Finite Group Databasebeta

← #13072273

Subgroups — #13072273

Order 1408.

852 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 173 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×Q8×D11C211yesmaximal, characteristic, complemented
2C2×#67000C211yesmaximal, characteristic, complemented
3C2×#66948C211yesmaximal, characteristic
4C2×#66928C211yesmaximal, characteristic, complemented
5C11×#391C211yesmaximal, characteristic, complemented
6#10622587C211yesmaximal, characteristic, complemented
7C4×(C2×D11)⋊C4C211yesmaximal, characteristic, complemented
8#10622286C214yesmaximal, complemented
9#10622278C214yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11C2×Q8×D11—24no—
12C2×Q8×D11—24no—
13C2×Q8×D11—24no—
14C22×D44C2211yescharacteristic
15C22×D44C2212yes—
16C22×C4×D11C2212yes—
17C22×C4×D11C2211yescharacteristic
18C22×Dic22C2212yes—
19C22×Dic22C2211yescharacteristic
20C2×C11×C4⋊C4C2211yescharacteristic
21C2×C42×C11C2211yescharacteristic
22#67000—24no—
23C2×(C2×D11)⋊C4C2214yes—
24C2×(C2×D11)⋊C4C2212yes—
25C2×(C2×D11)⋊C4C2212yes—
26C2×(C2×D11)⋊C4C2214yes—
27C2×C4×Dic11C2211yescharacteristic
28#66948—24no—
29#66928—24no—
30C11×#95C2214yes—
31C11⋊#95C2214yes—
32C11⋊#95C2212yes—
33C23×D11C2312yes—
34C23×D11C2311yescharacteristic
35C2×Q8×C11—22no—
36C2×Q8×C11—22no—
37C2×Q8×C11D414yescomplemented
38C22×C4×C11C2312yes—
39C22×C4×C11C2312yes—
40C22×C4×C11C2312yes—
41C22×C4×C11C2311yescharacteristic
42C22×Dic11C2312yes—
43C22×Dic11C2311yescharacteristic
44C22×Dic11C2312yes—
45Q8×D11—232no—
46C2×D44—24no—
47C2×D44—28no—
48C2×D44—22no—
49C2×D44—22no—
50C2×D44—22no—
51C2×C4×D11—24no—
52C2×C4×D11—24no—
53C2×C4×D11—24no—
54C2×C4×D11—22no—
55C2×C4×D11—22no—
56C2×C4×D11D414yes—
57C2×Dic22—24no—
58C2×Dic22—24no—
59C2×Dic22—24no—
60C2×Dic22—22no—
61C2×Dic22—22no—
62C2×Dic22D414yescomplemented
63C11×C4⋊C4—22no—
64C42×C11—22no—
65(C2×D11)⋊C4—28no—
66(C2×D11)⋊C4—24no—
67(C2×D11)⋊C4—24no—
68(C2×D11)⋊C4—28no—
69C4×Dic11—22no—
70#2436—111nomaximal, Sylow
71C23×C11C2411yescharacteristic
72C22×D11—21no—
73C22×D11—21no—
74C22×D11—22no—
75C22×D11—22no—
76C22×D11—22no—
77C22×D11—22no—
78C22×D11—24no—
79C22×D11—24no—
80C22×D11C2×D412yes—
81C22×D11#4114yes—
82Q8×C11—28no—
83C2×C4×C11—22no—
84C2×C4×C11—22no—
85C2×C4×C11—22no—
86C2×C4×C11—22no—
87C2×C4×C11—22no—
88C2×C4×C11—24no—
89C2×C4×C11—22no—
90C2×C4×C11C2×D414yes—
91C2×C4×C11C2×D412yes—
92C2×C4×C11C2×D412yes—
93C2×C4×C11C2×D412yes—
94C2×Dic11—21no—
95C2×Dic11—22no—
96C2×Dic11—22no—
97C2×Dic11—21no—
98C2×Dic11—22no—
99C2×Dic11—22no—
100C2×Dic11—22no—
101C2×Dic11C2×D412yes—
102C2×Dic11C2×D414yes—
103D44—48no—
104D44—44no—
105C4×D11—216no—
106C4×D11—28no—
107Dic22—216no—
108Dic22—28no—
109C23×Q8—111no—
110C2×#124—111no—
111C2×#124—111no—
112C2×#121—111no—
113#393—114no—
114#391—111no—
115#391D1111yescharacteristic, complemented
116#386—114no—
117C4×C22⋊C4—111no—
118C22×C11C22×D411yescharacteristic
119C22×C11C22×D411yescharacteristic
120C22×C11C22×D411yescharacteristic
121C22×C11C2×#4112yes—
122C22×C112−1+412yes—
123C2×D11—22no—
124C2×D11—22no—
125C2×D11—24no—
126C2×D11—24no—
127C2×D11—42no—
128C2×D11—42no—
129C2×D11—42no—
130C2×D11—44no—
131C2×D11—44no—
132C2×D11#12414yescomplemented
