Finite Group Databasebeta

← #27161718

Subgroups — #27161718

Order 12608.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#15691357C211yesmaximal, characteristic, complemented
2C2×D4×D197C211yesmaximal, characteristic, complemented
3#20263312C211yesmaximal, characteristic, complemented
4#20263311C211yesmaximal, characteristic, complemented
5#20263307C211yesmaximal, characteristic, complemented
6#20263304C211yesmaximal, characteristic, complemented
7#20263299C211yesmaximal, characteristic, complemented
8#20263298C211yesmaximal, characteristic, complemented
9#20263294C211yesmaximal, characteristic, complemented
10#20263294C211yesmaximal, characteristic, complemented
11C4×C22⋊D197C211yesmaximal, characteristic, complemented
12#20263286C211yesmaximal, characteristic, complemented
13C2×#15691335C211yesmaximal, characteristic, complemented
14C22⋊C4×D197C211yesmaximal, characteristic, complemented
15#121×C197C211yesmaximal, characteristic, complemented
16C23×D197C2211yescharacteristic
17C23×D197C2211yescharacteristic
18#15691357—24no—
19C2×Dic394C2211yescharacteristic, complemented
20C2×D788C2211yescharacteristic
21C2×C22⋊D197C2211yescharacteristic
22C2×C22⋊D197C2211yescharacteristic
23C2×C22⋊D197C2211yescharacteristic
24C2×C22⋊D197C2211yescharacteristic, complemented
25C2×C22⋊D197C2211yescharacteristic, complemented
26D4×D197—24no—
27C22×Dic197C2211yescharacteristic, complemented
28C22×Dic197C2211yescharacteristic, complemented
29C2×C4×D197C2211yescharacteristic
30C2×C4×D197C2211yescharacteristic, complemented
31C2×C4×D197C2211yescharacteristic, complemented
32#15691337C2211yescharacteristic, complemented
33#15691337C2211yescharacteristic, complemented
34#15691337C2211yescharacteristic, complemented
35C22⋊Dic197C2211yescharacteristic, complemented
36C22⋊Dic197C2211yescharacteristic, complemented
37C22⋊Dic197C2211yescharacteristic, complemented
38#15691335C2211yescharacteristic
39#15691335C2211yescharacteristic
40#15691335C2211yescharacteristic
41#15691335C2211yescharacteristic, complemented
42#15691335C2211yescharacteristic, complemented
43#15691335C2211yescharacteristic, complemented
44#15691335C2211yescharacteristic, complemented
45C4×Dic197C2211yescharacteristic, complemented
46C2×D4×C197C2211yescharacteristic
47C2×D4×C197C2211yescharacteristic, complemented
48C2×D4×C197C2211yescharacteristic, complemented
49C22×C4×C197C2211yescharacteristic, complemented
50C4⋊C4×C197C2211yescharacteristic, complemented
51C22⋊C4×C197C2211yescharacteristic, complemented
52C22⋊C4×C197C2211yescharacteristic, complemented
53C22×D197—22no—
54C22×D197—22no—
55C22×D197—22no—
56C22×D197—22no—
57C22×D197—41no—
58C22×D197—41no—
59C22×D197C2311yescharacteristic
60C22×D197C2311yescharacteristic
61C22×D197C2311yescharacteristic
62Dic394—22no—
63D788—22no—
64C22⋊D197—22no—
65C22⋊D197—22no—
66C22⋊D197—22no—
67C22⋊D197—41no—
68C22⋊D197D414yescomplemented
69C2×Dic197—22no—
70C2×Dic197—22no—
71C2×Dic197C2311yescharacteristic
72C2×Dic197C2311yescharacteristic
73C2×Dic197C2311yescharacteristic
74C2×Dic197C2311yescharacteristic
75C2×Dic197C2311yescharacteristic
76C4×D197—22no—
77C4×D197—22no—
78C4×D197—41no—
79C23×C197C2311yescharacteristic
80C23×C197C2311yescharacteristic
81C23×C197C2311yescharacteristic
82D4×C197—22no—
83D4×C197—22no—
84D4×C197—41no—
85C2×C4×C197—41no—
86C2×C4×C197C2311yescharacteristic
87C2×C4×C197C2311yescharacteristic
88C2×C4×C197C2311yescharacteristic
89C2×C4×C197C2311yescharacteristic
90C22×C197—21no—
91C22×C197—21no—
92C22×C197—21no—
93C22×C197—21no—
94C22×C197—21no—
95C22×C197—21no—
96C22×C197—21no—
97C22×C197—21no—
98C22×C197C2×D412yes—
99C22×C197C2411yescharacteristic
100C2×D197—21no—
101C2×D197—21no—
102C2×D197—21no—
103C2×D197—21no—
104C2×D197—21no—
105C2×D197—21no—
106C2×D197—21no—
107C2×D197—44no—
108C2×D197—44no—
109C2×D197C2×D412yes—
110C2×D197#4112yescomplemented
111C4×C197—21no—
112C4×C197—21no—
113C4×C197—21no—
114C4×C197—21no—
115Dic197—21no—
116Dic197—21no—
117Dic197—21no—
118Dic197—21no—
119Dic197C2×D412yes—
120C2×C197—22no—
121C2×C197—41no—
122C2×C197—41no—
123C2×C197C22×D411yescharacteristic
124C2×C197C2×#4111yescharacteristic
125C2×C197H(5,𝔽2)11yescharacteristic
126D197—22no—
