Finite Group Databasebeta

← D4×#14581935 #36335811

Subgroups — D4×#14581935 #36335811

Order 21376.

620 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581935C212yesmaximal, complemented
2D4×D668C211yesmaximal, characteristic, complemented
3#24923645C211yesmaximal, characteristic, complemented
4#24923503C212yesmaximal, complemented
5#24923464C211yesmaximal, characteristic, complemented
6#24923461C211yesmaximal, characteristic, complemented
7#24923403C212yesmaximal, complemented
8#24923402C212yesmaximal, complemented
9C4×#14581935C211yesmaximal, characteristic, complemented
10D4×C167⋊C8C211yesmaximal, characteristic, complemented
11D42×C167C211yesmaximal, characteristic, complemented
12C22×D668C2212yescomplemented
13C2×D4×D167—21no—
14C2×#14581935—28no—
15C2×#14581935C2211yescharacteristic
16C2×#14581935C2212yescomplemented
17C2×#14581935C2212yescomplemented
18C2×#14581935C2212yescomplemented
19C2×#14581935C2212yescomplemented
20C22×C167⋊C8C2212yescomplemented
21#19128829C2211yescharacteristic, complemented
22#19128807—21no—
23#19128806C2212yescomplemented
24#19128802—22no—
25C4×D668C2211yescharacteristic, complemented
26#19128765C2211yescharacteristic, complemented
27C4⋊(C167⋊C8)C2211yescharacteristic, complemented
28#19128753C2211yescharacteristic, complemented
29#19128753C2212yescomplemented
30#19128752C2211yescharacteristic, complemented
31#19128752C2212yescomplemented
32C22⋊(C167⋊C8)C2212yescomplemented
33C4×C167⋊C8C2211yescharacteristic, complemented
34C22×D4×C167—21no—
35C22×D4×C167C2212yescomplemented
36#127×C167C2211yescharacteristic, complemented
37#121×C167—21no—
38#121×C167C2212yescomplemented
39#120×C167—22no—
40C4×D4×C167C2211yescharacteristic
41C4×D4×C167C2211yescharacteristic, complemented
42C23×D167—22no—
43C2×D668—24no—
44C2×D668—21no—
45C2×D668—22no—
46C2×D668—21no—
47C2×D668—42no—
48C2×D668C2312yes—
49C2×D668C2311yescharacteristic
50C2×C22⋊D167—22no—
51D4×D167—44no—
52C2×C4×D167—21no—
53#14581935—24no—
54#14581935—24no—
55#14581935—24no—
56#14581935—42no—
57#14581935D414yesdirect product factor
58C2×C167⋊C8—24no—
59C2×C167⋊C8C2312yes—
60C2×C167⋊C8C2311yescharacteristic
61C2×C167⋊C8C2311yescharacteristic
62C4⋊Dic167C2311yescharacteristic
63(C2×D167)⋊C4—21no—
64(C2×D167)⋊C4—22no—
65C24×C167—22no—
66C2×D4×C167—24no—
67C2×D4×C167—22no—
68C2×D4×C167—22no—
69C2×D4×C167—21no—
70C2×D4×C167—21no—
71C2×D4×C167—42no—
72C2×D4×C167—44no—
73C2×D4×C167D411yescharacteristic
74C2×D4×C167C2312yes—
75C2×D4×C167C2311yescharacteristic
76C22×C4×C167—21no—
77C22×C4×C167C2312yes—
78C4⋊C4×C167D411yescharacteristic
79C4⋊C4×C167C2311yescharacteristic
80C22⋊C4×C167—21no—
81C22⋊C4×C167—22no—
82C22⋊C4×C167D412yes—
83C42×C167C2311yescharacteristic
84C22×D167—21no—
85C22×D167—22no—
86C22×D167—44no—
87C22×D167—44no—
88C22×D167—82no—
89D668—22no—
90D668—21no—
91D668—22no—
92D668—44no—
93D668—42no—
94D668—81no—
95D668C2×D412yescomplemented
96C22⋊D167—44no—
97C2×Dic167—21no—
98C4×D167—42no—
99C167⋊C8—22no—
