Finite Group Databasebeta

← D4×#15691343 #38754385

Subgroups — D4×#15691343 #38754385

Order 25216.

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×#15691343C212yesmaximal, complemented
2D4×D788C211yesmaximal, characteristic, complemented
3#27161455C211yesmaximal, characteristic, complemented
4#27161313C212yesmaximal, complemented
5#27161274C211yesmaximal, characteristic, complemented
6#27161271C211yesmaximal, characteristic, complemented
7#27161213C212yesmaximal, complemented
8#27161212C212yesmaximal, complemented
9C4×#15691343C211yesmaximal, characteristic, complemented
10D4×#13096802C211yesmaximal, characteristic, complemented
11D42×C197C211yesmaximal, characteristic, complemented
12C22×D788C2212yescomplemented
13C2×D4×D197—21no—
14C2×#15691343—28no—
15C2×#15691343C2211yescharacteristic
16C2×#15691343C2212yescomplemented
17C2×#15691343C2212yescomplemented
18C2×#15691343C2212yescomplemented
19C2×#15691343C2212yescomplemented
20C22×#13096802C2212yescomplemented
21#20263323C2211yescharacteristic, complemented
22#20263301—21no—
23#20263300C2212yescomplemented
24#20263296—22no—
25C4×D788C2211yescharacteristic, complemented
26#20263259C2211yescharacteristic, complemented
27C4⋊#13096802C2211yescharacteristic, complemented
28#20263247C2211yescharacteristic, complemented
29#20263247C2212yescomplemented
30#20263246C2211yescharacteristic, complemented
31#20263246C2212yescomplemented
32C22⋊#13096802C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
34C22×D4×C197—21no—
35C22×D4×C197C2212yescomplemented
36#127×C197C2211yescharacteristic, complemented
37#121×C197—21no—
38#121×C197C2212yescomplemented
39#120×C197—22no—
40C4×D4×C197C2211yescharacteristic
41C4×D4×C197C2211yescharacteristic, complemented
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#15691343—24no—
54#15691343—24no—
55#15691343—24no—
56#15691343—42no—
57#15691343D414yesdirect product factor
58C2×#13096802—24no—
59C2×#13096802C2311yescharacteristic
60C2×#13096802C2312yes—
61C2×#13096802C2311yescharacteristic
62C4⋊Dic197C2311yescharacteristic
63#15691335—21no—
64#15691335—22no—
65C24×C197—22no—
66C2×D4×C197—24no—
67C2×D4×C197—22no—
68C2×D4×C197—22no—
69C2×D4×C197—21no—
70C2×D4×C197—21no—
71C2×D4×C197—42no—
72C2×D4×C197—44no—
73C2×D4×C197D411yescharacteristic
74C2×D4×C197C2312yes—
75C2×D4×C197C2311yescharacteristic
76C22×C4×C197—21no—
77C22×C4×C197C2312yes—
78C4⋊C4×C197D411yescharacteristic
79C4⋊C4×C197C2311yescharacteristic
80C22⋊C4×C197—21no—
81C22⋊C4×C197—22no—
82C22⋊C4×C197D412yes—
83C42×C197C2311yescharacteristic
84C22×D197—21no—
85C22×D197—22no—
86C22×D197—44no—
87C22×D197—44no—
88C22×D197—82no—
89D788—22no—
90D788—21no—
91D788—22no—
92D788—44no—
93D788—42no—
94D788—81no—
95D788C2×D412yescomplemented
96C22⋊D197—44no—
97C2×Dic197—21no—
98C4×D197—42no—
99#13096802—22no—
100#13096802—21no—
101#13096802C2×D412yescomplemented
102C23×C197—21no—
103C23×C197—22no—
104C23×C197—44no—
