Finite Group Databasebeta

← #13069713

Subgroups — #13069713

Order 1408.

390 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 91 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×Dic44C211yesmaximal, characteristic, complemented
2C22×#4314C211yesmaximal, characteristic
3C2×#66928C211yesmaximal, characteristic
4C2×#66885C211yesmaximal, characteristic
5C2×#66880C211yesmaximal, characteristic, complemented
6#10622542C211yesmaximal, characteristic, complemented
7C11×#336C211yesmaximal, characteristic, complemented
8C22×D44C2211yescharacteristic
9C22×Dic22C2211yescharacteristic
10C22×Dic22C2211yescharacteristic
11C22×C8×C11C2211yescharacteristic
12C22×C8×C11C2211yescharacteristic
13C2×C42×C11C2211yescharacteristic
14C2×(C2×D11)⋊C4—22no—
15C2×C4⋊Dic11C2211yescharacteristic
16C2×Dic44—22no—
17C2×Dic44—22no—
18C2×Dic44—22no—
19C2×#4314—22no—
20C2×#4314—22no—
21C2×#4314—22no—
22#66928—22no—
23#66928—22no—
24C11⋊#95—22no—
25#66885—22no—
26#66880—22no—
27C23×D11—21no—
28C22×C4×C11D412yes—
29C22×C4×C11C2311yescharacteristic
30C22×Dic11—21no—
31C22×Dic11—21no—
32C22×Dic11—21no—
33C2×D44—21no—
34C2×D44—21no—
35C2×D44—21no—
36C2×D44—21no—
37C2×D44—21no—
38C2×D44D412yes—
39C2×Dic22—21no—
40C2×Dic22—21no—
41C2×Dic22—21no—
42C2×Dic22—21no—
43C2×Dic22—21no—
44C2×Dic22—21no—
45C2×Dic22—21no—
46C2×Dic22—21no—
47C2×Dic22—21no—
48C2×Dic22—21no—
49C2×Dic22D412yes—
50C2×Dic22D412yes—
51C2×C8×C11—21no—
52C2×C8×C11—21no—
53C2×C8×C11—21no—
54C2×C8×C11—21no—
55C2×C8×C11D412yes—
56C2×C8×C11D412yes—
57C42×C11—21no—
58C42×C11—21no—
59(C2×D11)⋊C4—28no—
60C4⋊Dic11—21no—
61C4⋊Dic11—21no—
62Dic44—28no—
63#4314—28no—
64#2015—111nomaximal, Sylow
65C23×C11C2×D411yescharacteristic
66C22×D11—22no—
67C22×D11—41no—
68C22×D11—41no—
69C22×D11—42no—
70C22×D11—42no—
71C2×C4×C11—22no—
72C2×C4×C11—22no—
73C2×C4×C11—22no—
74C2×C4×C11C2×D411yescharacteristic
75C2×C4×C11C2×D411yescharacteristic
76C2×C4×C11C2×D411yescharacteristic
77C2×C4×C11C2×D411yescharacteristic
78C2×C4×C11C2×D411yescharacteristic
79C2×C4×C11#4111yescharacteristic
80C2×Dic11—22no—
81C2×Dic11—22no—
82C2×Dic11—41no—
83C2×Dic11—41no—
84C2×Dic11—41no—
85C2×Dic11—41no—
86C2×Dic11—41no—
87C2×Dic11—42no—
88D44—22no—
89D44—22no—
90D44—41no—
91D44—41no—
92Dic22—22no—
93Dic22—22no—
94Dic22—22no—
95Dic22—22no—
96Dic22—41no—
97Dic22—41no—
98Dic22—41no—
99Dic22—41no—
100C8×C11—22no—
101C8×C11—22no—
102C22×Q16—111no—
103C22×SD16—111no—
104C2×#124—111no—
105C2×#103—111no—
106C2×#102—111no—
107#391—111no—
108#336D1111yescharacteristic, complemented
109C22×C11#12011yescharacteristic
110C22×C11#12111yescharacteristic
111C22×C11#12711yescharacteristic
112C22×C11#13512yes—
113C22×C11#13712yes—
114C2×D11—42no—
115C2×D11—44no—
116C2×D11—82no—
117C2×D11—81no—
118C2×D11—81no—
119C4×C11—42no—
120C4×C11#12011yescharacteristic
121C4×C11#12011yescharacteristic
122C4×C11#12111yescharacteristic
123C4×C11#12111yescharacteristic
124Dic11—42no—
125Dic11—42no—
126Dic11—81no—
