Finite Group Databasebeta

← #13074387

Subgroups — #13074387

Order 1408.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#66999C211yesmaximal, characteristic
2C2×#66997C211yesmaximal, characteristic, complemented
3C2×#66950C211yesmaximal, characteristic
4C2×#66925C211yesmaximal, characteristic
5C11×#391C211yesmaximal, characteristic, complemented
6#10622590C211yesmaximal, characteristic, complemented
7#10622542C211yesmaximal, characteristic, complemented
8#10622288C214yesmaximal
9#10622270C214yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11C22×C4×D11C2211yescharacteristic
12C22×Dic22C2211yescharacteristic
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×C42×C11C2211yescharacteristic
15#66999—24no—
16#66997—24no—
17C2×(C2×D11)⋊C4C2212yes—
18C2×(C2×D11)⋊C4C2212yes—
19C2×(C2×D11)⋊C4C2212yes—
20C2×C4⋊Dic11C2212yes—
21C2×C4⋊Dic11C2211yescharacteristic
22C2×Dic11⋊C4C2212yes—
23C2×Dic11⋊C4C2212yes—
24C2×Dic11⋊C4C2214yes—
25C2×C4×Dic11C2211yescharacteristic
26#66950—24no—
27#66925—24no—
28C11×#95C2214yes—
29C11⋊#95C2214yes—
30C11⋊#95C2214yes—
31C23×D11C2311yescharacteristic
32C2×Q8×C11—22no—
33C2×Q8×C11—22no—
34C2×Q8×C11—22no—
35C22×C4×C11C2312yes—
36C22×C4×C11C2312yes—
37C22×C4×C11C2312yes—
38C22×C4×C11C2311yescharacteristic
39C22×Dic11C2312yes—
40C22×Dic11C2312yes—
41C22×Dic11C2312yes—
42C22×Dic11C2311yescharacteristic
43C2×C4×D11—22no—
44C2×C4×D11—22no—
45C2×C4×D11D414yes—
46C2×Dic22—22no—
47C2×Dic22—22no—
48C2×Dic22—22no—
49C11×C4⋊C4—22no—
50C42×C11—22no—
51(C2×D11)⋊C4—24no—
52(C2×D11)⋊C4—24no—
53(C2×D11)⋊C4—24no—
54C4⋊Dic11—24no—
55C4⋊Dic11—22no—
56Dic11⋊C4—24no—
57Dic11⋊C4—28no—
58Dic11⋊C4—24no—
59C4×Dic11D414yes—
60#2621—111nomaximal, Sylow
61C23×C11C2411yescharacteristic
62C22×D11—21no—
63C22×D11—21no—
64C22×D11—22no—
65C22×D11—22no—
66C22×D11C2×D412yes—
67Q8×C11—44no—
68C2×C4×C11—22no—
69C2×C4×C11—22no—
70C2×C4×C11—22no—
71C2×C4×C11—22no—
72C2×C4×C11—22no—
73C2×C4×C11—22no—
74C2×C4×C11—22no—
75C2×C4×C11—24no—
76C2×C4×C11C2×D412yes—
77C2×C4×C11C2×D412yes—
78C2×C4×C11#4112yes—
79C2×Dic11—21no—
80C2×Dic11—21no—
81C2×Dic11—22no—
82C2×Dic11—22no—
83C2×Dic11—22no—
84C2×Dic11—22no—
85C2×Dic11—22no—
86C2×Dic11—22no—
87C2×Dic11—24no—
88C2×Dic11C2×D414yes—
89C2×Dic11C2×D412yes—
90C4×D11—28no—
91Dic22—44no—
92C2×C4⋊Q8—111no—
93C2×C22⋊Q8—111no—
94C2×C22⋊Q8—111no—
95C2×#117—111no—
96#398—114no—
97#397—114no—
98#391—111no—
99#391D1111yescharacteristic, complemented
100#382—111no—
101C22×C11C22×D411yescharacteristic
102C22×C11C22×D411yescharacteristic
103C22×C11C2×#4111yescharacteristic
104C22×C112−1+412yes—
105C22×C112−1+412yes—
106C2×D11—22no—
107C2×D11—22no—
108C2×D11—22no—
109C2×D11—42no—
