Finite Group Databasebeta

← #13072259

Subgroups — #13072259

Order 1408.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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
2C2×#66997C211yesmaximal, characteristic, complemented
3C2×#66950C211yesmaximal, characteristic
4C2×#66922C211yesmaximal, characteristic, complemented
5C11×#391C211yesmaximal, characteristic, complemented
6#10622587C211yesmaximal, characteristic, complemented
7C4×(C2×D11)⋊C4C211yesmaximal, characteristic
8#10622281C214yesmaximal
9#10622269C214yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11C2×Q8×D11—24no—
12C2×Q8×D11—24no—
13C2×Q8×D11—24no—
14C22×C4×D11C2212yes—
15C22×C4×D11C2211yescharacteristic
16C22×Dic22C2211yescharacteristic
17C22×Dic22C2212yes—
18C22×Dic22C2211yescharacteristic
19C2×C11×C4⋊C4C2211yescharacteristic
20C2×C42×C11C2211yescharacteristic
21#66997—24no—
22C2×(C2×D11)⋊C4C2212yes—
23C2×(C2×D11)⋊C4C2212yes—
24C2×C4⋊Dic11C2214yes—
25C2×C4⋊Dic11C2212yes—
26C2×Dic11⋊C4C2214yes—
27C2×C4×Dic11C2211yescharacteristic
28#66950—24no—
29#66922—24no—
30C11×#95C2214yes—
31C11⋊#95C2212yes—
32C11⋊#95C2214yes—
33C23×D11C2311yescharacteristic
34C2×Q8×C11—22no—
35C2×Q8×C11—22no—
36C2×Q8×C11D414yes—
37C22×C4×C11C2312yes—
38C22×C4×C11C2312yes—
39C22×C4×C11C2312yes—
40C22×C4×C11C2311yescharacteristic
41C22×Dic11C2312yes—
42C22×Dic11C2311yescharacteristic
43C22×Dic11C2312yes—
44C22×Dic11C2312yes—
45Q8×D11—232no—
46C2×C4×D11—24no—
47C2×C4×D11—24no—
48C2×C4×D11—24no—
49C2×C4×D11—22no—
50C2×C4×D11—22no—
51C2×C4×D11D414yes—
52C2×Dic22—24no—
53C2×Dic22—24no—
54C2×Dic22—24no—
55C2×Dic22—22no—
56C2×Dic22—22no—
57C2×Dic22—22no—
58C2×Dic22—22no—
59C2×Dic22—22no—
60C2×Dic22D414yes—
61C11×C4⋊C4—22no—
62C42×C11—22no—
63(C2×D11)⋊C4—24no—
64(C2×D11)⋊C4Q818yescomplemented
65C4⋊Dic11—28no—
66C4⋊Dic11—24no—
67Dic11⋊C4—28no—
68C4×Dic11—22no—
69#2434—111nomaximal, Sylow
70C23×C11C2411yescharacteristic
71C22×D11—21no—
72C22×D11—21no—
73C22×D11—22no—
74C22×D11C2×D412yes—
75C22×D11C2×Q814yes—
76Q8×C11—28no—
77C2×C4×C11—22no—
78C2×C4×C11—22no—
79C2×C4×C11—22no—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11—24no—
83C2×C4×C11C2×D414yes—
84C2×C4×C11C2×D412yes—
85C2×C4×C11C2×D412yes—
86C2×C4×C11C2×D412yes—
87C2×C4×C11C2×Q814yescomplemented
88C2×Dic11—22no—
89C2×Dic11—21no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—21no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11—22no—
96C2×Dic11—22no—
97C2×Dic11C2×D412yes—
98C2×Dic11C2×D414yes—
99C2×Dic11C2×Q814yescomplemented
100C4×D11—216no—
101C4×D11—28no—
102Dic22—216no—
103Dic22—28no—
104Dic22—44no—
105C23×Q8—111no—
106C2×C4⋊Q8—111no—
107C2×C4⋊Q8—111no—
108C2×C22⋊Q8—111no—
109#395—114no—
110#391—111no—
111#391D1111yescharacteristic, complemented
112#385—114no—
113C4×C22⋊C4—111no—
114C22×C11C22×D411yescharacteristic
115C22×C11C22×D411yescharacteristic
116C22×C11C22×D411yescharacteristic
117C22×C11C22×Q812yes—
118C22×C112−1+412yes—
119C2×D11—22no—
120C2×D11—22no—
121C2×D11—24no—
122C2×D11—24no—
123C2×D11C4⋊Q814yescomplemented
124C4×C11—24no—
125C4×C11—24no—