105C23×C197—44no—
106C23×C197—82no—
107C23×C197C2×D412yes—
108D4×C197—22no—
109D4×C197—21no—
110D4×C197—22no—
111D4×C197—44no—
112D4×C197—44no—
113D4×C197—42no—
114D4×C197—81no—
115D4×C197D814yescomplemented
116D4×C197C2×D412yescomplemented
117C2×C4×C197—22no—
118C2×C4×C197—22no—
119C2×C4×C197—21no—
120C2×C4×C197—42no—
121C2×C4×C197C2×D411yescharacteristic
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2×D411yescharacteristic
124C2×C4×C197C2411yescharacteristic
125C22×C197—22no—
126C22×C197—22no—
127C22×C197—22no—
128C22×C197—41no—
129C22×C197—41no—
130C22×C197—42no—
131C22×C197—42no—
132C22×C197—42no—
133C22×C197—88no—
134C22×C197C2×D814yescomplemented
135C22×C197C22×D411yescharacteristic
136C2×D197—22no—
137C2×D197—41no—
138C2×D197—41no—
139C2×D197—42no—
140C2×D197—42no—
141C2×D197—42no—
142C2×D197—88no—
143C4×C197—22no—
144C4×C197—21no—
145C4×C197—41no—
146C4×C197C2×D812yescomplemented
147C4×C197C22×D411yescharacteristic
148C4×C197H(5,𝔽2)11yescharacteristic
149Dic197—41no—
150C2×C197—24no—
151C2×C197—42no—
152C2×C197—82no—
153C2×C197D4211yescharacteristic
154C2×C197C22×D811yescharacteristic
155C2×C197#57611yescharacteristic
156D197—42no—
157D197—82no—
158C197D4×D811yescharacteristic, complemented, Sylow
159D4×D8—1971nomaximal, Sylow
160C22×D8—1972no—
161D42—1971no—
162D42D19711yescharacteristic, complemented
163#493—1971no—
164#466—1972no—
165#459—1971no—
166#459—1971no—
167#447—1972no—
168#447—1972no—
169C4×D8—1971no—
170C8×D4—1971no—
171C22×D4—21no—
172C22×D4—3941no—
173C22×D4—1972no—
174C22×D4C2×D19712yescomplemented
175C2×D8—3948no—
176C2×D8—1972no—
177C2×D8—1972no—
178C2×D8—1972no—
179C2×D8—1972no—
180C2×D8—1971no—
181C22×C8—1972no—
182#127—1971no—
183#127C2×D19711yescharacteristic, complemented
184#121—21no—
185#121—3941no—
186#121—1972no—
187#121C2×D19712yescomplemented
188#120—22no—
189#120—3942no—
190C4×D4—1971no—
191C4×D4C2×D19711yescharacteristic
192C4×D4C2×D19711yescharacteristic, complemented
193#107—1971no—
194C4⋊C8—1971no—
195#102—1971no—
196#102—1972no—
197#102—1971no—
198#102—1972no—
199C22⋊C8—1972no—
200C4×C8—1971no—
201C24—22no—
202C24—3942no—
203C2×D4—24no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—44no—
210C2×D4—7882no—
211C2×D4—7884no—
212C2×D4—3944no—
213C2×D4—3942no—
214C2×D4—3942no—
215C2×D4—3941no—
216C2×D4—3941no—
217C2×D4—1972no—
218C2×D4—1971no—
219C2×D4C22⋊D19711yescharacteristic
220C2×D4C22×D19712yes—
221C2×D4C22×D19711yescharacteristic
222C22×C4—21no—
223C22×C4—3941no—
224C22×C4C22×D19712yes—
225D8—7882no—
226D8—3944no—
227D8—3944no—
228D8—3944no—
229D8—1974no—
230C2×C8—3944no—
231C2×C8—1971no—
232C2×C8—1972no—
233C2×C8—1971no—
234C4⋊C4—1971no—
235C4⋊C4C22⋊D19711yescharacteristic
236C4⋊C4C22×D19711yescharacteristic
237C22⋊C4—21no—
238C22⋊C4—22no—
239C22⋊C4—3941no—
240C22⋊C4—3942no—
241C22⋊C4C22⋊D19712yes—
242C42C22×D19711yescharacteristic
243C23—21no—
244C23—22no—
245C23—44no—
246C23—44no—
247C23—82no—
248C23—7884no—
