Finite Group Databasebeta

← #27164657

Subgroups — #27164657

Order 12736.

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×#15691885C211yesmaximal, characteristic, complemented
2C2×D4×D199C211yesmaximal, characteristic, complemented
3#20263941C211yesmaximal, characteristic, complemented
4#20263940C211yesmaximal, characteristic, complemented
5#20263936C211yesmaximal, characteristic, complemented
6#20263933C211yesmaximal, characteristic, complemented
7#20263928C211yesmaximal, characteristic, complemented
8#20263927C211yesmaximal, characteristic, complemented
9#20263923C211yesmaximal, characteristic, complemented
10#20263923C211yesmaximal, characteristic, complemented
11C4×C22⋊D199C211yesmaximal, characteristic, complemented
12(C22⋊D199)⋊C4C211yesmaximal, characteristic, complemented
13C2×(C2×D199)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D199C211yesmaximal, characteristic, complemented
15#121×C199C211yesmaximal, characteristic, complemented
16C23×D199C2211yescharacteristic
17C23×D199C2211yescharacteristic
18#15691885—24no—
19C2×Dic398C2211yescharacteristic, complemented
20C2×D796C2211yescharacteristic
21C2×C22⋊D199C2211yescharacteristic
22C2×C22⋊D199C2211yescharacteristic
23C2×C22⋊D199C2211yescharacteristic
24C2×C22⋊D199C2211yescharacteristic, complemented
25C2×C22⋊D199C2211yescharacteristic, complemented
26D4×D199—24no—
27C22×Dic199C2211yescharacteristic, complemented
28C22×Dic199C2211yescharacteristic, complemented
29C2×C4×D199C2211yescharacteristic
30C2×C4×D199C2211yescharacteristic, complemented
31C2×C4×D199C2211yescharacteristic, complemented
32Dic199⋊C4C2211yescharacteristic, complemented
33Dic199⋊C4C2211yescharacteristic, complemented
34Dic199⋊C4C2211yescharacteristic, complemented
35C22⋊Dic199C2211yescharacteristic, complemented
36C22⋊Dic199C2211yescharacteristic, complemented
37C22⋊Dic199C2211yescharacteristic, complemented
38(C2×D199)⋊C4C2211yescharacteristic
39(C2×D199)⋊C4C2211yescharacteristic
40(C2×D199)⋊C4C2211yescharacteristic
41(C2×D199)⋊C4C2211yescharacteristic, complemented
42(C2×D199)⋊C4C2211yescharacteristic, complemented
43(C2×D199)⋊C4C2211yescharacteristic, complemented
44(C2×D199)⋊C4C2211yescharacteristic, complemented
45C4×Dic199C2211yescharacteristic, complemented
46C2×D4×C199C2211yescharacteristic
47C2×D4×C199C2211yescharacteristic, complemented
48C2×D4×C199C2211yescharacteristic, complemented
49C22×C4×C199C2211yescharacteristic, complemented
50C4⋊C4×C199C2211yescharacteristic, complemented
51C22⋊C4×C199C2211yescharacteristic, complemented
52C22⋊C4×C199C2211yescharacteristic, complemented
53C22×D199—22no—
54C22×D199—22no—
55C22×D199—22no—
56C22×D199—22no—
57C22×D199—41no—
58C22×D199—41no—
59C22×D199C2311yescharacteristic
60C22×D199C2311yescharacteristic
61C22×D199C2311yescharacteristic
62Dic398—22no—
63D796—22no—
64C22⋊D199—22no—
65C22⋊D199—22no—
66C22⋊D199—22no—
67C22⋊D199—41no—
68C22⋊D199D414yescomplemented
69C2×Dic199—22no—
70C2×Dic199—22no—
71C2×Dic199C2311yescharacteristic
72C2×Dic199C2311yescharacteristic
73C2×Dic199C2311yescharacteristic
74C2×Dic199C2311yescharacteristic
75C2×Dic199C2311yescharacteristic
76C4×D199—22no—
77C4×D199—22no—
78C4×D199—41no—
79C23×C199C2311yescharacteristic
80C23×C199C2311yescharacteristic
81C23×C199C2311yescharacteristic
82D4×C199—22no—
83D4×C199—22no—
84D4×C199—41no—
85C2×C4×C199—41no—
86C2×C4×C199C2311yescharacteristic
87C2×C4×C199C2311yescharacteristic
88C2×C4×C199C2311yescharacteristic
89C2×C4×C199C2311yescharacteristic
90C22×C199—21no—
91C22×C199—21no—
92C22×C199—21no—
93C22×C199—21no—
94C22×C199—21no—
95C22×C199—21no—
96C22×C199—21no—
97C22×C199—21no—
98C22×C199C2×D412yes—
99C22×C199C2411yescharacteristic
100C2×D199—21no—
101C2×D199—21no—
102C2×D199—21no—
103C2×D199—21no—
104C2×D199—21no—
105C2×D199—21no—
106C2×D199—21no—
107C2×D199—44no—
108C2×D199—44no—
109C2×D199C2×D412yes—
110C2×D199#4112yescomplemented
111C4×C199—21no—
112C4×C199—21no—
113C4×C199—21no—
114C4×C199—21no—
115Dic199—21no—
116Dic199—21no—
117Dic199—21no—
118Dic199—21no—
119Dic199C2×D412yes—
120C2×C199—22no—
121C2×C199—41no—
122C2×C199—41no—
123C2×C199C22×D411yescharacteristic
124C2×C199C2×#4111yescharacteristic
125C2×C199H(5,𝔽2)11yescharacteristic
126D199—22no—
