Finite Group Databasebeta

← D8×C22⋊D167 #36335809

Subgroups — D8×C22⋊D167 #36335809

Order 21376.

620 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×D8×D167C211yesmaximal, characteristic, complemented
2D4×C22⋊D167C212yesmaximal, complemented
3#24923644C211yesmaximal, characteristic, complemented
4#24923505C211yesmaximal, characteristic, complemented
5#24923502C211yesmaximal, characteristic, complemented
6#24923462C212yesmaximal, complemented
7#24923403C212yesmaximal, complemented
8#24923401C212yesmaximal, complemented
9D8×Dic167C211yesmaximal, characteristic, complemented
10C8×C22⋊D167C211yesmaximal, characteristic, complemented
11C22×D8×C167C211yesmaximal, characteristic, complemented
12C22×C22⋊D167—22no—
13C2×D4×D167C2212yescomplemented
14C2×D1336C2211yescharacteristic, complemented
15C2×#14581935C2212yescomplemented
16C2×#14581935C2212yescomplemented
17D8×D167—24no—
18C2×C8×D167C2211yescharacteristic, complemented
19#19128828C2212yescomplemented
20#19128806C2212yescomplemented
21#19128804—22no—
22#19128803C2212yescomplemented
23#19128801—22no—
24#19128800—22no—
25D4×Dic167C2212yescomplemented
26C4×C22⋊D167C2211yescharacteristic
27#19128766C2211yescharacteristic, complemented
28Dic167⋊C8C2211yescharacteristic, complemented
29#19128754C2212yescomplemented
30#19128753C2212yescomplemented
31#19128753C2212yescomplemented
32C22⋊(C167⋊C8)C2211yescharacteristic, complemented
33(C2×D167)⋊C8C2211yescharacteristic, complemented
34C8×Dic167C2211yescharacteristic, complemented
35C22×D4×C167C2212yescomplemented
36C2×D8×C167—24no—
37C2×D8×C167C2211yescharacteristic
38C2×D8×C167C2212yescomplemented
39C2×D8×C167C2211yescharacteristic, complemented
40C22×C8×C167C2211yescharacteristic, complemented
41C23×D167—22no—
42C2×D668C2312yes—
43C2×C22⋊D167—22no—
44C2×C22⋊D167—22no—
45C2×C22⋊D167—22no—
46C2×C22⋊D167—22no—
47C2×C22⋊D167—22no—
48C2×C22⋊D167—48no—
49C2×C22⋊D167D411yescharacteristic
50D4×D167—24no—
51D4×D167—42no—
52C22×Dic167—22no—
53C2×C4×D167C2311yescharacteristic
54D1336—22no—
55#14581935—24no—
56#14581935—42no—
57C2×C167⋊C8C2311yescharacteristic
58C2×C167⋊C8C2311yescharacteristic
59C8×D167—22no—
60C4⋊Dic167C2312yes—
61Dic167⋊C4D411yescharacteristic
62C22⋊Dic167—22no—
63C22⋊Dic167—22no—
64C22⋊Dic167D411yescharacteristic
65(C2×D167)⋊C4—22no—
66(C2×D167)⋊C4D411yescharacteristic
67C4×Dic167C2311yescharacteristic
68C24×C167—22no—
69C2×D4×C167—24no—
70C2×D4×C167—22no—
71C2×D4×C167—42no—
72C2×D4×C167C2312yes—
73C2×D4×C167C2312yes—
74C22×C4×C167C2311yescharacteristic
75D8×C167—24no—
76D8×C167—22no—
77D8×C167D414yescomplemented
78C2×C8×C167—22no—
79C2×C8×C167C2311yescharacteristic
80C2×C8×C167C2311yescharacteristic
81C22×D167—22no—
82C22×D167—44no—
83C22×D167—44no—
84C22×D167—82no—
85C22×D167C2×D411yescharacteristic
86D668—22no—
87D668—22no—
88C22⋊D167—44no—
89C22⋊D167—44no—
90C22⋊D167—44no—
91C22⋊D167—44no—
92C22⋊D167—82no—
93C22⋊D167D814yesdirect product factor
94C2×Dic167—22no—
95C2×Dic167—44no—
