Finite Group Databasebeta

← D4×#15691345 #38754408

Subgroups — D4×#15691345 #38754408

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×#15691345C212yesmaximal, complemented
2D4×D788C211yesmaximal, characteristic
3#27161453C211yesmaximal, characteristic, complemented
4#27161310C212yesmaximal, complemented
5#27161281C211yesmaximal, characteristic, complemented
6#27161279C211yesmaximal, characteristic, complemented
7#27161229C212yesmaximal, complemented
8#27161218C212yesmaximal, complemented
9C4×#15691345C211yesmaximal, characteristic, complemented
10D4×#13096802C211yesmaximal, characteristic, complemented
11D4×Q8×C197C211yesmaximal, characteristic, complemented
12C22×D788C2212yes—
13C2×D4×D197—21no—
14C2×#15691345—28no—
15C2×#15691345C2212yes—
16C2×#15691345C2211yescharacteristic
17C2×#15691345C2212yescomplemented
18C2×#15691345C2212yescomplemented
19C2×#15691345C2212yescomplemented
20C22×#13096802C2212yescomplemented
21#20263323C2211yescharacteristic
22#20263301—21no—
23#20263300C2212yes—
24#20263296—22no—
25C4×D788C2211yescharacteristic
26#20263257C2211yescharacteristic, complemented
27C4⋊#13096802C2211yescharacteristic, complemented
28#20263250C2211yescharacteristic, complemented
29#20263250C2212yescomplemented
30#20263246C2211yescharacteristic
31#20263246C2212yescomplemented
32C22⋊#13096802C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
34C22×Q8×C197C2212yescomplemented
35C4⋊Q8×C197—21no—
36C4⋊Q8×C197C2211yescharacteristic, complemented
37C22⋊Q8×C197—22no—
38C22⋊Q8×C197C2212yescomplemented
39C4×Q8×C197C2211yescharacteristic, complemented
40C4×D4×C197—21no—
41C4×D4×C197C2211yescharacteristic
42C23×D197—22no—
43C2×D788—24no—
44C2×D788—22no—
45C2×D788—21no—
46C2×D788—21no—
47C2×D788—42no—
48C2×D788C2312yes—
49C2×D788C2311yescharacteristic
50C2×C22⋊D197—22no—
51D4×D197—44no—
52C2×C4×D197—21no—
53#15691345—24no—
54#15691345—24no—
55#15691345—24no—
56#15691345—42no—
57#15691345D414yesdirect product factor
58C2×#13096802—24no—
59C2×#13096802C2311yescharacteristic
60C2×#13096802C2312yes—
61C2×#13096802C2311yescharacteristic
62C4⋊Dic197C2311yescharacteristic
63#15691335—21no—
64#15691335—22no—
65C2×Q8×C197—22no—
66C2×Q8×C197—24no—
67C2×Q8×C197—42no—
68C2×Q8×C197C2312yes—
69C2×Q8×C197C2311yescharacteristic
70C2×D4×C197D411yescharacteristic
71C22×C4×C197—22no—
72C22×C4×C197C2312yes—
73C4⋊C4×C197—21no—
74C4⋊C4×C197—21no—
75C4⋊C4×C197—21no—
76C4⋊C4×C197—22no—
77C4⋊C4×C197D411yescharacteristic
78C4⋊C4×C197C2311yescharacteristic
79C22⋊C4×C197—22no—
80C22⋊C4×C197D412yes—
81C42×C197—21no—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—44no—
86C22×D197—44no—
87C22×D197—82no—
88D788—22no—
89D788—21no—
90D788—22no—
91D788—44no—
92D788—42no—
93D788—81no—
94D788C2×D412yes—
95C22⋊D197—44no—
96C2×Dic197—21no—
97C4×D197—42no—
98#13096802—22no—
99#13096802—21no—
100#13096802C2×D412yescomplemented
101C23×C197C2×D412yes—
102Q8×C197—21no—
103Q8×C197—22no—
104Q8×C197—22no—
105Q8×C197—44no—
106Q8×C197C2×D412yescomplemented
107D4×C197SD1614yescomplemented
108C2×C4×C197—21no—
109C2×C4×C197—22no—
110C2×C4×C197—22no—
111C2×C4×C197—21no—
112C2×C4×C197—22no—
113C2×C4×C197—44no—
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2411yescharacteristic
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—22no—
131C4×C197—21no—
132C4×C197—42no—
133C4×C197—41no—
134C4×C197C2×SD1612yescomplemented
135C4×C197C22×D411yescharacteristic
136C4×C197H(5,𝔽2)11yescharacteristic
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×Q8D19711yescharacteristic, complemented
148D42—1971no—
149#492—1971no—
150#465—1972no—
151#461—1971no—
152#460—1971no—
153#450—1972no—
154#448—1972no—
155C4×SD16—1971no—
156C8×D4—1971no—
157C22×Q8C2×D19712yescomplemented
