Finite Group Databasebeta

← #38754386

Subgroups — #38754386

Order 25216.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 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×D1576C211yesmaximal, characteristic, complemented
2C2×D8×D197C211yesmaximal, characteristic, complemented
3#27161713C212yesmaximal, complemented
4#27161454C211yesmaximal, characteristic, complemented
5#27161316C211yesmaximal, characteristic, complemented
6#27161273C212yesmaximal, complemented
7#27161214C212yesmaximal, complemented
8#27161212C212yesmaximal, complemented
9#27161161C211yesmaximal, characteristic, complemented
10C8×C22⋊D197C211yesmaximal, characteristic, complemented
11#466×C197C211yesmaximal, characteristic, complemented
12C22×D788C2212yescomplemented
13C2×D4×D197—22no—
14C2×D4×D197C2212yescomplemented
15C2×D1576—24no—
16C2×D1576C2211yescharacteristic
17C2×D1576C2211yescharacteristic, complemented
18C2×D1576C2212yescomplemented
19C2×#15691343C2212yescomplemented
20C2×#15691343C2212yescomplemented
21D8×D197—24no—
22C2×C8×D197C2211yescharacteristic, complemented
23#20263322C2212yescomplemented
24#20263301C2212yescomplemented
25#20263299—22no—
26#20263296—22no—
27#20263294—22no—
28#20263288C2212yescomplemented
29C4×C22⋊D197C2211yescharacteristic
30#20263255C2211yescharacteristic, complemented
31#20263246C2212yescomplemented
32#20263246C2212yescomplemented
33C22⋊#13096802C2211yescharacteristic, complemented
34#20263238C2211yescharacteristic, complemented
35C8×Dic197C2211yescharacteristic, complemented
36C2×D8×C197C2211yescharacteristic, complemented
37C22×C8×C197C2211yescharacteristic, complemented
38#121×C197C2212yescomplemented
39#107×C197C2211yescharacteristic, complemented
40#102×C197C2212yescomplemented
41C23×D197—22no—
42C23×D197—22no—
43C2×D788—24no—
44C2×D788—22no—
45C2×D788—22no—
46C2×D788—42no—
47C2×D788C2312yes—
48C2×D788C2312yes—
49C2×C22⋊D197—22no—
50C2×C22⋊D197—22no—
51C2×C22⋊D197—22no—
52C2×C22⋊D197D411yescharacteristic
53D4×D197—24no—
54D4×D197—42no—
55D4×D197—48no—
56C2×C4×D197—22no—
57C2×C4×D197C2311yescharacteristic
58D1576—24no—
59D1576—22no—
60D1576D414yescomplemented
61#15691343—24no—
62#15691343—42no—
63C2×#13096802C2311yescharacteristic
64C2×#13096802C2311yescharacteristic
65C8×D197—22no—
66#15691337D411yescharacteristic
67C22⋊Dic197D411yescharacteristic
68#15691335—22no—
69#15691335—22no—
70#15691335D411yescharacteristic
71C4×Dic197C2311yescharacteristic
72C2×D4×C197—22no—
73C2×D4×C197C2312yes—
74C22×C4×C197C2311yescharacteristic
75D8×C197—22no—
76C2×C8×C197—22no—
77C2×C8×C197C2311yescharacteristic
78C2×C8×C197C2311yescharacteristic
79C4⋊C4×C197C2312yes—
80C22⋊C4×C197—22no—
81C22×D197—22no—
82C22×D197—22no—
83C22×D197—44no—
84C22×D197—44no—
85C22×D197—44no—
86C22×D197—44no—
87C22×D197—82no—
88C22×D197—82no—
89C22×D197C2×D411yescharacteristic
90D788—22no—
91D788—22no—
92D788—22no—
93D788—44no—
94D788—44no—
95D788C2×D414yescomplemented
