Finite Group Databasebeta

← #28412219

Subgroups — #28412219

Order 14656.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#20300984C211yesmaximal, characteristic, complemented
2#20300983C211yesmaximal, characteristic, complemented
3#20300981C211yesmaximal, characteristic, complemented
4#20300973C211yesmaximal, characteristic, complemented
5#20300972C211yesmaximal, characteristic, complemented
6#20300971C211yesmaximal, characteristic, complemented
7#20300962C211yesmaximal, characteristic, complemented
8#20300961C211yesmaximal, characteristic, complemented
9#20300960C211yesmaximal, characteristic, complemented
10#20300960C211yesmaximal, characteristic, complemented
11#20300959C211yesmaximal, characteristic, complemented
12#20300958C211yesmaximal, characteristic, complemented
13#20300956C211yesmaximal, characteristic, complemented
14#20300956C211yesmaximal, characteristic, complemented
15#121×C229C211yesmaximal, characteristic, complemented
16C23×D229C2211yescharacteristic
17C2×D916C2211yescharacteristic
18C2×D916C2211yescharacteristic
19C2×C22⋊D229C2211yescharacteristic
20C2×C22⋊D229C2211yescharacteristic
21C2×C22⋊D229C2211yescharacteristic
22C2×C22⋊D229C2211yescharacteristic
23C2×C22⋊D229C2211yescharacteristic
24C2×C22⋊D229C2211yescharacteristic, complemented
25C2×C22⋊D229C2211yescharacteristic, complemented
26C22×Dic229C2211yescharacteristic, complemented
27C2×C4×D229C2211yescharacteristic, complemented
28C4⋊Dic229C2211yescharacteristic, complemented
29C4⋊Dic229C2211yescharacteristic, complemented
30#16814645C2211yescharacteristic, complemented
31#16814645C2211yescharacteristic, complemented
32#16814645C2211yescharacteristic, complemented
33C22⋊Dic229C2211yescharacteristic
34C22⋊Dic229C2211yescharacteristic
35C22⋊Dic229C2211yescharacteristic, complemented
36#16814643C2211yescharacteristic
37#16814643C2211yescharacteristic
38#16814643C2211yescharacteristic
39#16814643C2211yescharacteristic
40#16814643C2211yescharacteristic
41#16814643C2211yescharacteristic
42#16814643C2211yescharacteristic, complemented
43C4×Dic229C2211yescharacteristic, complemented
44C2×D4×C229C2211yescharacteristic
45C2×D4×C229C2211yescharacteristic
46C2×D4×C229C2211yescharacteristic, complemented
47C22×C4×C229C2211yescharacteristic, complemented
48C4⋊C4×C229C2211yescharacteristic, complemented
49C22⋊C4×C229C2211yescharacteristic
50C22⋊C4×C229C2211yescharacteristic, complemented
51C22×D229—41no—
52C22×D229—41no—
53C22×D229—41no—
54C22×D229C2311yescharacteristic
55C22×D229C2311yescharacteristic
56C22×D229C2311yescharacteristic
57D916—41no—
58D916—41no—
59C22⋊D229—41no—
60C22⋊D229—41no—
61C22⋊D229—41no—
62C22⋊D229—41no—
63C22⋊D229—41no—
64C22⋊D229—41no—
65C22⋊D229—41no—
66C2×Dic229—41no—
67C2×Dic229C2311yescharacteristic
68C2×Dic229C2311yescharacteristic
69C2×Dic229C2311yescharacteristic
70C2×Dic229C2311yescharacteristic
71C2×Dic229C2311yescharacteristic
72C4×D229—41no—
73C23×C229C2311yescharacteristic
74C23×C229C2311yescharacteristic
75C23×C229C2311yescharacteristic
76D4×C229—41no—
77D4×C229—41no—
78D4×C229—41no—
79C2×C4×C229—41no—
80C2×C4×C229C2311yescharacteristic
81C2×C4×C229C2311yescharacteristic
82C2×C4×C229C2311yescharacteristic
83C2×C4×C229C2311yescharacteristic
84C22×C229—21no—
85C22×C229—21no—
86C22×C229—21no—
87C22×C229—21no—
88C22×C229—21no—
89C22×C229—21no—
90C22×C229—21no—
91C22×C229—21no—
92C22×C229—21no—
93C22×C229C2411yescharacteristic
94C2×D229—21no—
95C2×D229—21no—
96C2×D229—21no—
97C2×D229—21no—
98C2×D229—21no—
99C2×D229—21no—
100C2×D229—21no—
101C2×D229—21no—
102C2×D229—21no—
103C2×D229—44no—
104C4×C229—21no—
105C4×C229—21no—
106C4×C229—21no—
107C4×C229—21no—
108Dic229—21no—
109Dic229—21no—
110Dic229—21no—
111Dic229—21no—
112Dic229—21no—
113C2×C229—41no—
114C2×C229—41no—
115C2×C229—41no—
116C2×C229H(5,𝔽2)11yescharacteristic