126C4×C11—42no—
127C4×C11#12014yes—
128Dic11—22no—
129Dic11—24no—
130Dic11—24no—
131Dic11—42no—
132C22×Q8—111no—
133C22×Q8—111no—
134C22×Q8—112no—
135C22×Q8—224no—
136C22×Q8—224no—
137C22×Q8—224no—
138C22×Q8C2×D1111yescharacteristic
139C23×C4—111no—
140C23×C4—112no—
141C4⋊Q8—224no—
142C4⋊Q8—224no—
143C22⋊Q8—224no—
144C2×C4⋊C4—112no—
145C2×C4⋊C4—114no—
146C2×C4⋊C4—114no—
147C2×C4⋊C4C2×D1111yescharacteristic
148C2×C22⋊C4—112no—
149C2×C22⋊C4—112no—
150C2×C42—111no—
151C2×C42C2×D1111yescharacteristic
152#95—112no—
153#95—114no—
154#95C2×D1114yes—
155C2×C11C2×#12011yescharacteristic
156C2×C11C2×C4⋊Q811yescharacteristic
157C2×C11#53611yescharacteristic
158C2×C11D4×Q812yes—
159C2×C11D4×Q812yes—
160D11—24no—
161C24—111no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—114no—
165C2×Q8—224no—
166C2×Q8—224no—
167C2×Q8—2232no—
168C2×Q8—224no—
169C2×Q8—222no—
170C2×Q8—222no—
171C2×Q8—222no—
172C2×Q8—222no—
173C2×Q8—222no—
174C2×Q8D4414yes—
175C22×C4—112no—
176C22×C4—111no—
177C22×C4—112no—
178C22×C4—114no—
179C22×C4—112no—
180C22×C4—224no—
181C22×C4—224no—
182C22×C4—224no—
183C22×C4—222no—
184C22×C4—222no—
185C22×C4C22×D1112yes—
186C22×C4C22×D1112yes—
187C22×C4C22×D1112yes—
188C22×C4C22×D1111yescharacteristic
189C4⋊C4—22no—
190C4⋊C4—224no—
191C4⋊C4—228no—
192C4⋊C4—228no—
193C22⋊C4—118no—
194C22⋊C4—224no—
195C42—22no—
196C42—222no—
197C11#243411yescharacteristic, complemented, Sylow
198C23—112no—
199C23—114no—
200C23—221no—
201C23—221no—
202C23—222no—
203C23C23×D1111yescharacteristic
204Q8—28no—
205Q8—2216no—
206Q8—228no—
207Q8—444no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—112no—
215C2×C4—114no—
216C2×C4—114no—
217C2×C4—2216no—
218C2×C4—222no—
219C2×C4—221no—
220C2×C4—222no—
221C2×C4—222no—
222C2×C4—221no—
223C2×C4—222no—
224C2×C4—228no—
225C2×C4—222no—
226C2×C4—222no—
227C2×C4—222no—
228C2×C4C2×D4414yes—
229C2×C4C2×D4412yes—
230C2×C4D4×D1112yes—
231C2×C4D4×D1112yes—
232C2×C4Q8×D1114yescomplemented
233C22—114no—
234C22—222no—
235C22—224no—
236C22—224no—
237C22—222no—
238C22C22×D4411yescharacteristic
239C22C2×D4×D1111yescharacteristic
240C22C2×D4×D1111yescharacteristic
241C22C2×Q8×D1112yes—
242C22#6704312yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—222no—
247C4—224no—
248C4—224no—
249C4—442no—
250C4#6693514yes—
251C2—224no—
252C2C2×#6693511yescharacteristic
253C2Q8×D4412yes—
254C2#1062316312yes—
255C2#1062320611yescharacteristic
256C2D11×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
prime radicalC23×D11C23
hypercenter#391D11
hypocenterC11#2434
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#66997C2semidirect
4. C2×#66922C2semidirect
5. C11×#391C2semidirect
6. #10622587C2semidirect
9. #10622269C2semidirect
64. (C2×D11)⋊C4Q8semidirect
87. C2×C4×C11C2×Q8semidirect
99. C2×Dic11C2×Q8semidirect
111. #391D11semidirect
123. C2×D11C4⋊Q8semidirect
197. C11#2434semidirect
232. C2×C4Q8×D11semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D112
C2×#669972
C2×#669502
C2×#669222
C11×#3912
#106225872
C4×(C2×D11)⋊C42
#106222812
#106222692
#243411