Finite Group Databasebeta

← C22×#10622856 #15620490

Subgroups — C22×#10622856 #15620490

Order 2816.

5294 conjugacy classes of subgroups; only the 827 normal subgroups are shown.

normal subgroups only

This group has 5294 conjugacy classes of subgroups — too many to draw. Only its 827 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
1C22×#66988C213yesmaximal, complemented
2C2×#10622856C2124yesmaximal, direct product factor
3C22×#66898C213yesmaximal, complemented
4C22×C11×H(5,𝔽2)C211yesmaximal, characteristic
5C2×C11×H(5,𝔽2)—218no—
6C2×C11×H(5,𝔽2)C22112yes—
7C22×C11×#41—23no—
8C23×D4×C11—23no—
9C23×D4×C11C2213yes—
10C24×Dic11C2211yescharacteristic, complemented
11C22×C22⋊Dic11—23no—
12C22×C22⋊Dic11C2213yescomplemented
13C2×#66988—218no—
14C2×#66988C22136yescomplemented
15C22×Dic11⋊C4—23no—
16#10622856C22164yesdirect product factor
17C2×#66898C22136yescomplemented
18C25×C11—21no—
19C25×C11D413yes—
20C25×C11C2311yescharacteristic
21C11×H(5,𝔽2)—248no—
22C11×H(5,𝔽2)—272no—
23C11×H(5,𝔽2)C23116yes—
24C2×C11×#41—236no—
25C2×C11×#41—236no—
26C22×Q8×C11—21no—
27C22×D4×C11—23no—
28C22×D4×C11—236no—
29C22×D4×C11—26no—
30C22×D4×C11—236no—
31C22×D4×C11—218no—
32C22×D4×C11D4118yes—
33C22×D4×C11C23118yes—
34C23×C4×C11—23no—
35C23×C4×C11D413yes—
36C23×Dic11—23no—
37C23×Dic11—23no—
38C23×Dic11—29no—
39C23×Dic11C2316yescomplemented
40C2×C22⋊Dic11—29no—
41C2×C22⋊Dic11—236no—
42C2×C22⋊Dic11C23118yescomplemented
43#66988—272no—
44#66988C23148yescomplemented
45C2×Dic11⋊C4—236no—
46#66898C23148yescomplemented
47C22×#458—111nomaximal, Sylow
48C24×C11—212no—
49C24×C11—23no—
50C24×C11—29no—
51C24×C11—44no—
52C24×C11—412no—
53C24×C11—84no—
54C24×C11C2×D413yes—
55C24×C11C2×D4118yes—
56C24×C11C2×D419yes—
57C24×C11C2413yes—
58C11×#41—2144no—
59C11×#41—248no—
60C2×Q8×C11—212no—
61C2×D4×C11—236no—
62C2×D4×C11—2144no—
63C2×D4×C11—236no—
64C2×D4×C11—236no—
65C2×D4×C11—236no—
66C2×D4×C11—248no—
67C2×D4×C11—236no—
68C2×D4×C11C2×D4136yes—
69C2×D4×C11C24112yes—
70C22×C4×C11—236no—
71C22×C4×C11—23no—
72C22×C4×C11—29no—
73C22×C4×C11C2×D4118yes—
74C22×Dic11—218no—
75C22×Dic11—218no—
76C22×Dic11—218no—
77C22×Dic11—412no—
78C22×Dic11—49no—
79C22×Dic11—46no—
80C22×Dic11—41no—
81C22×Dic11C2414yescomplemented
82C22⋊Dic11—218no—
83C22⋊Dic11—248no—
84C22⋊Dic11C24112yescomplemented
85Dic11⋊C4—248no—
86C22×H(5,𝔽2)D1111yescharacteristic
87C22×#123—113no—
88C2×#458—1124no—
89C22×#99—113no—
90C23×C11—212no—
91C23×C11—216no—
92C23×C11—218no—
93C23×C11—218no—
94C23×C11—418no—
95C23×C11—46no—
96C23×C11—436no—
97C23×C11—412no—
98C23×C11—818no—
99C23×C11—824no—
100C23×C11#12019yes—
101C23×C11#12016yes—
102C23×C11#12011yescharacteristic
103C23×C11C22×D419yes—
104C23×C11C22×D4112yes—
105C23×C11C22×D419yes—
106C23×C11C2511yescharacteristic
107Q8×C11—216no—
108D4×C11—248no—
109D4×C11—272no—
110D4×C11—224no—
111C2×C4×C11—248no—
112C2×C4×C11—218no—
113C2×C4×C11—218no—
114C2×C4×C11C22×D4112yes—
115C2×Dic11—212no—
116C2×Dic11—212no—
117C2×Dic11—436no—
118C2×Dic11—418no—
119C2×Dic11—46no—
120C2×H(5,𝔽2)—218no—
121C2×H(5,𝔽2)C2×D11112yes—
122C22×#41—23no—
123C23×D4—23no—
124C23×D4C2×D1113yes—
125C24×C4—111no—
126C2×#123—1136no—
127C2×#123—2218no—
128C22×C4⋊C4—223no—
129C22×C22⋊C4—113no—
130C22×C22⋊C4—223no—
131#458—1164no—
132C2×#99—1136no—
133C22×C11—218no—
134C22×C11—218no—
135C22×C11—212no—
136C22×C11—46no—
137C22×C11—49no—
138C22×C11—41no—
139C22×C11—412no—
140C22×C11—436no—
141C22×C11—818no—
