Finite Group Databasebeta

← #13080004

Subgroups — #13080004

Order 1408.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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#10623008C211yesmaximal, characteristic
2#10623001C211yesmaximal, characteristic, complemented
3C11×#423C211yesmaximal, characteristic, complemented
4#10622747C214yesmaximal, complemented
5#10622463C212yesmaximal, complemented
6#10622461C212yesmaximal, complemented
7#10622459C212yesmaximal, complemented
8#10622454C212yesmaximal, complemented
9C2×#4339—21no—
10C2×C11×M16C2212yes—
11C2×C42×C11C2211yescharacteristic
12#66983—21no—
13#66983C2212yes—
14#66979—21no—
15#66977—21no—
16#66977C2212yes—
17C2×#4314C2214yes—
18#66928—21no—
19#66927C2211yescharacteristic
20C4×D44—21no—
21C4×D44C2212yes—
22#66923—21no—
23#66922C2211yescharacteristic
24#66922C2211yescharacteristic
25#66922C2211yescharacteristic
26C4×Dic22—21no—
27C4×Dic22C2212yes—
28C11×C4⋊C8C2212yes—
29C11×C4⋊C8C2212yes—
30#66885C2214yes—
31#66882C2214yes—
32#66880C2214yes—
33C22×C4×C11D411yescharacteristic
34C22×C4×C11D411yescharacteristic
35C22×C4×C11C2311yescharacteristic
36C2×C22⋊D11—22no—
37#4339—44no—
38C2×D44—21no—
39C2×D44—21no—
40C2×D44C2312yes—
41C2×C4×D11—22no—
42C2×Dic22—21no—
43C2×Dic22—21no—
44C2×Dic22C2312yes—
45C11×M16—42no—
46C2×C8×C11C2314yes—
47C42×C11D411yescharacteristic
48C42×C11D411yescharacteristic
49C42×C11C2311yescharacteristic
50C42×C11C2311yescharacteristic
51C22⋊Dic11—22no—
52(C2×D11)⋊C4—22no—
53(C2×D11)⋊C4—22no—
54C4⋊Dic11—21no—
55C4⋊Dic11—21no—
56C4⋊Dic11—21no—
57C4⋊Dic11—21no—
58C4⋊Dic11C2312yes—
59C4⋊Dic11C2312yes—
60Dic11⋊C4—22no—
61Dic11⋊C4—22no—
62C4×Dic11—22no—
63#4314—44no—
64#3254—111nomaximal, Sylow
65C23×C11C2×D411yescharacteristic
66C22×D11—22no—
67C2×C4×C11—22no—
68C2×C4×C11—22no—
69C2×C4×C11—21no—
70C2×C4×C11—21no—
71C2×C4×C11C2×D411yescharacteristic
72C2×C4×C11C2×D411yescharacteristic
73C2×C4×C11C2×D411yescharacteristic
74C2×C4×C11C2×D411yescharacteristic
75C2×C4×C11C2×D411yescharacteristic
76C2×C4×C11C2411yescharacteristic
77C22⋊D11—44no—
78C2×Dic11—22no—
79C2×Dic11—22no—
80C2×Dic11—22no—
81D44—22no—
82D44—22no—
83D44—42no—
84D44—42no—
85C4×D11—44no—
86Dic22—22no—
87Dic22—22no—
88Dic22—42no—
89Dic22—42no—
90C8×C11—24no—
91#533—111no—
92#532—111no—
93#477—112no—
94#474—112no—
95#468—114no—
96#461—112no—
97#460—112no—
98#423D1111yescharacteristic, complemented
99C22×C11—21no—
100C22×C11—21no—
101C22×C11—21no—
102C22×C11C22×D411yescharacteristic
103C2×D11—42no—
104C2×D11—42no—
105C2×D11—42no—
106C4×C11—21no—
107C4×C11—21no—
108C4×C11—21no—
109C4×C11AGL(1,ℤ/8)12yes—
110C4×C11#13712yes—
111C4×C11H(5,𝔽2)11yescharacteristic
112C4×C112−1+411yescharacteristic
113Dic11—42no—
114Dic11—42no—
115Dic11—42no—
116C2×#41—221no—