117C2×C229H(5,𝔽2)11yescharacteristic
118C2×C229H(5,𝔽2)11yescharacteristic
119D229—41no—
120D229—41no—
121D229—41no—
122C229#56011yescharacteristic, complemented, Sylow
123#560—2291nomaximal, Sylow
124#127—2291no—
125#126—2291no—
126#126—2291no—
127#123—2291no—
128#123—2291no—
129#123—2291no—
130#121—2291no—
131#121—2291no—
132#121—2291no—
133#121—2291no—
134#121—2291no—
135#121D22911yescharacteristic, complemented
136#120—2291no—
137#120—2291no—
138#120—2291no—
139C24—2291no—
140C2×D4—2291no—
141C2×D4—2291no—
142C2×D4—2291no—
143C2×D4—2291no—
144C2×D4—2291no—
145C2×D4—2291no—
146C2×D4—2291no—
147C2×D4—2291no—
148C2×D4—2291no—
149C2×D4C2×D22911yescharacteristic
150C2×D4C2×D22911yescharacteristic
151C2×D4C2×D22911yescharacteristic, complemented
152C22×C4—2291no—
153C22×C4—2291no—
154C22×C4C2×D22911yescharacteristic, complemented
155C4⋊C4—2291no—
156C4⋊C4—2291no—
157C4⋊C4—2291no—
158C4⋊C4—2291no—
159C4⋊C4—2291no—
160C4⋊C4C2×D22911yescharacteristic, complemented
161C22⋊C4—2291no—
162C22⋊C4—2291no—
163C22⋊C4—2291no—
164C22⋊C4—2291no—
165C22⋊C4—2291no—
166C22⋊C4—2291no—
167C22⋊C4—2291no—
168C22⋊C4—2291no—
169C22⋊C4—2291no—
170C22⋊C4—2291no—
171C22⋊C4C2×D22911yescharacteristic
172C22⋊C4C2×D22911yescharacteristic, complemented
173C42—2291no—
174C23—9161no—
175C23—9161no—
176C23—9161no—
177C23—2291no—
178C23—2291no—
179C23—2291no—
180C23C22×D22911yescharacteristic
181C23C22×D22911yescharacteristic
182C23C22×D22911yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—9161no—
187D4—9161no—
188D4—9161no—
189D4—9161no—
190D4—9161no—
191D4—9161no—
192D4—9161no—
193D4—9161no—
194D4—9161no—
195C2×C4—41no—
196C2×C4—9161no—
197C2×C4—9161no—
198C2×C4—2291no—
199C2×C4—2291no—
200C2×C4—2291no—
201C2×C4—2291no—
202C2×C4—2291no—
203C2×C4C22×D22911yescharacteristic
204C2×C4C22×D22911yescharacteristic
205C2×C4C22×D22911yescharacteristic
206C2×C4C22×D22911yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—9164no—
217C22—4581no—
218C22—4581no—
219C22—4581no—
220C22—4581no—
221C22—4581no—
222C22—4581no—
223C22—4581no—
224C22—4581no—
225C22—4581no—
226C22C23×D22911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—4581no—
232C4—4581no—
233C4—4581no—
234C4—4581no—
235C4—4581no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—9161no—
240C2—9161no—
241C2—9161no—
242C2#2030103311yescharacteristic
243C2#2030103311yescharacteristic
244C2#2030103411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D229
commutator, socle, cosocleC22×C229C24
Fitting#121×C229C2
hypercenter#121D229
hypocenterC229#560
LCSG ▷ C22×C229 ▷ C229
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20300984C2semidirect
2. #20300983C2semidirect
3. #20300981C2semidirect
4. #20300973C2semidirect
5. #20300972C2semidirect
6. #20300971C2semidirect
7. #20300962C2semidirect
8. #20300961C2semidirect
9. #20300960C2semidirect
10. #20300960C2semidirect
11. #20300959C2semidirect
12. #20300958C2semidirect
13. #20300956C2semidirect
14. #20300956C2semidirect
15. #121×C229C2semidirect
24. C2×C22⋊D229C22semidirect
25. C2×C22⋊D229C22semidirect
26. C22×Dic229C22semidirect
27. C2×C4×D229C22semidirect
28. C4⋊Dic229C22semidirect
29. C4⋊Dic229C22semidirect
30. #16814645C22semidirect
31. #16814645C22semidirect
32. #16814645C22semidirect
35. C22⋊Dic229C22semidirect
42. #16814643C22semidirect
43. C4×Dic229C22semidirect
46. C2×D4×C229C22semidirect
47. C22×C4×C229C22semidirect
48. C4⋊C4×C229C22semidirect
50. C22⋊C4×C229C22semidirect
122. C229#560semidirect
135. #121D229semidirect
151. C2×D4C2×D229semidirect
154. C22×C4C2×D229semidirect
160. C4⋊C4C2×D229semidirect
172. C22⋊C4C2×D229semidirect

Maximal subgroups

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

subgroupindex
#203009842
#203009832
#203009812
#203009732
#203009722
#203009712
#203009622
#203009612
#203009602
#203009592
#203009582
#203009562
#121×C2292
#560229