Finite Group Databasebeta

← #13076920

Subgroups — #13076920

Order 1408.

696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#67000C211yesmaximal, characteristic, complemented
2C2×#66999C211yesmaximal, characteristic, complemented
3C2×#66948C211yesmaximal, characteristic
4C2×C42×D11C211yesmaximal, characteristic, complemented
5C11×#391C211yesmaximal, characteristic, complemented
6#10622590C211yesmaximal, characteristic, complemented
7#10622552C211yesmaximal, characteristic, complemented
8#10622282C214yesmaximal, complemented
9#10622278C214yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11C22×D44C2212yes—
12C22×D44C2211yescharacteristic
13C22×C4×D11C2212yes—
14C22×C4×D11C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C42×C11C2211yescharacteristic
17#67000—24no—
18#66999—24no—
19C2×(C2×D11)⋊C4C2212yes—
20C2×(C2×D11)⋊C4C2212yes—
21C2×(C2×D11)⋊C4C2214yes—
22C2×(C2×D11)⋊C4C2214yes—
23C2×C4⋊Dic11C2211yescharacteristic
24C2×Dic11⋊C4C2212yes—
25C2×C4×Dic11C2212yes—
26C2×C4×Dic11C2211yescharacteristic
27#66948—24no—
28C42×D11—24no—
29C11×#95C2214yes—
30C11⋊#95C2214yes—
31C23×D11C2312yes—
32C23×D11C2311yescharacteristic
33C2×Q8×C11—22no—
34C2×Q8×C11—22no—
35C2×Q8×C11—22no—
36C22×C4×C11C2312yes—
37C22×C4×C11C2312yes—
38C22×C4×C11C2312yes—
39C22×C4×C11C2311yescharacteristic
40C22×Dic11C2312yes—
41C22×Dic11C2312yes—
42C22×Dic11C2311yescharacteristic
43C2×D44—24no—
44C2×D44—28no—
45C2×D44—22no—
46C2×D44—22no—
47C2×D44—22no—
48C2×C4×D11—24no—
49C2×C4×D11—24no—
50C2×C4×D11—22no—
51C2×C4×D11—22no—
52C2×C4×D11—24no—
53C2×C4×D11D414yes—
54C11×C4⋊C4—22no—
55C42×C11—22no—
56(C2×D11)⋊C4—24no—
57(C2×D11)⋊C4—28no—
58(C2×D11)⋊C4—24no—
59(C2×D11)⋊C4—28no—
60C4⋊Dic11—22no—
61Dic11⋊C4—24no—
62C4×Dic11—24no—
63C4×Dic11D414yescomplemented
64#2850—111nomaximal, Sylow
65C23×C11C2411yescharacteristic
66C22×D11—21no—
67C22×D11—21no—
68C22×D11—22no—
69C22×D11—22no—
70C22×D11—22no—
71C22×D11—24no—
72C22×D11—22no—
73C22×D11—24no—
74C22×D11C2×D412yes—
75C22×D11#4114yes—
76Q8×C11—44no—
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×C11—22no—
84C2×C4×C11C2×D412yes—
85C2×C4×C11C2×D412yes—
86C2×C4×C11#4112yes—
87C2×C4×C11#4114yes—
88C2×Dic11—21no—
89C2×Dic11—21no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11—24no—
94C2×Dic11C2×D414yes—
95C2×Dic11C2×D412yes—
96D44—48no—
97D44—44no—
98C4×D11—216no—
99C4×D11—28no—
100C2×#124—111no—
101C2×C22⋊Q8—111no—
102C2×#121—111no—
103C22×C42—111no—
104#391D1111yescharacteristic, complemented
105#390—111no—
106#388—114no—
107#386—114no—
108#382—111no—
109C22×C11C22×D411yescharacteristic
110C22×C11C22×D411yescharacteristic
111C22×C11C2×#4111yescharacteristic
112C22×C11C2×#4112yes—
113C22×C11C2×#4112yes—
114C2×D11—22no—
115C2×D11—24no—
116C2×D11—24no—
117C2×D11—22no—
118C2×D11—42no—
119C2×D11—44no—
120C2×D11—42no—
121C2×D11—44no—
122C2×D11—42no—
123C2×D11#12414yescomplemented
124C4×C11—24no—
125C4×C11—42no—
126C4×C11—42no—
127C4×C11#12114yes—
128Dic11—22no—
129Dic11—24no—