127D197—22no—
128D197—41no—
129C197#54611yescharacteristic, complemented, Sylow
130#546—1971nomaximal, Sylow
131C2×#41—1971no—
132C22×D4—1971no—
133#124—1971no—
134#123—1971no—
135#123—1971no—
136C22⋊Q8—1971no—
137#121—1971no—
138#121—1971no—
139#121D19711yescharacteristic, complemented
140#120—1971no—
141#120—1971no—
142C4×D4—1971no—
143C4×D4—1971no—
144C2×C22⋊C4—1971no—
145C2×C22⋊C4—1971no—
146C24—1971no—
147C24—1971no—
148#41—3944no—
149C2×Q8—1971no—
150C2×D4—3944no—
151C2×D4—1971no—
152C2×D4—1971no—
153C2×D4—1971no—
154C2×D4—1971no—
155C2×D4—1971no—
156C2×D4—1971no—
157C2×D4C2×D19711yescharacteristic
158C2×D4C2×D19711yescharacteristic, complemented
159C2×D4C2×D19711yescharacteristic, complemented
160C22×C4—1971no—
161C22×C4—1971no—
162C22×C4—1971no—
163C22×C4—1971no—
164C22×C4—1971no—
165C22×C4C2×D19711yescharacteristic, complemented
166C4⋊C4—1971no—
167C4⋊C4—1971no—
168C4⋊C4—1971no—
169C4⋊C4C2×D19711yescharacteristic, complemented
170C22⋊C4—1971no—
171C22⋊C4—1971no—
172C22⋊C4—1971no—
173C22⋊C4—1971no—
174C22⋊C4—1971no—
175C22⋊C4—1971no—
176C22⋊C4—1971no—
177C22⋊C4—1971no—
178C22⋊C4—1971no—
179C22⋊C4—1971no—
180C22⋊C4C2×D19711yescharacteristic, complemented
181C22⋊C4C2×D19711yescharacteristic, complemented
182C42—1971no—
183C23—7881no—
184C23—7881no—
185C23—3942no—
186C23—3942no—
187C23—3942no—
188C23—3942no—
189C23—1971no—
190C23—1971no—
191C23—1971no—
192C23C22×D19711yescharacteristic
193C23C22×D19711yescharacteristic
194C23C22×D19711yescharacteristic
195Q8—3942no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—7881no—
200D4—3942no—
201D4—3942no—
202D4—3942no—
203D4—3942no—
204D4—1974no—
205C2×C4—41no—
206C2×C4—7881no—
207C2×C4—3942no—
208C2×C4—3942no—
209C2×C4—3942no—
210C2×C4—3942no—
211C2×C4—1971no—
212C2×C4—1971no—
213C2×C4—1971no—
214C2×C4—1971no—
215C2×C4—1971no—
216C2×C4C22×D19711yescharacteristic
217C2×C4C22×D19711yescharacteristic
218C2×C4C22×D19711yescharacteristic
219C2×C4C22×D19711yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—7884no—
229C22—7884no—
230C22—3941no—
231C22—3941no—
232C22—3941no—
233C22—3941no—
234C22—3941no—
235C22—3941no—
236C22—3941no—
237C22—1972no—
238C22—1972no—
239C22D4×D19712yes—
240C22C23×D19711yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—3941no—
246C4—3941no—
247C4—3941no—
248C4—3941no—
249C4—1972no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—7881no—
254C2—3942no—
255C2—3942no—
256C2C2×D4×D19711yescharacteristic
257C2#41×D19711yescharacteristic
258C2#2026337111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#121×C197C2
hypercenter#121D197
hypocenterC197#546
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15691357C2semidirect
2. C2×D4×D197C2semidirect
3. #20263312C2semidirect
4. #20263311C2semidirect
5. #20263307C2semidirect
6. #20263304C2semidirect
7. #20263299C2semidirect
8. #20263298C2semidirect
9. #20263294C2semidirect
10. #20263294C2semidirect
11. C4×C22⋊D197C2semidirect
12. #20263286C2semidirect
13. C2×#15691335C2semidirect
14. C22⋊C4×D197C2semidirect
15. #121×C197C2semidirect
19. C2×Dic394C22semidirect
24. C2×C22⋊D197C22semidirect
25. C2×C22⋊D197C22semidirect
27. C22×Dic197C22semidirect
28. C22×Dic197C22semidirect
30. C2×C4×D197C22semidirect
31. C2×C4×D197C22semidirect
32. #15691337C22semidirect
33. #15691337C22semidirect
34. #15691337C22semidirect
35. C22⋊Dic197C22semidirect
36. C22⋊Dic197C22semidirect
37. C22⋊Dic197C22semidirect
41. #15691335C22semidirect
42. #15691335C22semidirect
43. #15691335C22semidirect
44. #15691335C22semidirect
45. C4×Dic197C22semidirect
47. C2×D4×C197C22semidirect
48. C2×D4×C197C22semidirect
49. C22×C4×C197C22semidirect
50. C4⋊C4×C197C22semidirect
51. C22⋊C4×C197C22semidirect
52. C22⋊C4×C197C22semidirect
68. C22⋊D197D4semidirect
110. C2×D197#41semidirect
129. C197#546semidirect
139. #121D197semidirect
158. C2×D4C2×D197semidirect
159. C2×D4C2×D197semidirect
165. C22×C4C2×D197semidirect
169. C4⋊C4C2×D197semidirect
180. C22⋊C4C2×D197semidirect
181. C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#156913572
C2×D4×D1972
#202633122
#202633112
#202633072
#202633042
#202632992
#202632982
#202632942
C4×C22⋊D1972
#202632862
C2×#156913352
C22⋊C4×D1972
#121×C1972
#546197