127D199—22no—
128D199—41no—
129C199#54611yescharacteristic, complemented, Sylow
130#546—1991nomaximal, Sylow
131C2×#41—1991no—
132C22×D4—1991no—
133#124—1991no—
134#123—1991no—
135#123—1991no—
136C22⋊Q8—1991no—
137#121—1991no—
138#121—1991no—
139#121D19911yescharacteristic, complemented
140#120—1991no—
141#120—1991no—
142C4×D4—1991no—
143C4×D4—1991no—
144C2×C22⋊C4—1991no—
145C2×C22⋊C4—1991no—
146C24—1991no—
147C24—1991no—
148#41—3984no—
149C2×Q8—1991no—
150C2×D4—3984no—
151C2×D4—1991no—
152C2×D4—1991no—
153C2×D4—1991no—
154C2×D4—1991no—
155C2×D4—1991no—
156C2×D4—1991no—
157C2×D4C2×D19911yescharacteristic
158C2×D4C2×D19911yescharacteristic, complemented
159C2×D4C2×D19911yescharacteristic, complemented
160C22×C4—1991no—
161C22×C4—1991no—
162C22×C4—1991no—
163C22×C4—1991no—
164C22×C4—1991no—
165C22×C4C2×D19911yescharacteristic, complemented
166C4⋊C4—1991no—
167C4⋊C4—1991no—
168C4⋊C4—1991no—
169C4⋊C4C2×D19911yescharacteristic, complemented
170C22⋊C4—1991no—
171C22⋊C4—1991no—
172C22⋊C4—1991no—
173C22⋊C4—1991no—
174C22⋊C4—1991no—
175C22⋊C4—1991no—
176C22⋊C4—1991no—
177C22⋊C4—1991no—
178C22⋊C4—1991no—
179C22⋊C4—1991no—
180C22⋊C4C2×D19911yescharacteristic, complemented
181C22⋊C4C2×D19911yescharacteristic, complemented
182C42—1991no—
183C23—7961no—
184C23—7961no—
185C23—3982no—
186C23—3982no—
187C23—3982no—
188C23—3982no—
189C23—1991no—
190C23—1991no—
191C23—1991no—
192C23C22×D19911yescharacteristic
193C23C22×D19911yescharacteristic
194C23C22×D19911yescharacteristic
195Q8—3982no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—7961no—
200D4—3982no—
201D4—3982no—
202D4—3982no—
203D4—3982no—
204D4—1994no—
205C2×C4—41no—
206C2×C4—7961no—
207C2×C4—3982no—
208C2×C4—3982no—
209C2×C4—3982no—
210C2×C4—3982no—
211C2×C4—1991no—
212C2×C4—1991no—
213C2×C4—1991no—
214C2×C4—1991no—
215C2×C4—1991no—
216C2×C4C22×D19911yescharacteristic
217C2×C4C22×D19911yescharacteristic
218C2×C4C22×D19911yescharacteristic
219C2×C4C22×D19911yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—7964no—
229C22—7964no—
230C22—3981no—
231C22—3981no—
232C22—3981no—
233C22—3981no—
234C22—3981no—
235C22—3981no—
236C22—3981no—
237C22—1992no—
238C22—1992no—
239C22D4×D19912yes—
240C22C23×D19911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—3981no—
246C4—3981no—
247C4—3981no—
248C4—3981no—
249C4—1992no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—7961no—
254C2—3982no—
255C2—3982no—
256C2C2×D4×D19911yescharacteristic
257C2#41×D19911yescharacteristic
258C2#2026400011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#121×C199C2
hypercenter#121D199
hypocenterC199#546
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15691885C2semidirect
2. C2×D4×D199C2semidirect
3. #20263941C2semidirect
4. #20263940C2semidirect
5. #20263936C2semidirect
6. #20263933C2semidirect
7. #20263928C2semidirect
8. #20263927C2semidirect
9. #20263923C2semidirect
10. #20263923C2semidirect
11. C4×C22⋊D199C2semidirect
12. (C22⋊D199)⋊C4C2semidirect
13. C2×(C2×D199)⋊C4C2semidirect
14. C22⋊C4×D199C2semidirect
15. #121×C199C2semidirect
19. C2×Dic398C22semidirect
24. C2×C22⋊D199C22semidirect
25. C2×C22⋊D199C22semidirect
27. C22×Dic199C22semidirect
28. C22×Dic199C22semidirect
30. C2×C4×D199C22semidirect
31. C2×C4×D199C22semidirect
32. Dic199⋊C4C22semidirect
33. Dic199⋊C4C22semidirect
34. Dic199⋊C4C22semidirect
35. C22⋊Dic199C22semidirect
36. C22⋊Dic199C22semidirect
37. C22⋊Dic199C22semidirect
41. (C2×D199)⋊C4C22semidirect
42. (C2×D199)⋊C4C22semidirect
43. (C2×D199)⋊C4C22semidirect
44. (C2×D199)⋊C4C22semidirect
45. C4×Dic199C22semidirect
47. C2×D4×C199C22semidirect
48. C2×D4×C199C22semidirect
49. C22×C4×C199C22semidirect
50. C4⋊C4×C199C22semidirect
51. C22⋊C4×C199C22semidirect
52. C22⋊C4×C199C22semidirect
68. C22⋊D199D4semidirect
110. C2×D199#41semidirect
129. C199#546semidirect
139. #121D199semidirect
158. C2×D4C2×D199semidirect
159. C2×D4C2×D199semidirect
165. C22×C4C2×D199semidirect
169. C4⋊C4C2×D199semidirect
180. C22⋊C4C2×D199semidirect
181. C22⋊C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
C2×#156918852
C2×D4×D1992
#202639412
#202639402
#202639362
#202639332
#202639282
#202639272
#202639232
C4×C22⋊D1992
(C22⋊D199)⋊C42
C2×(C2×D199)⋊C42
C22⋊C4×D1992
#121×C1992
#546199