Finite Group Databasebeta

← #13078023

Subgroups — #13078023

Order 1408.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#10623453C211yesmaximal, characteristic
2#10623356C211yesmaximal, characteristic, complemented
3#10623057C212yesmaximal, complemented
4C11×#410C211yesmaximal, characteristic, complemented
5#10622619C212yesmaximal, complemented
6#10622355C218yesmaximal, complemented
7C2×C11×#41C2211yescharacteristic
8#67043—22no—
9#67042—22no—
10#41×D11—24no—
11#67040—21no—
12#67037—21no—
13#67037C2214yes—
14C2×#4339—22no—
15C2×#4339C2212yes—
16C22×Dic22C2211yescharacteristic, complemented
17C11×#117C2212yescomplemented
18#66997—21no—
19#66989—21no—
20#66988C2214yescomplemented
21#66979C2211yescharacteristic, complemented
22#66977—22no—
23#66950—24no—
24#66939C2218yescomplemented
25#66928—22no—
26#66922—22no—
27C11×#99C2214yescomplemented
28#66870C2218yescomplemented
29C11×#41—22no—
30C11×#41—22no—
31C2×Q8×C11D411yescharacteristic
32C2×D4×C11D411yescharacteristic
33C2×D4×C11C2312yes—
34C22×C4×C11D412yes—
35C22×C4×C11C2311yescharacteristic
36C2×C22⋊D11—24no—
37C2×C22⋊D11C2314yes—
38C22×Dic11C2312yes—
39#4343—24no—
40Q8×D11—24no—
41#4341—24no—
42#4341—28no—
43D4×D11—24no—
44D4×D11—28no—
45#4339—24no—
46#4339—24no—
47#4339—22no—
48#4339—24no—
49#4339—22no—
50#4339—22no—
51#4339—22no—
52C2×D44—22no—
53C2×D44D412yes—
54C2×C4×D11—24no—
55C2×C4×D11D414yes—
56C2×Dic22—22no—
57C2×Dic22—22no—
58C2×Dic22—22no—
59C2×Dic22—21no—
60C2×Dic22—21no—
61C2×Dic22D412yes—
62C11×C4⋊C4—22no—
63C11×C22⋊C4C2314yes—
64C42×C11—22no—
65C22⋊Dic11—24no—
66C22⋊Dic11C2312yes—
67(C2×D11)⋊C4—28no—
68C4⋊Dic11—28no—
69C4⋊Dic11—21no—
70Dic11⋊C4—24no—
71C4×Dic11—21no—
72#3028—111nomaximal, Sylow
73C23×C11C2×D412yes—
74C23×C11C2411yescharacteristic
75C22×D11—24no—
76C22×D11C2×D414yes—
77Q8×C11—22no—
78D4×C11—24no—
79D4×C11—22no—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11—24no—
83C2×C4×C11C2×D412yes—
84C2×C4×C11C2×D411yescharacteristic
85C2×C4×C11C2×D411yescharacteristic
86C2×C4×C11C2×D412yes—
87C2×C4×C11C2×D412yes—
88C22⋊D11—28no—
89C22⋊D11—24no—
90C22⋊D11—24no—
91C2×Dic11—24no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—24no—
95C2×Dic11C2×D414yes—
96D44—24no—
97D44—22no—
98D44—22no—
99C4×D11—28no—
100C4×D11—24no—
101C4×D11—24no—
102Dic22—24no—
103Dic22—22no—
104Dic22—22no—
105Dic22—41no—
106Dic22—41no—
107Dic22—42no—
108#585—111no—
109#536—111no—
110#536—112no—
111#458—118no—
112#410—112no—
113#410D1111yescharacteristic, complemented
114C22×C11—22no—
115C22×C11—41no—
116C22×C11—42no—
117C22×C11C22×D411yescharacteristic
118C22×C11C22×D412yes—
119C2×D11—24no—
120C2×D11—44no—
121C2×D11—84no—
122C4×C11—22no—
123C4×C11—44no—
124C4×C11#12011yescharacteristic
125C4×C11#12011yescharacteristic
126C4×C11#12012yes—
127Dic11—24no—
128Dic11—42no—
129Dic11—42no—
1302−1+4—222no—
1312−1+4—221no—
132H(5,𝔽2)—114no—
133H(5,𝔽2)—222no—
134H(5,𝔽2)—221no—
135C2×#41—112no—
136C2×#41—224no—