96C2×Dic167C2×D411yescharacteristic
97C2×Dic167C2×D411yescharacteristic
98C2×Dic167C2×D411yescharacteristic
99C4×D167—21no—
100C4×D167—21no—
101C167⋊C8—21no—
102C167⋊C8—21no—
103C23×C167—22no—
104C23×C167—22no—
105C23×C167—44no—
106C23×C167—44no—
107C23×C167—82no—
108C23×C167C2×D411yescharacteristic
109D4×C167—22no—
110D4×C167—22no—
111D4×C167—22no—
112D4×C167—44no—
113D4×C167C2×D414yescomplemented
114C2×C4×C167—21no—
115C2×C4×C167—21no—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2411yescharacteristic
118C8×C167—21no—
119C8×C167C2×D412yescomplemented
120C22×C167—21no—
121C22×C167—21no—
122C22×C167—24no—
123C22×C167—42no—
124C22×C167—42no—
125C22×C167—42no—
126C22×C167—42no—
127C22×C167—42no—
128C22×C167—88no—
129C22×C167C2×D812yescomplemented
130C22×C167C22×D411yescharacteristic
131C2×D167—21no—
132C2×D167—21no—
133C2×D167—42no—
134C2×D167—42no—
135C2×D167—42no—
136C2×D167—88no—
137C2×D167C2×D812yescomplemented
138C4×C167—21no—
139C4×C167C22×D411yescharacteristic
140C4×C167H(5,𝔽2)11yescharacteristic
141Dic167—21no—
142Dic167—21no—
143Dic167—42no—
144Dic167C2×D812yescomplemented
145C2×C167—22no—
146C2×C167—44no—
147C2×C167—82no—
148C2×C167D4211yescharacteristic
149C2×C167C22×D811yescharacteristic
150C2×C167#57611yescharacteristic
151D167—22no—
152D167—82no—
153C167D4×D811yescharacteristic, complemented, Sylow
154D4×D8—1671nomaximal, Sylow
155C22×D8—1671no—
156C22×D8D16711yescharacteristic, complemented
157D42—1672no—
158#493—1671no—
159#466—1671no—
160#466—1671no—
161#459—1672no—
162#447—1672no—
163#447—1672no—
164C4×D8—1671no—
165C8×D4—1671no—
166C22×D4—3342no—
167C22×D4—1672no—
168C22×D4C2×D16712yescomplemented
169C2×D8—24no—
170C2×D8—3344no—
171C2×D8—1672no—
172C2×D8—1672no—
173C2×D8—1671no—
174C2×D8C2×D16711yescharacteristic
175C2×D8C2×D16712yescomplemented
176C2×D8C2×D16711yescharacteristic, complemented
177C22×C8—1671no—
178C22×C8C2×D16711yescharacteristic, complemented
179#127—1672no—
180#121—3342no—
181#121—1672no—
182#121—1672no—
183#120—3342no—
184#120—3342no—
185C4×D4—1672no—
186C4×D4—1671no—
187#107—1671no—
188C4⋊C8—1671no—
189#102—1672no—
190#102—1672no—
191#102—1672no—
192C22⋊C8—1671no—
193C22⋊C8—1671no—
194C4×C8—1671no—
195C24—22no—
196C24—3342no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—42no—
200C2×D4—3344no—
201C2×D4—3342no—
202C2×D4—3342no—
203C2×D4—3342no—
204C2×D4—3342no—
205C2×D4—3342no—
206C2×D4—6682no—
207C2×D4—6688no—
208C2×D4—1671no—
209C2×D4—1672no—
210C2×D4C22×D16712yes—
211C2×D4C22×D16712yes—
212C22×C4—3342no—
213C22×C4—1671no—
214C22×C4C22×D16711yescharacteristic
215D8—24no—
216D8—22no—
217D8—3344no—
218D8—3342no—
219D8—6682no—
220D8C22⋊D16714yesdirect product factor
221C2×C8—22no—
222C2×C8—3342no—
223C2×C8—1671no—
224C2×C8—1671no—
225C2×C8C22×D16711yescharacteristic
226C2×C8C22×D16711yescharacteristic
227C4⋊C4—1672no—
228C4⋊C4—1671no—
229C22⋊C4—3342no—
230C22⋊C4—3342no—
231C22⋊C4—3342no—
232C22⋊C4—1671no—
233C22⋊C4—1671no—
234C42—1671no—