127C22×Q8—111no—
128C22×Q8—111no—
129C22×D4—111no—
130C2×Q16—222no—
131C2×Q16—222no—
132C2×Q16—222no—
133C2×SD16—222no—
134C2×SD16—222no—
135C2×SD16—222no—
136C22×C8C2×D1111yescharacteristic
137C22×C8C2×D1111yescharacteristic
138#124—222no—
139#124—222no—
140C2×C4⋊C4—111no—
141C2×C22⋊C4—222no—
142C2×C42C2×D1111yescharacteristic
143#103—222no—
144#102—222no—
145#95—222no—
146C2×C11#39211yescharacteristic
147C2×C11#45212yes—
148C2×C11#46412yes—
149C2×C11#49511yescharacteristic
150C2×C11#49711yescharacteristic
151D11—82no—
152C24—221no—
153C2×Q8—112no—
154C2×Q8—112no—
155C2×Q8—221no—
156C2×Q8—221no—
157C2×Q8—221no—
158C2×Q8—221no—
159C2×Q8—221no—
160C2×Q8—221no—
161C2×Q8—221no—
162C2×Q8—221no—
163C2×Q8—221no—
164C2×Q8—221no—
165C2×D4—112no—
166C2×D4—221no—
167C2×D4—221no—
168C2×D4—221no—
169C2×D4—221no—
170C2×D4—221no—
171C22×C4—221no—
172C22×C4—221no—
173C22×C4—221no—
174C22×C4D4412yes—
175C22×C4C22×D1111yescharacteristic
176Q16—228no—
177SD16—228no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8D4412yes—
183C2×C8D4412yes—
184C4⋊C4—221no—
185C4⋊C4—221no—
186C22⋊C4—228no—
187C42—21no—
188C42—21no—
189C11#201511yescharacteristic, complemented, Sylow
190C23—222no—
191C23—441no—
192C23—441no—
193C23—442no—
194C23—442no—
195C23C2×D4411yescharacteristic
196Q8—222no—
197Q8—222no—
198Q8—222no—
199Q8—222no—
200Q8—441no—
201Q8—441no—
202Q8—441no—
203Q8—441no—
204D4—222no—
205D4—222no—
206D4—441no—
207D4—441no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—222no—
212C2×C4—222no—
213C2×C4—441no—
214C2×C4—441no—
215C2×C4—441no—
216C2×C4—441no—
217C2×C4—441no—
218C2×C4—442no—
219C2×C4C2×D4411yescharacteristic
220C2×C4C2×D4411yescharacteristic
221C2×C4D4×D1111yescharacteristic
222C2×C4D4×D1111yescharacteristic
223C2×C4D4×D1111yescharacteristic
224C2×C4#434311yescharacteristic
225C8—22no—
226C8—22no—
227C22—442no—
228C22—444no—
229C22—881no—
230C22—881no—
231C22—882no—
232C22#6692711yescharacteristic
233C22#6693511yescharacteristic
234C22#6694811yescharacteristic
235C22#6695712yes—
236C22#6696212yes—
237C4—42no—
238C4—442no—
239C4—442no—
240C4—881no—
241C4#6693511yescharacteristic
242C4#6693511yescharacteristic
243C4#6694811yescharacteristic
244C4#6694811yescharacteristic
245C2—882no—
246C2#1062228411yescharacteristic
247C2#1062230911yescharacteristic
248C2#1062232811yescharacteristic
249C2#1062234612yes—
250C2#1062245812yes—
subgroupG/H
centerC23C2×D44
commutator, cosocleC22×C4×C11C23
FrattiniC22×C4C22×D11
FittingC11×#336C2
socleC23×C11C2×D4
prime radicalC22×D44C22
hypercenter#336D11
hypocenterC11#2015
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
1. C22×Dic44C2semidirect
5. C2×#66880C2semidirect
6. #10622542C2semidirect
7. C11×#336C2semidirect
108. #336D11semidirect
189. C11#2015semidirect

Maximal subgroups

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

subgroupindex
C22×Dic442
C22×#43142
C2×#669282
C2×#668852
C2×#668802
#106225422
C11×#3362
#201511