Finite Group Databasebeta

← D4×#15691347 #38754410

Subgroups — D4×#15691347 #38754410

Order 25216.

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

normal subgroups only

This group has 546 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×#15691347C212yesmaximal, complemented
2D4×Dic394C211yesmaximal, characteristic, complemented
3D4×D788C211yesmaximal, characteristic
4#27161452C211yesmaximal, characteristic, complemented
5#27161308C212yesmaximal, complemented
6#27161285C211yesmaximal, characteristic, complemented
7#27161280C211yesmaximal, characteristic, complemented
8#27161232C212yesmaximal, complemented
9#27161220C212yesmaximal, complemented
10C4×#15691347C211yesmaximal, characteristic, complemented
11C8×D4×C197C211yesmaximal, characteristic, complemented
12C22×Dic394C2212yescomplemented
13C22×D788C2212yes—
14C2×D4×D197—21no—
15C2×#15691347—28no—
16C2×#15691347C2212yes—
17C2×#15691347C2211yescharacteristic
18C2×#15691347C2212yescomplemented
19C2×#15691347C2212yescomplemented
20C2×#15691347C2212yescomplemented
21#20263326C2211yescharacteristic, complemented
22#20263325—21no—
23#20263323C2211yescharacteristic
24#20263305C2212yescomplemented
25#20263303—22no—
26#20263301—21no—
27#20263300C2212yes—
28#20263296—22no—
29C4×Dic394C2211yescharacteristic, complemented
30C4×D788C2211yescharacteristic
31D4×Dic197—21no—
32#20263258C2211yescharacteristic, complemented
33#20263251C2211yescharacteristic, complemented
34#20263251C2212yescomplemented
35#20263248C2211yescharacteristic
36#20263248C2212yescomplemented
37C22×C8×C197C2212yescomplemented
38C4×D4×C197C2211yescharacteristic
39C4⋊C8×C197C2211yescharacteristic, complemented
40C22⋊C8×C197C2212yescomplemented
41C4×C8×C197C2211yescharacteristic, complemented
42C23×D197—22no—
43C2×Dic394—24no—
44C2×Dic394—22no—
45C2×Dic394—42no—
46C2×Dic394C2312yes—
47C2×Dic394C2311yescharacteristic
48C2×D788—24no—
49C2×D788—22no—
50C2×D788—21no—
51C2×D788—21no—
52C2×D788—42no—
53C2×D788C2312yes—
54C2×D788C2311yescharacteristic
55C2×C22⋊D197—22no—
56D4×D197—44no—
57C22×Dic197—22no—
58C2×C4×D197—21no—
59#15691347—24no—
60#15691347—24no—
61#15691347—24no—
62#15691347—42no—
63#15691347D414yesdirect product factor
64C4⋊Dic197—21no—
65C4⋊Dic197—21no—
66C4⋊Dic197C2311yescharacteristic
67C4⋊Dic197C2311yescharacteristic
68#15691337—21no—
69#15691337—22no—
70C22⋊Dic197—22no—
71#15691335—21no—
72#15691335—22no—
73C4×Dic197—21no—
74C2×D4×C197D411yescharacteristic
75C22×C4×C197C2312yes—
76C2×C8×C197—24no—
77C2×C8×C197C2311yescharacteristic
78C2×C8×C197C2312yes—
79C2×C8×C197C2311yescharacteristic
80C4⋊C4×C197D411yescharacteristic
81C22⋊C4×C197D412yes—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—44no—
86C22×D197—44no—
87C22×D197—82no—
88Dic394—21no—
89Dic394—22no—
90Dic394—22no—
91Dic394—44no—