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×C8—1972no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D19711yescharacteristic, complemented
169#127—1971no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D19712yescomplemented
172#121—3941no—
173#121—1972no—
174#120—3942no—
175C4×Q8C2×D19711yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—1971no—
178C4×D4C2×D19711yescharacteristic
179#106—1971no—
180C4⋊C8—1971no—
181#103—1971no—
182#103—1972no—
183#102—1971no—
184#102—1972no—
185C22⋊C8—1972no—
186C4×C8—1971no—
187C24—3942no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D19712yes—
192C2×Q8C22×D19711yescharacteristic
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×D4C22⋊D19711yescharacteristic
203C22×C4—22no—
204C22×C4—3941no—
205C22×C4C22×D19712yes—
206SD16—7882no—
207SD16—3944no—
208SD16—3944no—
209SD16—3944no—
210SD16—1974no—
211C2×C8—3944no—
212C2×C8—1971no—
213C2×C8—1972no—
214C2×C8—1971no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—1971no—
220C4⋊C4C22⋊D19711yescharacteristic
221C4⋊C4C22×D19711yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—3941no—
224C22⋊C4—3942no—
225C22⋊C4C22⋊D19712yes—
226C42—21no—
227C42C22×D19711yescharacteristic
228C23—7884no—
229C23—7884no—
230C23—15762no—
231C23—3942no—
232C23—3941no—
233C23C2×C22⋊D19712yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D19712yescomplemented
239D4—7884no—
240D4—7884no—
241D4—7882no—
242D4—15761no—
243D4—3942no—
244D4—3942no—
245D4—3941no—
246D4—1972no—
247D4#1569134514yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—7882no—
255C2×C4—3941no—
256C2×C4C2×C22⋊D19711yescharacteristic
257C2×C4C2×C22⋊D19712yes—
258C2×C4C2×C22⋊D19711yescharacteristic
259C2×C4C23×D19711yescharacteristic
260C8—3942no—
261C8—3941no—
262C8—1972no—
263C22—22no—
264C22—22no—
265C22—7881no—
266C22—7881no—
267C22—7882no—
268C22—7882no—
269C22—7882no—
270C22—15768no—
271C22—3942no—
272C22C2×#1569134514yescomplemented
273C22C22×C22⋊D19711yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—7881no—
280C4C2×#1569134512yescomplemented
281C4C2×D4×D19711yescharacteristic
282C4#2026337111yescharacteristic
283C2—24no—
284C2—7882no—
285C2—15762no—
286C2D4×C22⋊D19711yescharacteristic
287C2C22×#1569134511yescharacteristic
288C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD4×Q8×C197C2
socleC22×C197C22×D4
prime radicalD4×D788C2
hypercenterD4×Q8D197
hypocenterC197D4×SD16
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#15691345C2semidirect
3. #27161453C2semidirect
4. #27161310C2semidirect
5. #27161281C2semidirect
6. #27161279C2semidirect
7. #27161229C2semidirect
8. #27161218C2semidirect
9. C4×#15691345C2semidirect
10. D4×#13096802C2semidirect
11. D4×Q8×C197C2semidirect
17. C2×#15691345C22semidirect
18. C2×#15691345C22semidirect
19. C2×#15691345C22semidirect
20. C22×#13096802C22semidirect
26. #20263257C22semidirect
27. C4⋊#13096802C22semidirect
28. #20263250C22semidirect
29. #20263250C22semidirect
31. #20263246C22semidirect
32. C22⋊#13096802C22semidirect
33. C4×#13096802C22semidirect
34. C22×Q8×C197C22semidirect
36. C4⋊Q8×C197C22semidirect
38. C22⋊Q8×C197C22semidirect
39. C4×Q8×C197C22semidirect
57. #15691345D4direct (N × H)
100. #13096802C2×D4semidirect
106. Q8×C197C2×D4semidirect
107. D4×C197SD16semidirect
120. C22×C197C2×SD16semidirect
134. C4×C197C2×SD16semidirect
144. C197D4×SD16semidirect
147. D4×Q8D197semidirect
157. C22×Q8C2×D197semidirect
168. C4⋊Q8C2×D197semidirect
171. C22⋊Q8C2×D197semidirect
175. C4×Q8C2×D197semidirect
238. Q8D4×D197semidirect
247. D4#15691345direct (N × H)
272. C22C2×#15691345semidirect
280. C4C2×#15691345semidirect

Maximal subgroups

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

subgroupindex
C22×#156913452
D4×D7882
#271614532
#271613102
#271612812
#271612792
#271612292
#271612182
C4×#156913452
D4×#130968022
D4×Q8×C1972
D4×SD16197