Finite Group Databasebeta

← C22×#10623242 #15660643

Subgroups — C22×#10623242 #15660643

Order 2816.

9606 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623242C2148yesmaximal, direct product factor
2C22×#66927C211yesmaximal, characteristic, complemented
3C22×#66929C211yesmaximal, characteristic, complemented
4C22×#66947C213yesmaximal, complemented
5C22×#66948C213yesmaximal, complemented
6C22×#66937C213yesmaximal, complemented
7C22×#66935C213yesmaximal, complemented
8C22×C11×#126C211yesmaximal, characteristic, complemented
9C25×D11C2211yescharacteristic
10C23×C22⋊D11C2213yes—
11C23×D44C2213yes—
12C23×D44C2213yes—
13C23×D44C2213yescomplemented
14C23×C4×D11C2213yescomplemented
15C2×C11×#126C22124yescomplemented
16C22×C11×C4⋊C4C2213yescomplemented
17C22×C11×C22⋊C4C2213yes—
18C22×C42×C11C2211yescharacteristic, complemented
19C22×(C2×D11)⋊C4C2213yes—
20C22×(C2×D11)⋊C4C2213yes—
21C22×(C2×D11)⋊C4C2213yescomplemented
22C22×Dic11⋊C4C2213yescomplemented
23#10623242C221256yesdirect product factor
24C2×#66948C22172yescomplemented
25C2×#66947C22172yescomplemented
26C2×#66937C22172yescomplemented
27C2×#66935C22172yescomplemented
28C2×#66929C22124yescomplemented
29C2×#66927C22124yescomplemented
30C25×C11C2311yescharacteristic
31C24×D11—412no—
32C24×D11C2311yescharacteristic
33C24×D11C2311yescharacteristic
34C24×D11C2313yes—
35C24×D11C23112yes—
36C23×C4×C11C2313yes—
37C23×C4×C11C2313yes—
38C22×C22⋊D11—412no—
39C22×C22⋊D11C23136yes—
40C23×Dic11C2313yes—
41C22×D44—412no—
42C22×D44—412no—
43C22×D44—412no—
44C22×D44C23136yes—
45C22×D44C23136yes—
46C22×D44C23136yescomplemented
47C22×C4×D11—412no—
48C22×C4×D11C23136yescomplemented
49C11×#126C23164yescomplemented
50C2×C11×C4⋊C4C23136yescomplemented
51C2×C11×C22⋊C4C23136yes—
52C2×C42×C11C23112yescomplemented
53C2×(C2×D11)⋊C4C23136yes—
54C2×(C2×D11)⋊C4C23136yes—
55C2×(C2×D11)⋊C4C23136yescomplemented
56C2×Dic11⋊C4C23136yescomplemented
57#66948C231192yescomplemented
58#66947C231192yescomplemented
59#66937C231192yescomplemented
60#66935C231192yescomplemented
61#66929C23164yescomplemented
62#66927C23164yescomplemented
63C22×#560—111nomaximal, Sylow
64C24×C11—212no—
65C24×C11C2416yes—
66C24×C11C2411yescharacteristic
67C23×D11—212no—
68C23×D11—212no—
69C23×D11—212no—
70C23×D11—212no—
71C23×D11—212no—
72C23×D11—472no—
73C23×D11—464no—
74C23×D11C2416yes—
75C23×D11C2416yes—
76C23×D11C24118yes—
77C23×D11C24116yes—
78C22×C4×C11—212no—
79C22×C4×C11—212no—
80C22×C4×C11C24118yes—
81C22×C4×C11C24118yes—
82C2×C22⋊D11—472no—
83C2×C22⋊D11C24148yes—
84C22×Dic11—212no—
85C22×Dic11C24118yes—
86C2×D44—472no—
87C2×D44—472no—
88C2×D44—472no—
89C2×D44C24148yes—
90C2×D44C24148yes—
91C2×D44C24148yescomplemented
92C2×C4×D11—472no—
93C2×C4×D11C24148yescomplemented
94C11×C4⋊C4C24148yescomplemented
95C11×C22⋊C4C24148yes—
96C42×C11C24116yescomplemented
97(C2×D11)⋊C4C24148yes—
98(C2×D11)⋊C4C24148yes—
99(C2×D11)⋊C4C24148yescomplemented
100Dic11⋊C4C24148yescomplemented