96C22⋊D197—44no—
97C22⋊D197—44no—
98C22⋊D197—82no—
99C22⋊D197D814yescomplemented
100C2×Dic197C2×D411yescharacteristic
101C2×Dic197C2×D411yescharacteristic
102C2×Dic197C2×D411yescharacteristic
103C4×D197—21no—
104C4×D197—21no—
105C4×D197—44no—
106#13096802—21no—
107#13096802—21no—
108C23×C197—22no—
109C23×C197C2×D411yescharacteristic
110D4×C197—22no—
111D4×C197—22no—
112D4×C197—44no—
113C2×C4×C197—21no—
114C2×C4×C197—21no—
115C2×C4×C197—22no—
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2411yescharacteristic
118C8×C197—21no—
119C8×C197C2×D412yescomplemented
120C22×C197—21no—
121C22×C197—21no—
122C22×C197—42no—
123C22×C197—42no—
124C22×C197—42no—
125C22×C197C2×D812yescomplemented
126C22×C197C22×D411yescharacteristic
127C2×D197—21no—
128C2×D197—21no—
129C2×D197—24no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—42no—
133C2×D197—42no—
134C2×D197—42no—
135C2×D197—88no—
136C2×D197—88no—
137C2×D197C2×D812yescomplemented
138C4×C197—21no—
139C4×C197—42no—
140C4×C197C22×D411yescharacteristic
141C4×C197H(5,𝔽2)11yescharacteristic
142Dic197—21no—
143Dic197—21no—
144Dic197C2×D812yescomplemented
145C2×C197—22no—
146C2×C197—82no—
147C2×C197D4211yescharacteristic
148C2×C197C22×D811yescharacteristic
149C2×C197#57611yescharacteristic
150D197—22no—
151D197—44no—
152D197—82no—
153C197D4×D811yescharacteristic, complemented, Sylow
154D4×D8—1971nomaximal, Sylow
155C22×D8—1971no—
156C22×D8—1971no—
157D42—1972no—
158#493—1971no—
159#466—1971no—
160#466D19711yescharacteristic, complemented
161#459—1972no—
162#447—1972no—
163#447—1972no—
164C4×D8—1971no—
165C8×D4—1971no—
166C22×D4—3942no—
167C22×D4—1972no—
168C22×D4—1972no—
169C2×D8—3944no—
170C2×D8—3944no—
171C2×D8—1972no—
172C2×D8—1972no—
173C2×D8—1972no—
174C2×D8—1971no—
175C2×D8—1971no—
176C2×D8C2×D19711yescharacteristic, complemented
177C22×C8—1971no—
178C22×C8C2×D19711yescharacteristic, complemented
179#127—1972no—
180#121—3942no—
181#121—1972no—
182#121C2×D19712yescomplemented
183#120—3942no—
184#120—3942no—
185C4×D4—1972no—
186C4×D4—1971no—
187#107C2×D19711yescharacteristic, complemented
188C4⋊C8—1971no—
189#102—1972no—
190#102—1972no—
191#102C2×D19712yescomplemented
192C22⋊C8—1971no—
193C22⋊C8—1971no—
194C4×C8—1971no—
195C24—3942no—
196C24—3942no—
197C2×D4—22no—
198C2×D4—7882no—
199C2×D4—7888no—
200C2×D4—7882no—
201C2×D4—3942no—
202C2×D4—3944no—
203C2×D4—3944no—
204C2×D4—3942no—
205C2×D4—3942no—
206C2×D4—3942no—
207C2×D4—3942no—
208C2×D4—1971no—
209C2×D4—1972no—
210C2×D4—1972no—
211C2×D4C22×D19712yes—
212C22×C4—3942no—
213C22×C4—1971no—
214C22×C4C22×D19711yescharacteristic
215D8—22no—
216D8—7882no—
217D8—3944no—
218D8—3944no—
219D8—3942no—
220D8—1974no—
221C2×C8—22no—
222C2×C8—3942no—
223C2×C8—1971no—
224C2×C8—1971no—
225C2×C8C22×D19711yescharacteristic
226C2×C8C22×D19711yescharacteristic
227C4⋊C4—1971no—
228C4⋊C4C22×D19712yes—
229C22⋊C4—22no—
230C22⋊C4—3942no—
231C22⋊C4—3942no—
232C22⋊C4—1971no—
233C22⋊C4—1971no—
