11yescharacteristic
142Dic167—21no—
143Dic167—21no—
144Dic167C2×D812yescomplemented
145C2×C167—22no—
146C2×C167—82no—
147C2×C167D4211yescharacteristic
148C2×C167C22×D811yescharacteristic
149C2×C167#57611yescharacteristic
150D167—22no—
151D167—44no—
152D167—82no—
153C167D4×D811yescharacteristic, complemented, Sylow
154D4×D8—1671nomaximal, Sylow
155C22×D8—1671no—
156C22×D8—1671no—
157D42—1672no—
158#493—1671no—
159#466—1671no—
160#466D16711yescharacteristic, complemented
161#459—1672no—
162#447—1672no—
163#447—1672no—
164C4×D8—1671no—
165C8×D4—1671no—
166C22×D4—3342no—
167C22×D4—1672no—
168C22×D4—1672no—
169C2×D8—3344no—
170C2×D8—3344no—
171C2×D8—1672no—
172C2×D8—1672no—
173C2×D8—1672no—
174C2×D8—1671no—
175C2×D8—1671no—
176C2×D8C2×D16711yescharacteristic, complemented
177C22×C8—1671no—
178C22×C8C2×D16711yescharacteristic, complemented
179#127—1672no—
180#121—3342no—
181#121—1672no—
182#121C2×D16712yescomplemented
183#120—3342no—
184#120—3342no—
185C4×D4—1672no—
186C4×D4—1671no—
187#107C2×D16711yescharacteristic, complemented
188C4⋊C8—1671no—
189#102—1672no—
190#102—1672no—
191#102C2×D16712yescomplemented
192C22⋊C8—1671no—
193C22⋊C8—1671no—
194C4×C8—1671no—
195C24—3342no—
196C24—3342no—
197C2×D4—22no—
198C2×D4—3344no—
199C2×D4—3342no—
200C2×D4—3344no—
201C2×D4—3342no—
202C2×D4—3342no—
203C2×D4—3342no—
204C2×D4—3342no—
205C2×D4—6682no—
206C2×D4—6682no—
207C2×D4—6688no—
208C2×D4—1671no—
209C2×D4—1672no—
210C2×D4—1672no—
211C2×D4C22×D16712yes—
212C22×C4—3342no—
213C22×C4—1671no—
214C22×C4C22×D16711yescharacteristic
215D8—22no—
216D8—3344no—
217D8—3344no—
218D8—3342no—
219D8—6682no—
220D8—1674no—
221C2×C8—22no—
222C2×C8—3342no—
223C2×C8—1671no—
224C2×C8—1671no—
225C2×C8C22×D16711yescharacteristic
226C2×C8C22×D16711yescharacteristic
227C4⋊C4—1671no—
228C4⋊C4C22×D16712yes—
229C22⋊C4—22no—
230C22⋊C4—3342no—
231C22⋊C4—3342no—
232C22⋊C4—1671no—
233C22⋊C4—1671no—
234C42—1671no—
235C23—22no—
236C23—13362no—
237C23—13362no—
238C23—3342no—
239C23—3342no—
240C23—6684no—
241C23—6684no—
242C23—6684no—
243C23—6684no—
244C23—1671no—
245C23D4×D16711yescharacteristic
246D4—22no—
247D4—22no—
248D4—44no—
249D4—13362no—
250D4—3342no—
251D4—3342no—
252D4—3342no—
253D4—6684no—
254D4—6684no—
255D4—6684no—
256D4—6684no—
257D4—1674no—
258D4—1674no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—3341no—
263C2×C4—3341no—
264C2×C4—6684no—
265C2×C4—1671no—
266C2×C4—1671no—
267C2×C4—1671no—
268C2×C4D4×D16711yescharacteristic
269C2×C4C23×D16711yescharacteristic
270C8—21no—
271C8—3341no—
272C8—3341no—
273C8D4×D16712yescomplemented
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—13368no—
280C22—13368no—
281C22—3341no—
282C22—3341no—
283C22—3344no—
284C22—6682no—
285C22—6682no—
286C22—6682no—
287C22—6682no—
288C22—6682no—
289C22—1672no—
290C22D8×D16712yescomplemented
291C22C2×D4×D16711yescharacteristic
292C4—21no—
293C4—42no—
294C4—3341no—
295C4—3341no—
296C4—1672no—
297C4C2×D4×D16711yescharacteristic
298C4#1912887811yescharacteristic
299C2—22no—
300C2—82no—
301C2—13362no—
302C2—3342no—
303C2—6684no—
304C2#2492390311yescharacteristic
305C2C2×D8×D16711yescharacteristic
306C2#2492407111yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#466×C167C2
socleC22×C167C22×D4
hypercenter#466D167
hypocenterC167D4×D8
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C22×D1336C2semidirect
2. C2×D8×D167C2semidirect
3. #24923903C2semidirect
4. #24923644C2semidirect
5. #24923506C2semidirect
6. #24923463C2semidirect
7. #24923404C2semidirect
8. #24923402C2semidirect
9. #24923351C2semidirect
10. C8×C22⋊D167C2semidirect
11. #466×C167C2semidirect
12. C22×D668C22semidirect
14. C2×D4×D167C22semidirect
17. C2×D1336C22semidirect
18. C2×D1336C22semidirect
19. C2×#14581935C22semidirect
20. C2×#14581935C22semidirect
22. C2×C8×D167C22semidirect
23. #19128828C22semidirect
24. #19128807C22semidirect
28. #19128794C22semidirect
30. Dic167⋊C8C22semidirect
31. #19128752C22semidirect
32. #19128752C22semidirect
33. C22⋊(C167⋊C8)C22semidirect
34. (C2×D167)⋊C8C22semidirect
35. C8×Dic167C22semidirect
36. C2×D8×C167C22semidirect
37. C22×C8×C167C22semidirect
38. #121×C167C22semidirect
39. #107×C167C22semidirect
40. #102×C167C22semidirect
60. D1336D4semidirect
95. D668C2×D4semidirect
99. C22⋊D167D8semidirect
119. C8×C167C2×D4semidirect
125. C22×C167C2×D8semidirect
137. C2×D167C2×D8semidirect
144. Dic167C2×D8semidirect
153. C167D4×D8semidirect
160. #466D167semidirect
176. C2×D8C2×D167semidirect
178. C22×C8C2×D167semidirect
182. #121C2×D167semidirect
187. #107C2×D167semidirect
191. #102C2×D167semidirect
273. C8D4×D167semidirect
290. C22D8×D167semidirect

Maximal subgroups

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

subgroupindex
C22×D13362
C2×D8×D1672
#249239032
#249236442
#249235062
#249234632
#249234042
#249234022
#249233512
C8×C22⋊D1672
#466×C1672
D4×D8167