101C2×#560—1148no—
102C22×#127—111no—
103C22×#126—111no—
104C22×#126D1111yescharacteristic, complemented
105C22×#123—113no—
106C22×#121—113no—
107C22×#121—113no—
108C22×#120—113no—
109C23×C11—236no—
110C23×C11—416no—
111C23×C11H(5,𝔽2)112yes—
112C23×C11C2514yes—
113C23×C11C2513yes—
114C22×D11—236no—
115C22×D11—236no—
116C22×D11—236no—
117C22×D11—236no—
118C22×D11—236no—
119C22×D11—448no—
120C22×D11—4192no—
121C22×D11—448no—
122C22×D11—416no—
123C22×D11—416no—
124C22×D11C2514yes—
125C22×D11C2514yes—
126C22×D11C25112yes—
127C2×C4×C11—236no—
128C2×C4×C11—236no—
129C2×C4×C11C25112yes—
130C2×C4×C11C25112yes—
131C22⋊D11—448no—
132C2×Dic11—236no—
133C2×Dic11C25112yes—
134D44—448no—
135D44—448no—
136D44—448no—
137C4×D11—448no—
138C26—111no—
139C23×D4—113no—
140C23×D4—113no—
141C23×D4—113no—
142C23×D4—113no—
143C24×C4—113no—
144#560—11256no—
145C2×#127—1124no—
146C2×#126—1124no—
147C2×#126C2×D11124yescomplemented
148C2×#123—1172no—
149C2×#121—1172no—
150C2×#121—1172no—
151C2×#120—1172no—
152C22×C4⋊C4—113no—
153C22×C4⋊C4C2×D1113yescomplemented
154C22×C22⋊C4—113no—
155C22×C22⋊C4—113no—
156C22×C22⋊C4—113no—
157C22×C22⋊C4C2×D1113yes—
158C22×C42C2×D1111yescharacteristic, complemented
159C22×C11—212no—
160C22×C11—424no—
161C22×C11#560116yescomplemented
162C22×C11C2×H(5,𝔽2)118yes—
163C22×C11C2611yescharacteristic
164C2×D11—212no—
165C2×D11—212no—
166C2×D11—212no—
167C2×D11—212no—
168C2×D11—212no—
169C2×D11—464no—
170C2×D11—472no—
171C2×D11—424no—
172C2×D11—424no—
173C4×C11—212no—
174C4×C11—212no—
175Dic11—212no—
176C25—111no—
177C25—111no—
178C25—113no—
179C25—1112no—
180C25—4412no—
181C25C22×D1111yescharacteristic
182C22×D4—1136no—
183C22×D4—1136no—
184C22×D4—1136no—
185C22×D4—1136no—
186C22×D4—4412no—
187C22×D4—4412no—
188C22×D4—4412no—
189C22×D4—4412no—
190C23×C4—113no—
191C23×C4—1136no—
192C23×C4—4412no—
193C23×C4C22×D1113yes—
194C23×C4C22×D1113yes—
195#127—1164no—
196#126—1164no—
197#126C22×D11164yescomplemented
198#123—11192no—
199#121—11192no—
200#121—11192no—
201#120—11192no—
202C2×C4⋊C4—1136no—
203C2×C4⋊C4C22×D11136yescomplemented
204C2×C22⋊C4—1136no—
205C2×C22⋊C4—1136no—
206C2×C22⋊C4—1136no—
207C2×C22⋊C4C22×D11136yes—
208C2×C42C22×D11112yescomplemented
209C2×C11—44no—
210C2×C11C2×#560112yescomplemented
211C2×C11C22×H(5,𝔽2)13yes—
212D11—412no—
213D11—44no—
214D11—44no—
215C24—212no—
216C24—116no—
217C24—116no—
218C24—1118no—
219C24—1116no—
220C24—2212no—
221C24—2212no—
222C24—2212no—
223C24—2212no—
224C24—2212no—
225C24—4472no—
226C24—4464no—
227C24C23×D1116yes—
228C24C23×D1111yescharacteristic
229C2×D4—1148no—
230C2×D4—1148no—
231C2×D4—1148no—
232C2×D4—1148no—
233C2×D4—4472no—
234C2×D4—4472no—
235C2×D4—4472no—
236C2×D4—4472no—
237C22×C4—212no—
238C22×C4—212no—
239C22×C4—1118no—
240C22×C4—1148no—
241C22×C4—2212no—
242C22×C4—4472no—
243C22×C4C23×D11118yes—
244C22×C4C23×D11118yes—
245C4⋊C4—1148no—
246C4⋊C4C23×D11148yescomplemented
247C22⋊C4—1148no—
