Finite Group Databasebeta

← #13075582

Subgroups — #13075582

Order 1408.

512 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 133 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×#66931C212yesmaximal, complemented
2C11×#394C211yesmaximal, characteristic
3#10622833C212yesmaximal
4#10622596C211yesmaximal, characteristic, complemented
5#10622587C211yesmaximal, characteristic, complemented
6#10622565C212yesmaximal, complemented
7#10622563C212yesmaximal, complemented
8#10622558C212yesmaximal, complemented
9#10622273C212yesmaximal, complemented
10C22×D4×C11C2211yescharacteristic
11C23×Dic11C2212yes—
12C22×Dic22C2211yescharacteristic
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×C11×C22⋊C4C2212yes—
15C2×C11×C22⋊C4C2212yes—
16C2×C22⋊Dic11C2212yes—
17C2×C22⋊Dic11C2212yes—
18C2×C22⋊Dic11C2212yes—
19C2×C4⋊Dic11C2212yes—
20C2×C4⋊Dic11C2211yescharacteristic
21C2×Dic11⋊C4C2212yes—
22C2×Dic11⋊C4C2212yes—
23C2×C4×Dic11C2211yescharacteristic
24#66931—28no—
25C11×#95C2211yescharacteristic
26C11⋊#95C2212yes—
27C11⋊#95C2212yes—
28C11⋊#95C2212yes—
29C11⋊#95C2212yes—
30C11⋊#95C2212yes—
31C11⋊#95C2211yescharacteristic
32C24×C11C2312yes—
33C2×D4×C11—24no—
34C2×D4×C11—41no—
35C22×C4×C11C2311yescharacteristic
36C22×C4×C11C2311yescharacteristic
37C22×C4×C11C2312yes—
38C22×C4×C11C2311yescharacteristic
39C22×Dic11—24no—
40C22×Dic11—42no—
41C22×Dic11—42no—
42C22×Dic11C2312yes—
43C22×Dic11C2312yes—
44C22×Dic11C2312yes—
45C22×Dic11C2312yes—
46C2×Dic22—41no—
47C2×Dic22—41no—
48C2×Dic22D414yescomplemented
49C11×C4⋊C4—41no—
50C11×C22⋊C4—24no—
51C11×C22⋊C4—24no—
52C22⋊Dic11—24no—
53C22⋊Dic11—24no—
54C22⋊Dic11—42no—
55C4⋊Dic11—24no—
56C4⋊Dic11—41no—
57Dic11⋊C4—24no—
58Dic11⋊C4—24no—
59C4×Dic11—41no—
60#2728—111nomaximal, Sylow
61C23×C11—22no—
62C23×C11—22no—
63C23×C11—22no—
64C23×C11—22no—
65C23×C11—22no—
66C23×C11—22no—
67C23×C11#4114yes—
68C23×C11C2411yescharacteristic
69D4×C11—44no—
70C2×C4×C11—21no—
71C2×C4×C11—21no—
72C2×C4×C11—22no—
73C2×C4×C11—22no—
74C2×C4×C11—22no—
75C2×C4×C11—22no—
76C2×C4×C11—21no—
77C2×C4×C11—21no—
78C2×C4×C11—22no—
79C2×C4×C11C2×D412yes—
80C2×Dic11—22no—
81C2×Dic11—22no—
82C2×Dic11—22no—
83C2×Dic11—22no—
84C2×Dic11—22no—
85C2×Dic11—22no—
86C2×Dic11—22no—
87C2×Dic11—22no—
88C2×Dic11—22no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—48no—
92C2×Dic11C2×D414yes—
93Dic22—44no—
94C2×C22⋊Q8—112no—
95#400—112no—
96#398—111no—
97#397—112no—
98#394D1111yescharacteristic
99#391—111no—
100#388—112no—
101#386—112no—
102#385—112no—
103C22×C11—24no—
104C22×C11—24no—
105C22×C11—24no—
106C22×C11—42no—
107C22×C11—42no—
108C22×C11—42no—
109C22×C11—42no—
110C22×C11C22×D411yescharacteristic
111C22×C11C2×#4112yes—
112C22×C11H(5,𝔽2)11yescharacteristic
113C22×C11H(5,𝔽2)11yescharacteristic
114C22×C112−1+412yes—