142C22×C11—816no—
143C22×C11#45814yescomplemented
144C22×C11C2×#12019yes—
145C22×C11C2×#12013yes—
146C22×C11C23×D413yes—
147C4×C11—212no—
148Dic11—412no—
149Dic11—44no—
150C25—21no—
151C25C22⋊D1113yes—
152C25C22×D1111yescharacteristic
153H(5,𝔽2)—248no—
154H(5,𝔽2)—272no—
155H(5,𝔽2)C22×D11116yes—
156C2×#41—236no—
157C2×#41—236no—
158C22×Q8—21no—
159C22×D4—23no—
160C22×D4—236no—
161C22×D4—26no—
162C22×D4—236no—
163C22×D4—218no—
164C22×D4C22⋊D11118yes—
165C22×D4C22×D11118yes—
166C23×C4—23no—
167C23×C4—116no—
168C23×C4—223no—
169C23×C4—223no—
170C23×C4—229no—
171C23×C4C22⋊D1113yes—
172#123—1148no—
173#123—2272no—
174C2×C4⋊C4—2236no—
175C2×C22⋊C4—1118no—
176C2×C22⋊C4—229no—
177C2×C22⋊C4—2236no—
178#99—1148no—
179C2×C11—23no—
180C2×C11—29no—
181C2×C11—412no—
182C2×C11C2×#45816yescomplemented
183C2×C11C22×#12011yescharacteristic
184C24—212no—
185C24—23no—
186C24—29no—
187C24—44no—
188C24—412no—
189C24—84no—
190C24C2×C22⋊D1113yes—
191C24C2×C22⋊D11118yes—
192C24C2×C22⋊D1119yes—
193C24C23×D1113yes—
194#41—2144no—
195#41—248no—
196C2×Q8—212no—
197C2×D4—236no—
198C2×D4—2144no—
199C2×D4—236no—
200C2×D4—236no—
201C2×D4—236no—
202C2×D4—248no—
203C2×D4—236no—
204C2×D4C2×C22⋊D11136yes—
205C2×D4C23×D11112yes—
206C22×C4—236no—
207C22×C4—29no—
208C22×C4—23no—
209C22×C4—114no—
210C22×C4—2218no—
211C22×C4—2218no—
212C22×C4—2218no—
213C22×C4—4412no—
214C22×C4—449no—
215C22×C4—446no—
216C22×C4—441no—
217C22×C4C2×C22⋊D11118yes—
218C4⋊C4—2248no—
219C22⋊C4—1112no—
220C22⋊C4—2218no—
221C22⋊C4—2248no—
222C11C22×#45811yescharacteristic, complemented, Sylow
223C23—212no—
224C23—216no—
225C23—218no—
226C23—218no—
227C23—418no—
228C23—46no—
229C23—436no—
230C23—412no—
231C23—818no—
232C23—824no—
233C23#6700619yes—
234C23#6700616yes—
235C23#6700611yescharacteristic
236C23C22×C22⋊D1119yes—
237C23C22×C22⋊D11112yes—
238C23C22×C22⋊D1119yes—
239C23C24×D1111yescharacteristic
240Q8—216no—
241D4—248no—
242D4—272no—
243D4—224no—
244C2×C4—248no—
245C2×C4—218no—
246C2×C4—218no—
247C2×C4—2212no—
248C2×C4—2212no—
249C2×C4—4436no—
250C2×C4—4418no—
251C2×C4—446no—
252C2×C4C22×C22⋊D11112yes—
253C22—218no—
254C22—218no—
255C22—212no—
256C22—46no—
257C22—49no—
258C22—41no—
259C22—412no—
260C22—436no—
261C22—818no—
262C22—816no—
263C22#1062285614yesdirect product factor
264C22C2×#6700619yes—
265C22C2×#6700613yes—
266C22C23×C22⋊D1113yes—
267C4—212no—
268C4—4412no—
269C4—444no—
270C2—23no—
271C2—29no—
272C2—412no—
273C2C2×#1062285616yesdirect product factor
274C2C22×#6700611yescharacteristic
subgroupG/H
centerC23#67006
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
Fitting, prime radicalC22×C11×H(5,𝔽2)C2
socleC23×C11#120
hypercenterC22×H(5,𝔽2)D11
hypocenterC11C22×#458
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C25 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×#66988C2semidirect
2. C2×#10622856C2direct (N × H)
3. C22×#66898C2semidirect
10. C24×Dic11C22semidirect
12. C22×C22⋊Dic11C22semidirect
14. C2×#66988C22semidirect
16. #10622856C22direct (N × H)
17. C2×#66898C22semidirect
39. C23×Dic11C23semidirect
42. C2×C22⋊Dic11C23semidirect
44. #66988C23semidirect
46. #66898C23semidirect
81. C22×Dic11C24semidirect
84. C22⋊Dic11C24semidirect
143. C22×C11#458semidirect
182. C2×C11C2×#458semidirect
222. C11C22×#458semidirect
263. C22#10622856direct (N × H)
273. C2C2×#10622856direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669882
C2×#106228562
C22×#668982
C22×C11×H(5,𝔽2)2
C22×#45811