Finite Group Databasebeta

← C22×#10623261 #15660741

Subgroups — C22×#10623261 #15660741

Order 2816.

8454 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623261C2148yesmaximal, direct product factor
2C22×#66943C212yesmaximal, complemented
3C22×#66950C212yesmaximal, complemented
4C22×#66999C214yesmaximal, complemented
5C22×C4×Dic22C211yesmaximal, characteristic, complemented
6C22×C4×D44C211yesmaximal, characteristic
7C22×#66946C212yesmaximal
8C22×C4⋊C4×D11C212yesmaximal, complemented
9C22×C11×C4⋊Q8C211yesmaximal, characteristic, complemented
10C23×Q8×C11C2212yes—
11C23×D44C2211yescharacteristic
12C23×C4×D11C2212yes—
13C23×C4×D11C2214yes—
14C23×Dic22C2211yescharacteristic
15C2×C11×C4⋊Q8C22124yescomplemented
16C22×C11×C4⋊C4C2212yes—
17C22×C11×C4⋊C4C2212yes—
18C22×C42×C11C2211yescharacteristic
19C2×#66999C22196yescomplemented
20C22×(C2×D11)⋊C4C2214yes—
21C22×(C2×D11)⋊C4C2212yes—
22C22×C4⋊Dic11C2211yescharacteristic
23C22×C4⋊Dic11C2214yes—
24C22×C4⋊Dic11C2211yescharacteristic
25C22×Dic11⋊C4C2214yes—
26C22×Dic11⋊C4C2212yes—
27C22×C4×Dic11C2212yes—
28#10623261C221256yesdirect product factor
29C2×#66950C22148yescomplemented
30C2×#66946C22148yes—
31C2×C4⋊C4×D11C22148yescomplemented
32C2×#66943C22148yescomplemented
33C2×C4×D44C22124yes—
34C2×C4×Dic22C22124yescomplemented
35C24×D11C2312yes—
36C22×Q8×C11—48no—
37C22×Q8×C11C23124yes—
38C23×C4×C11C2311yescharacteristic
39C23×C4×C11C2314yes—
40C23×C4×C11C2311yescharacteristic
41C23×C4×C11C2311yescharacteristic
42C23×Dic11C2312yes—
43C23×Dic11C2312yes—
44C23×Dic11C2312yes—
45C22×D44Q8116yescomplemented
46C22×D44C23112yes—
47C22×C4×D11—216no—
48C22×C4×D11—416no—
49C22×C4×D11C23124yes—
50C22×C4×D11C23148yes—
51C22×Dic22—28no—
52C22×Dic22C23112yes—
53C11×C4⋊Q8C23164yescomplemented
54C2×C11×C4⋊C4C23124yes—
55C2×C11×C4⋊C4C23124yes—
56C2×C42×C11C23112yes—
57#66999C231256yescomplemented
58C2×(C2×D11)⋊C4C23124yes—
59C2×(C2×D11)⋊C4C23148yes—
60C2×C4⋊Dic11C23112yes—
61C2×C4⋊Dic11C23112yes—
62C2×C4⋊Dic11C23148yes—
63C2×Dic11⋊C4C23124yes—
64C2×Dic11⋊C4C23148yes—
65C2×C4×Dic11C23124yes—
66#66950C231128yescomplemented
67#66946C231128yes—
68C4⋊C4×D11C231128yescomplemented
69#66943C231128yescomplemented
70C4×D44C23164yes—
71C4×Dic22C23164yescomplemented
72C22×#554—111nomaximal, Sylow
73C24×C11C2411yescharacteristic
74C23×D11—28no—
75C23×D11—28no—
76C23×D11C2×Q8116yes—
77C23×D11C24112yes—
78C2×Q8×C11—448no—
79C2×Q8×C11C24132yes—
80C22×C4×C11—216no—
81C22×C4×C11—24no—
82C22×C4×C11C2×Q818yes—
83C22×C4×C11#4118yes—
84C22×C4×C11C2416yes—
85C22×C4×C11C24124yes—
86C22×C4×C11C2416yes—
87C22×C4×C11C2416yes—
88C22×Dic11—28no—
89C22×Dic11—28no—
90C22×Dic11—28no—
91C22×Dic11C24112yes—
92C22×Dic11C24112yes—
93C22×Dic11C24112yes—
94C2×D44C2×Q8196yescomplemented
95C2×D44C24116yes—
96C2×C4×D11—296no—
97C2×C4×D11—496no—
98C2×C4×D11C24132yes—
99C2×C4×D11C24164yes—
100C2×Dic22—248no—
101C2×Dic22C24116yes—
102C11×C4⋊C4C24132yes—
103C11×C4⋊C4C24132yes—