248C22⋊C4—1148no—
249C22⋊C4—1148no—
250C22⋊C4C23×D11148yes—
251C42C23×D11116yescomplemented
252C11C22×#56011yescharacteristic, complemented, Sylow
253C23—236no—
254C23—416no—
255C23—114no—
256C23—114no—
257C23—1112no—
258C23—2236no—
259C23—2236no—
260C23—2236no—
261C23—2236no—
262C23—2236no—
263C23—4448no—
264C23—44192no—
265C23—4448no—
266C23—4416no—
267C23—4416no—
268C23#67042112yes—
269C23C24×D1114yes—
270C23C24×D1113yes—
271D4—4448no—
272D4—4448no—
273D4—4448no—
274D4—4448no—
275C2×C4—236no—
276C2×C4—236no—
277C2×C4—1112no—
278C2×C4—2236no—
279C2×C4—4448no—
280C2×C4C24×D11112yes—
281C2×C4C24×D11112yes—
282C22—212no—
283C22—424no—
284C22—2212no—
285C22—2212no—
286C22—2212no—
287C22—2212no—
288C22—2212no—
289C22—4464no—
290C22—4424no—
291C22—4424no—
292C22—4472no—
293C22#10623242116yesdirect product factor
294C22C2×#67042118yes—
295C22C25×D1111yescharacteristic
296C4—212no—
297C4—212no—
298C4—2212no—
299C2—44no—
300C2—444no—
301C2—444no—
302C2—4412no—
303C2C2×#10623242112yesdirect product factor
304C2C22×#6704213yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC22×C11C26
FrattiniC22C25×D11
FittingC22×C11×#126C2
socleC24×C11C24
hypercenterC22×#126D11
hypocenterC11C22×#560
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C2×#10623242C2direct (N × H)
2. C22×#66927C2semidirect
3. C22×#66929C2semidirect
4. C22×#66947C2semidirect
5. C22×#66948C2semidirect
6. C22×#66937C2semidirect
7. C22×#66935C2semidirect
8. C22×C11×#126C2semidirect
13. C23×D44C22semidirect
14. C23×C4×D11C22semidirect
15. C2×C11×#126C22semidirect
16. C22×C11×C4⋊C4C22semidirect
18. C22×C42×C11C22semidirect
21. C22×(C2×D11)⋊C4C22semidirect
22. C22×Dic11⋊C4C22semidirect
23. #10623242C22direct (N × H)
24. C2×#66948C22semidirect
25. C2×#66947C22semidirect
26. C2×#66937C22semidirect
27. C2×#66935C22semidirect
28. C2×#66929C22semidirect
29. C2×#66927C22semidirect
46. C22×D44C23semidirect
48. C22×C4×D11C23semidirect
49. C11×#126C23semidirect
50. C2×C11×C4⋊C4C23semidirect
52. C2×C42×C11C23semidirect
55. C2×(C2×D11)⋊C4C23semidirect
56. C2×Dic11⋊C4C23semidirect
57. #66948C23semidirect
58. #66947C23semidirect
59. #66937C23semidirect
60. #66935C23semidirect
61. #66929C23semidirect
62. #66927C23semidirect
91. C2×D44C24semidirect
93. C2×C4×D11C24semidirect
94. C11×C4⋊C4C24semidirect
96. C42×C11C24semidirect
99. (C2×D11)⋊C4C24semidirect
100. Dic11⋊C4C24semidirect
104. C22×#126D11semidirect
147. C2×#126C2×D11semidirect
153. C22×C4⋊C4C2×D11semidirect
158. C22×C42C2×D11semidirect
161. C22×C11#560semidirect
197. #126C22×D11semidirect
203. C2×C4⋊C4C22×D11semidirect
208. C2×C42C22×D11semidirect
210. C2×C11C2×#560semidirect
246. C4⋊C4C23×D11semidirect
251. C42C23×D11semidirect
252. C11C22×#560semidirect
293. C22#10623242direct (N × H)
303. C2C2×#10623242direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106232422
C22×#669272
C22×#669292
C22×#669472
C22×#669482
C22×#669372
C22×#669352
C22×C11×#1262
C22×#56011