115C4×C11—22no—
116C4×C11—42no—
117C4×C11—41no—
118C4×C11—41no—
119Dic11—24no—
120Dic11—42no—
121Dic11—42no—
122Dic11—42no—
123C22×Q8—111no—
124C22×D4C2×D1111yescharacteristic
125C23×C4—112no—
126C22⋊Q8—228no—
127C2×C4⋊C4—112no—
128C2×C4⋊C4—112no—
129C2×C4⋊C4—112no—
130C2×C4⋊C4—111no—
131C2×C4⋊C4C2×D1111yescharacteristic
132C2×C22⋊C4—112no—
133C2×C22⋊C4—112no—
134C2×C22⋊C4—112no—
135C2×C22⋊C4C2×D1112yes—
136C2×C22⋊C4C2×D1112yes—
137C2×C42—111no—
138#95—112no—
139#95—112no—
140#95—112no—
141#95—112no—
142#95—112no—
143#95—111no—
144#95C2×D1111yescharacteristic
145C2×C11—44no—
146C2×C11#53411yescharacteristic
147C2×C11#53912yes—
148C2×C11#54812yes—
149C2×C11#55111yescharacteristic
150C2×C11#56311yescharacteristic
151C24C22×D1112yes—
152C2×Q8—114no—
153C2×Q8—441no—
154C2×Q8—441no—
155C2×D4—24no—
156C2×D4—41no—
157C22×C4—112no—
158C22×C4—112no—
159C22×C4—112no—
160C22×C4—112no—
161C22×C4—224no—
162C22×C4—442no—
163C22×C4—442no—
164C22×C4C22×D1111yescharacteristic
165C22×C4C22×D1111yescharacteristic
166C22×C4C22×D1112yes—
167C22×C4C22×D1111yescharacteristic
168C4⋊C4—41no—
169C4⋊C4—224no—
170C4⋊C4—224no—
171C4⋊C4—224no—
172C4⋊C4—441no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—224no—
176C22⋊C4—224no—
177C22⋊C4—442no—
178C42—441no—
179C11#272811yescharacteristic, complemented, Sylow
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23#434114yes—
187C23C23×D1111yescharacteristic
188Q8—444no—
189D4—44no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—114no—
200C2×C4—222no—
201C2×C4—222no—
202C2×C4—222no—
203C2×C4—222no—
204C2×C4—222no—
205C2×C4—222no—
206C2×C4—222no—
207C2×C4—222no—
208C2×C4—222no—
209C2×C4—222no—
210C2×C4—222no—
211C2×C4—448no—
212C2×C4D4×D1112yes—
213C22—24no—
214C22—24no—
215C22—24no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22C2×D4×D1111yescharacteristic
221C22C2×#434112yes—
222C22#6703711yescharacteristic
223C22#6704211yescharacteristic
224C22#6704312yes—
225C4—22no—
226C4—41no—
227C4—41no—
228C4—42no—
229C4—224no—
230C4—442no—
231C4—442no—
232C4—442no—
233C2—44no—
234C2#1062311811yescharacteristic
235C2#1062312912yes—
236C2#1062313611yescharacteristic
237C2#1062319612yes—
238C2#1062320311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#394C2
prime radicalC22×D4×C11C22
hypercenter#394D11
hypocenterC11#2728
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#66931C2semidirect
4. #10622596C2semidirect
5. #10622587C2semidirect
6. #10622565C2semidirect
7. #10622563C2semidirect
8. #10622558C2semidirect
9. #10622273C2semidirect
48. C2×Dic22D4semidirect
179. C11#2728semidirect

Maximal subgroups

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

subgroupindex
C2×#669312
C11×#3942
#106228332
#106225962
#106225872
#106225652
#106225632
#106225582
#106222732
#272811