100C167⋊C8—21no—
101C167⋊C8C2×D412yescomplemented
102C23×C167—21no—
103C23×C167—22no—
104C23×C167—44no—
105C23×C167—44no—
106C23×C167—82no—
107C23×C167C2×D412yes—
108D4×C167—22no—
109D4×C167—21no—
110D4×C167—22no—
111D4×C167—44no—
112D4×C167—44no—
113D4×C167—42no—
114D4×C167—81no—
115D4×C167D814yescomplemented
116D4×C167C2×D412yescomplemented
117C2×C4×C167—22no—
118C2×C4×C167—22no—
119C2×C4×C167—21no—
120C2×C4×C167—42no—
121C2×C4×C167C2×D411yescharacteristic
122C2×C4×C167C2×D412yes—
123C2×C4×C167C2×D411yescharacteristic
124C2×C4×C167C2411yescharacteristic
125C22×C167—22no—
126C22×C167—22no—
127C22×C167—22no—
128C22×C167—41no—
129C22×C167—41no—
130C22×C167—42no—
131C22×C167—42no—
132C22×C167—42no—
133C22×C167—88no—
134C22×C167C2×D814yescomplemented
135C22×C167C22×D411yescharacteristic
136C2×D167—22no—
137C2×D167—41no—
138C2×D167—41no—
139C2×D167—42no—
140C2×D167—42no—
141C2×D167—42no—
142C2×D167—88no—
143C4×C167—22no—
144C4×C167—21no—
145C4×C167—41no—
146C4×C167C2×D812yescomplemented
147C4×C167C22×D411yescharacteristic
148C4×C167H(5,𝔽2)11yescharacteristic
149Dic167—41no—
150C2×C167—24no—
151C2×C167—42no—
152C2×C167—82no—
153C2×C167D4211yescharacteristic
154C2×C167C22×D811yescharacteristic
155C2×C167#57611yescharacteristic
156D167—42no—
157D167—82no—
158C167D4×D811yescharacteristic, complemented, Sylow
159D4×D8—1671nomaximal, Sylow
160C22×D8—1672no—
161D42—1671no—
162D42D16711yescharacteristic, complemented
163#493—1671no—
164#466—1672no—
165#459—1671no—
166#459—1671no—
167#447—1672no—
168#447—1672no—
169C4×D8—1671no—
170C8×D4—1671no—
171C22×D4—21no—
172C22×D4—3341no—
173C22×D4—1672no—
174C22×D4C2×D16712yescomplemented
175C2×D8—3348no—
176C2×D8—1672no—
177C2×D8—1672no—
178C2×D8—1672no—
179C2×D8—1672no—
180C2×D8—1671no—
181C22×C8—1672no—
182#127—1671no—
183#127C2×D16711yescharacteristic, complemented
184#121—21no—
185#121—3341no—
186#121—1672no—
187#121C2×D16712yescomplemented
188#120—22no—
189#120—3342no—
190C4×D4—1671no—
191C4×D4C2×D16711yescharacteristic
192C4×D4C2×D16711yescharacteristic, complemented
193#107—1671no—
194C4⋊C8—1671no—
195#102—1671no—
196#102—1672no—
197#102—1671no—
198#102—1672no—
199C22⋊C8—1672no—
200C4×C8—1671no—
201C24—22no—
202C24—3342no—
203C2×D4—24no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—44no—
210C2×D4—3344no—
211C2×D4—3342no—
212C2×D4—3341no—
213C2×D4—3342no—
214C2×D4—3341no—
215C2×D4—6682no—
216C2×D4—6684no—
217C2×D4—1672no—
218C2×D4—1671no—
219C2×D4C22⋊D16711yescharacteristic
220C2×D4C22×D16712yes—
221C2×D4C22×D16711yescharacteristic
222C22×C4—21no—
223C22×C4—3341no—
224C22×C4C22×D16712yes—
225D8—3344no—
226D8—3344no—
227D8—3344no—
228D8—6682no—
229D8—1674no—
230C2×C8—3344no—
231C2×C8—1671no—
232C2×C8—1672no—
233C2×C8—1671no—
234C4⋊C4—1671no—
235C4⋊C4C22⋊D16711yescharacteristic
236C4⋊C4C22×D16711yescharacteristic
237C22⋊C4—21no—
238C22⋊C4—22no—
239C22⋊C4—3341no—
240C22⋊C4—3342no—
241C22⋊C4C22⋊D16712yes—
242C42C22×D16711yescharacteristic
243C23—21no—
244C23—22no—