110C2×D11—42no—
111C4×C11—42no—
112C4×C11—42no—
113C4×C11—42no—
114C4×C11#12114yes—
115Dic11—22no—
116Dic11—24no—
117Dic11—42no—
118Dic11—42no—
119C22×Q8—111no—
120C22×Q8C2×D1111yescharacteristic
121C23×C4—111no—
122C4⋊Q8—224no—
123C22⋊Q8—224no—
124C22⋊Q8—224no—
125#117—224no—
126C2×C4⋊C4—112no—
127C2×C4⋊C4—111no—
128C2×C4⋊C4—112no—
129C2×C4⋊C4—112no—
130C2×C4⋊C4—114no—
131C2×C4⋊C4C2×D1111yescharacteristic
132C2×C22⋊C4—112no—
133C2×C22⋊C4—112no—
134C2×C22⋊C4—112no—
135C2×C42—111no—
136C2×C42C2×D1111yescharacteristic
137#95—114no—
138#95—114no—
139#95C2×D1114yes—
140C2×C11C2×#12111yescharacteristic
141C2×C11#53611yescharacteristic
142C2×C11#53612yes—
143C2×C11#53611yescharacteristic
144C2×C11#54112yes—
145D11—42no—
146C24—111no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—222no—
151C2×Q8—222no—
152C2×Q8—222no—
153C22×C4—112no—
154C22×C4—111no—
155C22×C4—114no—
156C22×C4—112no—
157C22×C4—112no—
158C22×C4—222no—
159C22×C4—222no—
160C22×C4C22×D1112yes—
161C22×C4C22×D1112yes—
162C22×C4C22×D1112yes—
163C22×C4C22×D1111yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—224no—
166C4⋊C4—224no—
167C4⋊C4—224no—
168C4⋊C4—228no—
169C4⋊C4—222no—
170C22⋊C4—224no—
171C22⋊C4—224no—
172C22⋊C4—224no—
173C42—22no—
174C42—114no—
175C11#262111yescharacteristic, complemented, Sylow
176C23—112no—
177C23—221no—
178C23—221no—
179C23—222no—
180C23—222no—
181C23C23×D1111yescharacteristic
182Q8—44no—
183Q8—444no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—114no—
193C2×C4—112no—
194C2×C4—221no—
195C2×C4—221no—
196C2×C4—222no—
197C2×C4—222no—
198C2×C4—228no—
199C2×C4—222no—
200C2×C4—222no—
201C2×C4—222no—
202C2×C4—222no—
203C2×C4—224no—
204C2×C4#433912yes—
205C2×C4D4×D1112yes—
206C2×C4D4×D1112yes—
207C22—222no—
208C22—222no—
209C22—222no—
210C22—442no—
211C22—442no—
212C22C2×#433911yescharacteristic
213C22C2×D4×D1111yescharacteristic
214C22C2×D4×D1111yescharacteristic
215C22#6704012yes—
216C22#6704312yes—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—222no—
221C4—224no—
222C4—442no—
223C4—442no—
224C4#6693714yes—
225C2—442no—
226C2C2×#6693711yescharacteristic
227C2#1062311412yes—
228C2#1062315912yes—
229C2#1062320611yescharacteristic
230C2#1062325511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
prime radicalC23×D11C23
hypercenter#391D11
hypocenterC11#2621
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#66997C2semidirect
5. C11×#391C2semidirect
6. #10622590C2semidirect
7. #10622542C2semidirect
9. #10622270C2semidirect
99. #391D11semidirect
175. C11#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#669992
C2×#669972
C2×#669502
C2×#669252
C11×#3912
#106225902
#106225422
#106222882
#106222702
#262111