137C2×#41—222no—
138C2×#41C2×D1111yescharacteristic
139C22×Q8—111no—
140C4⋊Q8—222no—
141C4⋊Q8—221no—
142#124—222no—
143#124—221no—
144#123—114no—
145#123—118no—
146C22⋊Q8—224no—
147C22⋊Q8—222no—
148#117—111no—
149#117C2×D1112yescomplemented
150#99—118no—
151#99C2×D1114yescomplemented
152C2×C11—21no—
153C2×C11—22no—
154C2×C11—42no—
155C2×C11C2×#12011yescharacteristic
156D11—44no—
157#41—22no—
158#41—22no—
159#41—224no—
160#41—224no—
161#41—228no—
162#41—224no—
163#41—224no—
164#41—222no—
165#41—222no—
166#41—224no—
167#41—222no—
168#41—222no—
169C2×Q8—112no—
170C2×Q8—222no—
171C2×Q8—224no—
172C2×Q8—221no—
173C2×Q8—221no—
174C2×Q8—222no—
175C2×Q8—222no—
176C2×Q8D4411yescharacteristic
177C2×D4—114no—
178C2×D4—112no—
179C2×D4—222no—
180C2×D4—228no—
181C2×D4—224no—
182C2×D4—224no—
183C2×D4D4411yescharacteristic
184C2×D4C22×D1112yes—
185C22×C4—114no—
186C22×C4—112no—
187C22×C4—224no—
188C22×C4D4412yes—
189C22×C4C22×D1111yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—228no—
192C4⋊C4—221no—
193C4⋊C4—224no—
194C22⋊C4—112no—
195C22⋊C4—228no—
196C22⋊C4—224no—
197C22⋊C4C22×D1114yes—
198C42—22no—
199C42—221no—
200C11#302811yescharacteristic, complemented, Sylow
201C23—114no—
202C23—224no—
203C23C2×D4412yes—
204C23C23×D1111yescharacteristic
205Q8—22no—
206Q8—224no—
207Q8—222no—
208Q8—222no—
209Q8—441no—
210Q8—441no—
211Q8—442no—
212D4—24no—
213D4—22no—
214D4—224no—
215D4—228no—
216D4—224no—
217D4—224no—
218D4—222no—
219D4—222no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—114no—
224C2×C4—228no—
225C2×C4—224no—
226C2×C4—224no—
227C2×C4—224no—
228C2×C4—222no—
229C2×C4—222no—
230C2×C4—224no—
231C2×C4C2×D4412yes—
232C2×C4C2×D4411yescharacteristic
233C2×C4C2×D4411yescharacteristic
234C2×C4D4×D1112yes—
235C2×C4D4×D1112yes—
236C22—22no—
237C22—41no—
238C22—42no—
239C22—224no—
240C22—444no—
241C22—884no—
242C22C22×D4411yescharacteristic
243C22C2×D4×D1112yes—
244C4—22no—
245C4—44no—
246C4—224no—
247C4—442no—
248C4—442no—
249C4#6693511yescharacteristic
250C4#6693511yescharacteristic
251C4#6693512yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—444no—
256C2C2×#6693511yescharacteristic
subgroupG/H
centerC2C2×#66935
commutator, cosocleC23×C11C24
FrattiniC23C23×D11
FittingC11×#410C2
socleC2×C11C2×#120
prime radical#10623453C2
hypercenter#410D11
hypocenterC11#3028
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #10623356C2semidirect
3. #10623057C2semidirect
4. C11×#410C2semidirect
5. #10622619C2semidirect
6. #10622355C2semidirect
16. C22×Dic22C22semidirect
17. C11×#117C22semidirect
20. #66988C22semidirect
21. #66979C22semidirect
24. #66939C22semidirect
27. C11×#99C22semidirect
28. #66870C22semidirect
113. #410D11semidirect
149. #117C2×D11semidirect
151. #99C2×D11semidirect
200. C11#3028semidirect

Maximal subgroups

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

subgroupindex
#106234532
#106233562
#106230572
C11×#4102
#106226192
#106223552
#302811