133C4×C11—24no—
134C4×C11—42no—
135C4×C11—42no—
136C4×C11#12014yes—
137Dic11—22no—
138Dic11—24no—
139Dic11—42no—
140C22×Q8—111no—
141C22×Q8—112no—
142C22×Q8—224no—
143C22×Q8—224no—
144C22×Q8—224no—
145C22×Q8C2×D1111yescharacteristic
146C22×D4—111no—
147C22×D4—112no—
148C23×C4—111no—
149C23×C4—112no—
150#124—224no—
151#124—224no—
152#121—224no—
153C2×C4⋊C4C2×D1111yescharacteristic
154C2×C22⋊C4—112no—
155C2×C22⋊C4—114no—
156C2×C22⋊C4—114no—
157C2×C22⋊C4—112no—
158C2×C42—111no—
159C2×C42C2×D1111yescharacteristic
160#95—112no—
161#95—114no—
162#95C2×D1114yes—
163C2×C11C2×#12011yescharacteristic
164C2×C11C2×#12411yescharacteristic
165C2×C11#53611yescharacteristic
166C2×C11#54812yes—
167C2×C11#54812yes—
168D11—24no—
169D11—82no—
170C24—111no—
171C24—112no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—114no—
175C2×Q8—224no—
176C2×Q8—224no—
177C2×Q8—2232no—
178C2×Q8—224no—
179C2×Q8—222no—
180C2×Q8—222no—
181C2×Q8D4414yescomplemented
182C2×D4—224no—
183C2×D4—228no—
184C2×D4—222no—
185C2×D4—222no—
186C2×D4—222no—
187C22×C4—111no—
188C22×C4—112no—
189C22×C4—114no—
190C22×C4—112no—
191C22×C4—224no—
192C22×C4—224no—
193C22×C4—224no—
194C22×C4—222no—
195C22×C4—222no—
196C22×C4C22×D1112yes—
197C22×C4C22×D1112yes—
198C22×C4C22×D1112yes—
199C22×C4C22×D1111yescharacteristic
200C4⋊C4—22no—
201C22⋊C4—224no—
202C22⋊C4—228no—
203C22⋊C4—224no—
204C22⋊C4—228no—
205C42—22no—
206C42—222no—
207C11#243611yescharacteristic, complemented, Sylow
208C23—112no—
209C23—114no—
210C23—221no—
211C23—221no—
212C23—222no—
213C23—222no—
214C23—222no—
215C23—222no—
216C23—224no—
217C23—224no—
218C23C23×D1111yescharacteristic
219Q8—28no—
220Q8—2216no—
221Q8—228no—
222D4—448no—
223D4—444no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—112no—
232C2×C4—114no—
233C2×C4—2216no—
234C2×C4—221no—
235C2×C4—222no—
236C2×C4—222no—
237C2×C4—221no—
238C2×C4—228no—
239C2×C4—222no—
240C2×C4—222no—
241C2×C4—222no—
242C2×C4C2×D4414yes—
243C2×C4C2×D4412yes—
244C2×C4D4×D1112yes—
245C2×C4D4×D1112yes—
246C22—114no—
247C22—222no—
248C22—222no—
249C22—224no—
250C22—224no—
251C22—442no—
252C22—442no—
253C22—444no—
254C22—444no—
255C22—442no—
256C22C22×D4411yescharacteristic
257C22C2×D4×D1111yescharacteristic
258C22C2×D4×D1111yescharacteristic
259C22#6704012yes—
260C22#41×D1112yes—
261C4—24no—
262C4—42no—
263C4—42no—
264C4—222no—
265C4—224no—
266C4—442no—
267C4#6693514yes—
268C2—224no—
269C2—882no—
270C2C2×#6693511yescharacteristic
271C2#1062310712yes—
272C2#1062316412yes—
273C2D11×#12411yescharacteristic
274C2#1062325511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
prime radicalC2×#66948C2
hypercenter#391D11
hypocenterC11#2436
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D11C2semidirect
2. C2×#67000C2semidirect
4. C2×#66928C2semidirect
5. C11×#391C2semidirect
6. #10622587C2semidirect
7. C4×(C2×D11)⋊C4C2semidirect
8. #10622286C2semidirect
9. #10622278C2semidirect
37. C2×Q8×C11D4semidirect
62. C2×Dic22D4semidirect
115. #391D11semidirect
132. C2×D11#124semidirect
181. C2×Q8D44semidirect
207. C11#2436semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D112
C2×#670002
C2×#669482
C2×#669282
C11×#3912
#106225872
C4×(C2×D11)⋊C42
#106222862
#106222782
#243611