104C42×C11C24116yes—
105(C2×D11)⋊C4C24132yes—
106(C2×D11)⋊C4C24164yes—
107C4⋊Dic11C24116yes—
108C4⋊Dic11C24116yes—
109C4⋊Dic11C24164yes—
110Dic11⋊C4C24132yes—
111Dic11⋊C4C24164yes—
112C4×Dic11C24132yes—
113C2×#554—1148no—
114C22×C4⋊Q8D1111yescharacteristic, complemented
115C22×#125—112no—
116C22×C22⋊Q8—114no—
117C22×C22⋊Q8—112no—
118C22×C4×Q8—111no—
119C22×C4×D4—111no—
120C22×C4×D4—112no—
121C23×C4⋊C4—112no—
122C23×C11C22×Q814yes—
123C23×C11C2×#4114yes—
124C23×C11H(5,𝔽2)14yes—
125C23×C11C2513yes—
126C22×D11—264no—
127C22×D11—224no—
128C22×D11—224no—
129C22×D11C22×Q8148yes—
130C22×D11C2518yes—
131Q8×C11—432no—
132C2×C4×C11—248no—
133C2×C4×C11—212no—
134C2×C4×C11C22×Q8124yes—
135C2×C4×C11C2×#41124yes—
136C2×C4×C11C2514yes—
137C2×C4×C11C25116yes—
138C2×C4×C11C2514yes—
139C2×C4×C11C2514yes—
140C2×Dic11—224no—
141C2×Dic11—224no—
142C2×Dic11—224no—
143C2×Dic11C2518yes—
144C2×Dic11C2518yes—
145C2×Dic11C2518yes—
146D44C22×Q8164yescomplemented
147C4×D11—264no—
148C4×D11—464no—
149Dic22—232no—
150C23×Q8—111no—
151C23×Q8C2×D1112yes—
152C23×D4—111no—
153C24×C4—112no—
154C24×C4—114no—
155#554—11256no—
156C2×C4⋊Q8C2×D11124yescomplemented
157C2×#125—1148no—
158C2×C22⋊Q8—1196no—
159C2×C22⋊Q8—1148no—
160C2×C4×Q8—1124no—
161C2×C4×D4—1148no—
162C2×C4×D4—1124no—
163C22×C4⋊C4—114no—
164C22×C4⋊C4—112no—
165C22×C4⋊C4—111no—
166C22×C4⋊C4—111no—
167C22×C4⋊C4—114no—
168C22×C4⋊C4—1148no—
169C22×C4⋊C4C2×D1112yes—
170C22×C4⋊C4C2×D1112yes—
171C22×C22⋊C4—114no—
172C22×C22⋊C4—112no—
173C22×C42—112no—
174C22×C42C2×D1111yescharacteristic
175C22×C11#554116yescomplemented
176C22×C11C23×Q816yes—
177C22×C11C22×#4116yes—
178C22×C11C2×H(5,𝔽2)16yes—
179C22×C11C2611yescharacteristic
180C2×D11—28no—
181C2×D11—28no—
182C2×D11—296no—
183C2×D11C23×Q8116yes—
184C4×C11—24no—
185C4×C11—216no—
186C4×C11C23×Q818yes—
187C4×C11C22×#4118yes—
188Dic11—28no—
189Dic11—28no—
190Dic11—28no—
191C25—112no—
192C22×Q8—48no—
193C22×Q8—1112no—
194C22×Q8—228no—
195C22×Q8C22×D11124yes—
196C22×D4—1116no—
197C22×D4—1112no—
198C23×C4—112no—
199C23×C4—112no—
200C23×C4—112no—
201C23×C4—1124no—
202C23×C4—1148no—
203C23×C4—2216no—
204C23×C4—4416no—
205C23×C4C22×D1111yescharacteristic
206C23×C4C22×D1114yes—
207C23×C4C22×D1111yescharacteristic
208C23×C4C22×D1111yescharacteristic
209C4⋊Q8C22×D11164yescomplemented
210#125—11128no—
211C22⋊Q8—11256no—
212C22⋊Q8—11128no—
213C4×Q8—1164no—
214C4×D4—11128no—
215C4×D4—1164no—
216C2×C4⋊C4—1112no—
217C2×C4⋊C4—1148no—
218C2×C4⋊C4—1112no—
219C2×C4⋊C4—1124no—
220C2×C4⋊C4—1148no—
221C2×C4⋊C4—11128no—
222C2×C4⋊C4C22×D11124yes—
223C2×C4⋊C4C22×D11124yes—
224C2×C22⋊C4—1148no—
225C2×C22⋊C4—1124no—
226C2×C42—1124no—
227C2×C42C22×D11112yes—
228C2×C11C2×#554112yescomplemented
229C2×C11C24×Q811yescharacteristic
230C2×C11C23×#4111yescharacteristic
231C2×C11C22×H(5,𝔽2)11yescharacteristic
232D11—216no—
233C24—1116no—
234C24—1112no—
235C24—228no—
236C24—228no—
237C24C23×D1111yescharacteristic
238C2×Q8—448no—