117C2×SD16—114no—
118C2×M16C2×D1112yes—
119C4⋊Q8—111no—
120C4⋊Q8—111no—
121C4⋊Q8—111no—
122#127—111no—
123#125—221no—
124#124—221no—
125C22⋊Q8—112no—
126C22⋊Q8—221no—
127#121—112no—
128#121—221no—
129C4×Q8—112no—
130C4×Q8—221no—
131C4×D4—112no—
132C4×D4—221no—
133#117—221no—
134C2×C42C2×D1111yescharacteristic
135#107—114no—
136C4⋊C8C2×D1112yes—
137C4⋊C8C2×D1112yes—
138#103—114no—
139#102—114no—
140C2×C11—41no—
141C2×C11#53711yescharacteristic
142C2×C11C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C11C2×#13711yescharacteristic
144D11—82no—
145#41—444no—
146C2×Q8—112no—
147C2×Q8—221no—
148C2×Q8—221no—
149C2×D4—112no—
150C2×D4—222no—
151C2×D4—221no—
152C2×D4—221no—
153C22×C4—222no—
154C22×C4D4411yescharacteristic
155C22×C4D4411yescharacteristic
156C22×C4C22×D1111yescharacteristic
157SD16—444no—
158M16—42no—
159C2×C8C22×D1114yes—
160C4⋊C4—112no—
161C4⋊C4—112no—
162C4⋊C4—221no—
163C4⋊C4—221no—
164C4⋊C4—221no—
165C4⋊C4—222no—
166C4⋊C4—222no—
167C4⋊C4—221no—
168C22⋊C4—222no—
169C22⋊C4—222no—
170C22⋊C4—222no—
171C42—222no—
172C42D4411yescharacteristic
173C42D4411yescharacteristic
174C42C22×D1111yescharacteristic
175C42C22×D1111yescharacteristic
176C11#325411yescharacteristic, complemented, Sylow
177C23—222no—
178C23C2×D4411yescharacteristic
179Q8—222no—
180Q8—222no—
181Q8—442no—
182Q8—442no—
183D4—222no—
184D4—222no—
185D4—442no—
186D4—444no—
187D4—442no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—222no—
193C2×C4—222no—
194C2×C4—222no—
195C2×C4—444no—
196C2×C4C2×D4411yescharacteristic
197C2×C4C2×D4411yescharacteristic
198C2×C4C2×D4411yescharacteristic
199C2×C4C2×D4411yescharacteristic
200C2×C4C2×D4411yescharacteristic
201C2×C4C23×D1111yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—442no—
207C22—442no—
208C22—442no—
209C22C22×D4411yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—442no—
214C4—442no—
215C4—442no—
216C4#6696112yes—
217C4#6696212yes—
218C4#6703711yescharacteristic
219C4#6704011yescharacteristic
220C2—41no—
221C2—882no—
222C2#1062304111yescharacteristic
223C2C2×#6696111yescharacteristic
224C2C2×#6696211yescharacteristic
subgroupG/H
centerC22C22×D44
commutator, cosocleC2×C4×C11C24
FrattiniC2×C4C23×D11
FittingC11×#423C2
socleC22×C11C22×D4
prime radical#10623008C2
hypercenter#423D11
hypocenterC11#3254
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
2. #10623001C2semidirect
3. C11×#423C2semidirect
4. #10622747C2semidirect
5. #10622463C2semidirect
6. #10622461C2semidirect
7. #10622459C2semidirect
8. #10622454C2semidirect
98. #423D11semidirect
176. C11#3254semidirect

Maximal subgroups

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

subgroupindex
#106230082
#106230012
C11×#4232
#106227472
#106224632
#106224612
#106224592
#106224542
#325411