130Dic11—42no—
131C22×Q8C2×D1111yescharacteristic
132C22×D4—111no—
133C22×D4—112no—
134C23×C4—111no—
135C23×C4—112no—
136#124—224no—
137C22⋊Q8—224no—
138#121—224no—
139C2×C4⋊C4—111no—
140C2×C4⋊C4—112no—
141C2×C4⋊C4C2×D1111yescharacteristic
142C2×C22⋊C4—112no—
143C2×C22⋊C4—112no—
144C2×C22⋊C4—114no—
145C2×C22⋊C4—114no—
146C2×C42—111no—
147C2×C42—112no—
148C2×C42—224no—
149C2×C42C2×D1111yescharacteristic
150#95—114no—
151#95C2×D1114yes—
152C2×C11C2×#12111yescharacteristic
153C2×C11#52512yes—
154C2×C11C2×#12411yescharacteristic
155C2×C11#52912yes—
156C2×C11#53211yescharacteristic
157D11—24no—
158D11—82no—
159C24—111no—
160C24—112no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×D4—224no—
165C2×D4—228no—
166C2×D4—222no—
167C2×D4—222no—
168C2×D4—222no—
169C22×C4—112no—
170C22×C4—111no—
171C22×C4—114no—
172C22×C4—112no—
173C22×C4—224no—
174C22×C4—224no—
175C22×C4—222no—
176C22×C4—222no—
177C22×C4—224no—
178C22×C4C22×D1112yes—
179C22×C4C22×D1112yes—
180C22×C4C22×D1112yes—
181C22×C4C22×D1111yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—224no—
184C4⋊C4—222no—
185C22⋊C4—224no—
186C22⋊C4—228no—
187C22⋊C4—224no—
188C22⋊C4—228no—
189C42—22no—
190C42—114no—
191C42—224no—
192C11#285011yescharacteristic, complemented, Sylow
193C23—112no—
194C23—114no—
195C23—221no—
196C23—221no—
197C23—222no—
198C23—222no—
199C23—222no—
200C23—222no—
201C23—224no—
202C23—224no—
203C23C23×D1111yescharacteristic
204Q8—44no—
205D4—448no—
206D4—444no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—114no—
215C2×C4—112no—
216C2×C4—2216no—
217C2×C4—221no—
218C2×C4—221no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—228no—
222C2×C4—222no—
223C2×C4—224no—
224C2×C4#433912yes—
225C2×C4D4×D1112yes—
226C2×C4D4×D1112yes—
227C2×C4#434314yes—
228C22—114no—
229C22—222no—
230C22—222no—
231C22—224no—
232C22—224no—
233C22—442no—
234C22—442no—
235C22—444no—
236C22—444no—
237C22—442no—
238C22C2×#433911yescharacteristic
239C22C2×D4×D1111yescharacteristic
240C22C2×D4×D1111yescharacteristic
241C22C2×#434312yes—
242C22#41×D1112yes—
243C4—24no—
244C4—42no—
245C4—42no—
246C4—222no—
247C4—224no—
248C4—442no—
249C4#6693714yes—
250C2—224no—
251C2—882no—
252C2C2×#6693711yescharacteristic
253C2#1062311112yes—
254C2#1062315812yes—
255C2D11×#12411yescharacteristic
256C2#1062325611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
prime radicalC2×#66948C2
hypercenter#391D11
hypocenterC11#2850
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#67000C2semidirect
2. C2×#66999C2semidirect
4. C2×C42×D11C2semidirect
5. C11×#391C2semidirect
6. #10622590C2semidirect
7. #10622552C2semidirect
8. #10622282C2semidirect
9. #10622278C2semidirect
63. C4×Dic11D4semidirect
104. #391D11semidirect
123. C2×D11#124semidirect
192. C11#2850semidirect

Maximal subgroups

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

subgroupindex
C2×#670002
C2×#669992
C2×#669482
C2×C42×D112
C11×#3912
#106225902
#106225522
#106222822
#106222782
#285011