92Dic394C2×D412yescomplemented
93D788—22no—
94D788—21no—
95D788—22no—
96D788—44no—
97D788—42no—
98D788—81no—
99D788C2×D412yes—
100C22⋊D197—44no—
101C2×Dic197—21no—
102C2×Dic197—21no—
103C2×Dic197—21no—
104C2×Dic197—22no—
105C2×Dic197—44no—
106C4×D197—42no—
107C23×C197C2×D412yes—
108D4×C197SD1614yescomplemented
109C2×C4×C197—22no—
110C2×C4×C197—22no—
111C2×C4×C197C2×D411yescharacteristic
112C2×C4×C197C2×D412yes—
113C2×C4×C197C2×D411yescharacteristic
114C2×C4×C197C2411yescharacteristic
115C8×C197—22no—
116C8×C197—21no—
117C8×C197C2×D412yescomplemented
118C22×C197—22no—
119C22×C197—22no—
120C22×C197C2×SD1614yescomplemented
121C22×C197C22×D411yescharacteristic
122C2×D197—22no—
123C2×D197—41no—
124C2×D197—41no—
125C2×D197—42no—
126C2×D197—42no—
127C2×D197—42no—
128C2×D197—88no—
129C4×C197—22no—
130C4×C197—21no—
131C4×C197C2×SD1612yescomplemented
132C4×C197C22×D411yescharacteristic
133C4×C197H(5,𝔽2)11yescharacteristic
134Dic197—22no—
135Dic197—42no—
136Dic197—41no—
137Dic197—41no—
138C2×C197—24no—
139C2×C197D4211yescharacteristic
140C2×C197C22×SD1611yescharacteristic
141C2×C197#57711yescharacteristic
142D197—42no—
143D197—82no—
144C197D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1971nomaximal, Sylow
146C22×SD16—1972no—
147D4×Q8—1971no—
148D42—1971no—
149#492—1971no—
150#465—1972no—
151#461—1971no—
152#460—1971no—
153#450—1972no—
154#448—1972no—
155C4×SD16—1971no—
156C8×D4D19711yescharacteristic, complemented
157C22×Q8—1972no—
158C22×D4—3941no—
159C22×D4—1972no—
160C2×SD16—3948no—
161C2×SD16—1972no—
162C2×SD16—1972no—
163C2×SD16—1972no—
164C2×SD16—1972no—
165C2×SD16—1971no—
166C22×C8C2×D19712yescomplemented
167C4⋊Q8—3941no—
168C4⋊Q8—1971no—
169#127—1971no—
170C22⋊Q8—3942no—
171C22⋊Q8—1972no—
172#121—3941no—
173#121—1972no—
174#120—3942no—
175C4×Q8—1971no—
176C4×D4—3941no—
177C4×D4—1971no—
178C4×D4C2×D19711yescharacteristic
179#106—1971no—
180C4⋊C8C2×D19711yescharacteristic, complemented
181#103—1971no—
182#103—1972no—
183#102—1971no—
184#102—1972no—
185C22⋊C8C2×D19712yescomplemented
186C4×C8C2×D19711yescharacteristic, complemented
187C24—3942no—
188C2×Q8—7882no—
189C2×Q8—3944no—
190C2×Q8—3942no—
191C2×Q8—1972no—
192C2×Q8—1971no—
193C2×D4—7882no—
194C2×D4—7884no—
195C2×D4—3944no—
196C2×D4—3942no—
197C2×D4—3942no—
198C2×D4—3941no—
199C2×D4—3941no—
200C2×D4—1972no—
201C2×D4—1971no—
202C2×D4D78811yescharacteristic
203C22×C4—3942no—
204C22×C4—3941no—
205C22×C4C22×D19712yes—
206SD16—7882no—
207SD16—3944no—
208SD16—3944no—
209SD16—3944no—
210SD16—1974no—
211C2×C8—24no—
212C2×C8C22×D19711yescharacteristic
213C2×C8C22×D19712yes—
214C2×C8C22×D19711yescharacteristic
215C4⋊C4—3941no—
216C4⋊C4—3942no—
217C4⋊C4—3941no—
218C4⋊C4—3941no—
219C4⋊C4—1971no—
220C4⋊C4—1971no—
221C4⋊C4D78811yescharacteristic
222C22⋊C4—3942no—
223C22⋊C4—3941no—