239C2×Q8—1116no—
240C2×Q8—2248no—
241C2×Q8C23×D11132yes—
242C2×D4—1196no—
243C2×D4—1116no—
244C22×C4—216no—
245C22×C4—24no—
246C22×C4—1112no—
247C22×C4—1112no—
248C22×C4—1112no—
249C22×C4—1132no—
250C22×C4—1164no—
251C22×C4—228no—
252C22×C4—228no—
253C22×C4—228no—
254C22×C4—2296no—
255C22×C4—4496no—
256C22×C4Q8×D1118yes—
257C22×C4#434318yes—
258C22×C4C23×D1116yes—
259C22×C4C23×D11124yes—
260C22×C4C23×D1116yes—
261C22×C4C23×D1116yes—
262C4⋊C4—1116no—
263C4⋊C4—1164no—
264C4⋊C4—1116no—
265C4⋊C4—1132no—
266C4⋊C4—1164no—
267C4⋊C4C23×D11132yes—
268C4⋊C4C23×D11132yes—
269C22⋊C4—1164no—
270C22⋊C4—1132no—
271C42—1132no—
272C42C23×D11116yes—
273C11C22×#55411yescharacteristic, complemented, Sylow
274C23—1148no—
275C23—118no—
276C23—2264no—
277C23—2224no—
278C23—2224no—
279C23#6703714yes—
280C23C2×Q8×D1114yes—
281C23C2×#434314yes—
282C23C24×D1113yes—
283Q8—432no—
284Q8—2232no—
285D4—1164no—
286C2×C4—248no—
287C2×C4—212no—
288C2×C4—118no—
289C2×C4—118no—
290C2×C4—118no—
291C2×C4—2224no—
292C2×C4—2224no—
293C2×C4—2224no—
294C2×C4—2264no—
295C2×C4—4464no—
296C2×C4C2×Q8×D11124yes—
297C2×C4C2×#4343124yes—
298C2×C4C24×D1114yes—
299C2×C4C24×D11116yes—
300C2×C4C24×D1114yes—
301C2×C4C24×D1114yes—
302C22—1116no—
303C22—228no—
304C22—228no—
305C22—2296no—
306C22#10623261116yesdirect product factor
307C22C2×#6703716yes—
308C22C22×Q8×D1116yes—
309C22C22×#434316yes—
310C22C25×D1111yescharacteristic
311C4—216no—
312C4—24no—
313C4—228no—
314C4—228no—
315C4—228no—
316C4C22×Q8×D1118yes—
317C4C22×#434318yes—
318C2—2216no—
319C2C2×#10623261112yesdirect product factor
320C2C23×Q8×D1111yescharacteristic
321C2C23×#434311yescharacteristic
322C2C22×#6703711yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC22×C11C26
FrattiniC22C25×D11
FittingC22×C11×C4⋊Q8C2
socleC24×C11C24
prime radicalC23×D44C22
hypercenterC22×C4⋊Q8D11
hypocenterC11C22×#554
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10623261C2direct (N × H)
2. C22×#66943C2semidirect
3. C22×#66950C2semidirect
4. C22×#66999C2semidirect
5. C22×C4×Dic22C2semidirect
8. C22×C4⋊C4×D11C2semidirect
9. C22×C11×C4⋊Q8C2semidirect
15. C2×C11×C4⋊Q8C22semidirect
19. C2×#66999C22semidirect
28. #10623261C22direct (N × H)
29. C2×#66950C22semidirect
31. C2×C4⋊C4×D11C22semidirect
32. C2×#66943C22semidirect
34. C2×C4×Dic22C22semidirect
45. C22×D44Q8semidirect
53. C11×C4⋊Q8C23semidirect
57. #66999C23semidirect
66. #66950C23semidirect
68. C4⋊C4×D11C23semidirect
69. #66943C23semidirect
71. C4×Dic22C23semidirect
94. C2×D44C2×Q8semidirect
114. C22×C4⋊Q8D11semidirect
146. D44C22×Q8semidirect
156. C2×C4⋊Q8C2×D11semidirect
175. C22×C11#554semidirect
209. C4⋊Q8C22×D11semidirect
228. C2×C11C2×#554semidirect
273. C11C22×#554semidirect
306. C22#10623261direct (N × H)
319. C2C2×#10623261direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106232612
C22×#669432
C22×#669502
C22×#669992
C22×C4×Dic222
C22×C4×D442
C22×#669462
C22×C4⋊C4×D112
C22×C11×C4⋊Q82
C22×#55411