249C23—7884no—
250C23—15762no—
251C23—3942no—
252C23—3941no—
253C23C2×C22⋊D19712yes—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—44no—
258D4—44no—
259D4—42no—
260D4—81no—
261D4—7884no—
262D4—7884no—
263D4—7882no—
264D4—15761no—
265D4—3942no—
266D4—3942no—
267D4—3941no—
268D4—1972no—
269D4#1569134314yesdirect product factor
270D4D4×D19712yescomplemented
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—7882no—
276C2×C4—3941no—
277C2×C4C2×C22⋊D19711yescharacteristic
278C2×C4C2×C22⋊D19712yes—
279C2×C4C2×C22⋊D19711yescharacteristic
280C2×C4C23×D19711yescharacteristic
281C8—3942no—
282C8—3941no—
283C8—1972no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—88no—
293C22—7881no—
294C22—7881no—
295C22—7882no—
296C22—7882no—
297C22—7882no—
298C22—15768no—
299C22—3942no—
300C22C2×#1569134314yescomplemented
301C22C22×C22⋊D19711yescharacteristic
302C4—21no—
303C4—22no—
304C4—41no—
305C4—7881no—
306C4C2×#1569134312yescomplemented
307C4C2×D4×D19711yescharacteristic
308C4#2026337111yescharacteristic
309C2—24no—
310C2—42no—
311C2—82no—
312C2—7882no—
313C2—15762no—
314C2D4×C22⋊D19711yescharacteristic
315C2C22×#1569134311yescharacteristic
316C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD42×C197C2
socleC22×C197C22×D4
hypercenterD42D197
hypocenterC197D4×D8
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#15691343C2semidirect
2. D4×D788C2semidirect
3. #27161455C2semidirect
4. #27161313C2semidirect
5. #27161274C2semidirect
6. #27161271C2semidirect
7. #27161213C2semidirect
8. #27161212C2semidirect
9. C4×#15691343C2semidirect
10. D4×#13096802C2semidirect
11. D42×C197C2semidirect
12. C22×D788C22semidirect
16. C2×#15691343C22semidirect
17. C2×#15691343C22semidirect
18. C2×#15691343C22semidirect
19. C2×#15691343C22semidirect
20. C22×#13096802C22semidirect
21. #20263323C22semidirect
23. #20263300C22semidirect
25. C4×D788C22semidirect
26. #20263259C22semidirect
27. C4⋊#13096802C22semidirect
28. #20263247C22semidirect
29. #20263247C22semidirect
30. #20263246C22semidirect
31. #20263246C22semidirect
32. C22⋊#13096802C22semidirect
33. C4×#13096802C22semidirect
35. C22×D4×C197C22semidirect
36. #127×C197C22semidirect
38. #121×C197C22semidirect
41. C4×D4×C197C22semidirect
57. #15691343D4direct (N × H)
95. D788C2×D4semidirect
101. #13096802C2×D4semidirect
115. D4×C197D8semidirect
116. D4×C197C2×D4semidirect
134. C22×C197C2×D8semidirect
146. C4×C197C2×D8semidirect
158. C197D4×D8semidirect
162. D42D197semidirect
174. C22×D4C2×D197semidirect
183. #127C2×D197semidirect
187. #121C2×D197semidirect
192. C4×D4C2×D197semidirect
269. D4#15691343direct (N × H)
270. D4D4×D197semidirect
300. C22C2×#15691343semidirect
306. C4C2×#15691343semidirect

Maximal subgroups

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

subgroupindex
C22×#156913432
D4×D7882
#271614552
#271613132
#271612742
#271612712
#271612132
#271612122
C4×#156913432
D4×#130968022
D42×C1972
D4×D8197