224C22⋊C4—3942no—
225C22⋊C4D78812yes—
226C42—3941no—
227C42C22×D19711yescharacteristic
228C23—7884no—
229C23—7884no—
230C23—15762no—
231C23—3942no—
232C23—3941no—
233C23C2×D78812yes—
234Q8—7884no—
235Q8—3942no—
236Q8—3942no—
237Q8—3941no—
238Q8—1972no—
239D4—7884no—
240D4—7884no—
241D4—7882no—
242D4—15761no—
243D4—3942no—
244D4—3942no—
245D4—3941no—
246D4—1972no—
247D4#1569134714yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—7884no—
251C2×C4—7882no—
252C2×C4—3941no—
253C2×C4—3941no—
254C2×C4—3942no—
255C2×C4—3941no—
256C2×C4C2×D78811yescharacteristic
257C2×C4C2×D78812yes—
258C2×C4C2×D78811yescharacteristic
259C2×C4C23×D19711yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D19712yescomplemented
263C22—22no—
264C22—22no—
265C22—7881no—
266C22—7881no—
267C22—7882no—
268C22—7882no—
269C22—7882no—
270C22—15768no—
271C22—3942no—
272C22C2×#1569134714yescomplemented
273C22C22×D78811yescharacteristic
274C4—21no—
275C4—22no—
276C4—7881no—
277C4—7882no—
278C4—7881no—
279C4—3942no—
280C4C2×#1569134712yescomplemented
281C4C2×D4×D19711yescharacteristic
282C4#2026337211yescharacteristic
283C2—24no—
284C2—7882no—
285C2—15762no—
286C2D4×D78811yescharacteristic
287C2C22×#1569134711yescharacteristic
288C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC8×D4×C197C2
socleC22×C197C22×D4
prime radicalD4×D788C2
hypercenterC8×D4D197
hypocenterC197D4×SD16
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#15691347C2semidirect
2. D4×Dic394C2semidirect
4. #27161452C2semidirect
5. #27161308C2semidirect
6. #27161285C2semidirect
7. #27161280C2semidirect
8. #27161232C2semidirect
9. #27161220C2semidirect
10. C4×#15691347C2semidirect
11. C8×D4×C197C2semidirect
12. C22×Dic394C22semidirect
18. C2×#15691347C22semidirect
19. C2×#15691347C22semidirect
20. C2×#15691347C22semidirect
21. #20263326C22semidirect
24. #20263305C22semidirect
29. C4×Dic394C22semidirect
32. #20263258C22semidirect
33. #20263251C22semidirect
34. #20263251C22semidirect
36. #20263248C22semidirect
37. C22×C8×C197C22semidirect
39. C4⋊C8×C197C22semidirect
40. C22⋊C8×C197C22semidirect
41. C4×C8×C197C22semidirect
63. #15691347D4direct (N × H)
92. Dic394C2×D4semidirect
108. D4×C197SD16semidirect
117. C8×C197C2×D4semidirect
120. C22×C197C2×SD16semidirect
131. C4×C197C2×SD16semidirect
144. C197D4×SD16semidirect
156. C8×D4D197semidirect
166. C22×C8C2×D197semidirect
180. C4⋊C8C2×D197semidirect
185. C22⋊C8C2×D197semidirect
186. C4×C8C2×D197semidirect
247. D4#15691347direct (N × H)
262. C8D4×D197semidirect
272. C22C2×#15691347semidirect
280. C4C2×#15691347semidirect

Maximal subgroups

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

subgroupindex
C22×#156913472
D4×Dic3942
D4×D7882
#271614522
#271613082
#271612852
#271612802
#271612322
#271612202
C4×#156913472
C8×D4×C1972
D4×SD16197