234C42—1971no—
235C23—22no—
236C23—7884no—
237C23—7884no—
238C23—7884no—
239C23—7884no—
240C23—15762no—
241C23—15762no—
242C23—3942no—
243C23—3942no—
244C23—1971no—
245C23D4×D19711yescharacteristic
246D4—22no—
247D4—22no—
248D4—44no—
249D4—7884no—
250D4—7884no—
251D4—7884no—
252D4—7884no—
253D4—15762no—
254D4—3942no—
255D4—3942no—
256D4—3942no—
257D4—1974no—
258D4—1974no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—7884no—
263C2×C4—3941no—
264C2×C4—3941no—
265C2×C4—1971no—
266C2×C4—1971no—
267C2×C4—1971no—
268C2×C4D4×D19711yescharacteristic
269C2×C4C23×D19711yescharacteristic
270C8—21no—
271C8—3941no—
272C8—3941no—
273C8D4×D19712yescomplemented
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—7882no—
280C22—7882no—
281C22—7882no—
282C22—7882no—
283C22—7882no—
284C22—15768no—
285C22—15768no—
286C22—3944no—
287C22—3941no—
288C22—3941no—
289C22—1972no—
290C22D8×D19712yescomplemented
291C22C2×D4×D19711yescharacteristic
292C4—21no—
293C4—42no—
294C4—3941no—
295C4—3941no—
296C4—1972no—
297C4C2×D4×D19711yescharacteristic
298C4#2026337211yescharacteristic
299C2—22no—
300C2—82no—
301C2—7884no—
302C2—15762no—
303C2—3942no—
304C2#2716171311yescharacteristic
305C2C2×D8×D19711yescharacteristic
306C2#2716188111yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#466×C197C2
socleC22×C197C22×D4
hypercenter#466D197
hypocenterC197D4×D8
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C22×D1576C2semidirect
2. C2×D8×D197C2semidirect
3. #27161713C2semidirect
4. #27161454C2semidirect
5. #27161316C2semidirect
6. #27161273C2semidirect
7. #27161214C2semidirect
8. #27161212C2semidirect
9. #27161161C2semidirect
10. C8×C22⋊D197C2semidirect
11. #466×C197C2semidirect
12. C22×D788C22semidirect
14. C2×D4×D197C22semidirect
17. C2×D1576C22semidirect
18. C2×D1576C22semidirect
19. C2×#15691343C22semidirect
20. C2×#15691343C22semidirect
22. C2×C8×D197C22semidirect
23. #20263322C22semidirect
24. #20263301C22semidirect
28. #20263288C22semidirect
30. #20263255C22semidirect
31. #20263246C22semidirect
32. #20263246C22semidirect
33. C22⋊#13096802C22semidirect
34. #20263238C22semidirect
35. C8×Dic197C22semidirect
36. C2×D8×C197C22semidirect
37. C22×C8×C197C22semidirect
38. #121×C197C22semidirect
39. #107×C197C22semidirect
40. #102×C197C22semidirect
60. D1576D4semidirect
95. D788C2×D4semidirect
99. C22⋊D197D8semidirect
119. C8×C197C2×D4semidirect
125. C22×C197C2×D8semidirect
137. C2×D197C2×D8semidirect
144. Dic197C2×D8semidirect
153. C197D4×D8semidirect
160. #466D197semidirect
176. C2×D8C2×D197semidirect
178. C22×C8C2×D197semidirect
182. #121C2×D197semidirect
187. #107C2×D197semidirect
191. #102C2×D197semidirect
273. C8D4×D197semidirect
290. C22D8×D197semidirect

Maximal subgroups

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

subgroupindex
C22×D15762
C2×D8×D1972
#271617132
#271614542
#271613162
#271612732
#271612142
#271612122
#271611612
C8×C22⋊D1972
#466×C1972
D4×D8197