245C23—44no—
246C23—44no—
247C23—82no—
248C23—13362no—
249C23—3341no—
250C23—3342no—
251C23—6684no—
252C23—6684no—
253C23C2×C22⋊D16712yes—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—44no—
258D4—44no—
259D4—42no—
260D4—81no—
261D4—13361no—
262D4—3342no—
263D4—3341no—
264D4—3342no—
265D4—6684no—
266D4—6684no—
267D4—6682no—
268D4—1672no—
269D4#1458193514yesdirect product factor
270D4D4×D16712yescomplemented
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—3341no—
276C2×C4—6682no—
277C2×C4C2×C22⋊D16711yescharacteristic
278C2×C4C2×C22⋊D16712yes—
279C2×C4C2×C22⋊D16711yescharacteristic
280C2×C4C23×D16711yescharacteristic
281C8—3342no—
282C8—3341no—
283C8—1672no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—88no—
293C22—13368no—
294C22—3342no—
295C22—6682no—
296C22—6682no—
297C22—6682no—
298C22—6681no—
299C22—6681no—
300C22C2×#1458193514yescomplemented
301C22C22×C22⋊D16711yescharacteristic
302C4—21no—
303C4—22no—
304C4—41no—
305C4—6681no—
306C4C2×#1458193512yescomplemented
307C4C2×D4×D16711yescharacteristic
308C4#1912887711yescharacteristic
309C2—24no—
310C2—42no—
311C2—82no—
312C2—13362no—
313C2—6682no—
314C2D4×C22⋊D16711yescharacteristic
315C2C22×#1458193511yescharacteristic
316C2#2492407211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingD42×C167C2
socleC22×C167C22×D4
hypercenterD42D167
hypocenterC167D4×D8
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#14581935C2semidirect
2. D4×D668C2semidirect
3. #24923645C2semidirect
4. #24923503C2semidirect
5. #24923464C2semidirect
6. #24923461C2semidirect
7. #24923403C2semidirect
8. #24923402C2semidirect
9. C4×#14581935C2semidirect
10. D4×C167⋊C8C2semidirect
11. D42×C167C2semidirect
12. C22×D668C22semidirect
16. C2×#14581935C22semidirect
17. C2×#14581935C22semidirect
18. C2×#14581935C22semidirect
19. C2×#14581935C22semidirect
20. C22×C167⋊C8C22semidirect
21. #19128829C22semidirect
23. #19128806C22semidirect
25. C4×D668C22semidirect
26. #19128765C22semidirect
27. C4⋊(C167⋊C8)C22semidirect
28. #19128753C22semidirect
29. #19128753C22semidirect
30. #19128752C22semidirect
31. #19128752C22semidirect
32. C22⋊(C167⋊C8)C22semidirect
33. C4×C167⋊C8C22semidirect
35. C22×D4×C167C22semidirect
36. #127×C167C22semidirect
38. #121×C167C22semidirect
41. C4×D4×C167C22semidirect
57. #14581935D4direct (N × H)
95. D668C2×D4semidirect
101. C167⋊C8C2×D4semidirect
115. D4×C167D8semidirect
116. D4×C167C2×D4semidirect
134. C22×C167C2×D8semidirect
146. C4×C167C2×D8semidirect
158. C167D4×D8semidirect
162. D42D167semidirect
174. C22×D4C2×D167semidirect
183. #127C2×D167semidirect
187. #121C2×D167semidirect
192. C4×D4C2×D167semidirect
269. D4#14581935direct (N × H)
270. D4D4×D167semidirect
300. C22C2×#14581935semidirect
306. C4C2×#14581935semidirect

Maximal subgroups

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

subgroupindex
C22×#145819352
D4×D6682
#249236452
#249235032
#249234642
#249234612
#249234032
#249234022
C4×#145819352
D4×C167⋊C82
D42×C1672
D4×D8167