235C23—22no—
236C23—22no—
237C23—44no—
238C23—44no—
239C23—82no—
240C23—13362no—
241C23—3342no—
242C23—6684no—
243C23—6684no—
244C23—1671no—
245C23D4×D16711yescharacteristic
246D4—22no—
247D4—22no—
248D4—22no—
249D4—44no—
250D4—13362no—
251D4—3342no—
252D4—3342no—
253D4—6684no—
254D4—6684no—
255D4—6684no—
256D4—6684no—
257D4—1674no—
258D4C2×C22⋊D16714yescomplemented
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—3341no—
262C2×C4—3341no—
263C2×C4—3342no—
264C2×C4—6684no—
265C2×C4—1671no—
266C2×C4—1671no—
267C2×C4—1671no—
268C2×C4D4×D16711yescharacteristic
269C2×C4C23×D16711yescharacteristic
270C8—21no—
271C8—3341no—
272C8—3341no—
273C8C2×C22⋊D16712yescomplemented
274C22—21no—
275C22—21no—
276C22—24no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—88no—
283C22—13368no—
284C22—3341no—
285C22—3341no—
286C22—6682no—
287C22—6682no—
288C22—6682no—
289C22—1672no—
290C22D8×D16712yescomplemented
291C22C2×D4×D16711yescharacteristic
292C4—21no—
293C4—3341no—
294C4—3341no—
295C4—6682no—
296C4—1672no—
297C4C22×C22⋊D16711yescharacteristic
298C4#1912887711yescharacteristic
299C2—22no—
300C2—44no—
301C2—82no—
302C2—13362no—
303C2—3342no—
304C2D4×C22⋊D16711yescharacteristic
305C2C2×D8×D16711yescharacteristic
306C2#2492407011yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC22×D8×C167C2
socleC22×C167C22×D4
hypercenterC22×D8D167
hypocenterC167D4×D8
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×D8×D167C2semidirect
2. D4×C22⋊D167C2semidirect
3. #24923644C2semidirect
4. #24923505C2semidirect
5. #24923502C2semidirect
6. #24923462C2semidirect
7. #24923403C2semidirect
8. #24923401C2semidirect
9. D8×Dic167C2semidirect
10. C8×C22⋊D167C2semidirect
11. C22×D8×C167C2semidirect
13. C2×D4×D167C22semidirect
14. C2×D1336C22semidirect
15. C2×#14581935C22semidirect
16. C2×#14581935C22semidirect
18. C2×C8×D167C22semidirect
19. #19128828C22semidirect
20. #19128806C22semidirect
22. #19128803C22semidirect
25. D4×Dic167C22semidirect
27. #19128766C22semidirect
28. Dic167⋊C8C22semidirect
29. #19128754C22semidirect
30. #19128753C22semidirect
31. #19128753C22semidirect
32. C22⋊(C167⋊C8)C22semidirect
33. (C2×D167)⋊C8C22semidirect
34. C8×Dic167C22semidirect
35. C22×D4×C167C22semidirect
38. C2×D8×C167C22semidirect
39. C2×D8×C167C22semidirect
40. C22×C8×C167C22semidirect
77. D8×C167D4semidirect
93. C22⋊D167D8direct (N × H)
113. D4×C167C2×D4semidirect
119. C8×C167C2×D4semidirect
129. C22×C167C2×D8semidirect
137. C2×D167C2×D8semidirect
144. Dic167C2×D8semidirect
153. C167D4×D8semidirect
156. C22×D8D167semidirect
168. C22×D4C2×D167semidirect
175. C2×D8C2×D167semidirect
176. C2×D8C2×D167semidirect
178. C22×C8C2×D167semidirect
220. D8C22⋊D167direct (N × H)
258. D4C2×C22⋊D167semidirect
273. C8C2×C22⋊D167semidirect
290. C22D8×D167semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1672
D4×C22⋊D1672
#249236442
#249235052
#249235022
#249234622
#249234032
#249234012
D8×Dic1672
C8×C22⋊D1672
C22×D8×C1672
D4×D8167