Finite Group Databasebeta

← #13074371

Subgroups — #13074371

Order 1408.

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

normal subgroups only

This group has 728 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×#66993C211yesmaximal, characteristic, complemented
2C2×#66991C211yesmaximal, characteristic, complemented
3C2×#66948C211yesmaximal, characteristic, complemented
4C2×#66925C211yesmaximal, characteristic, complemented
5C11×#390C211yesmaximal, characteristic, complemented
6#10622590C211yesmaximal, characteristic, complemented
7#10622573C214yesmaximal, complemented
8#10622572C214yesmaximal, complemented
9#10622552C211yesmaximal, characteristic, complemented
10C22×D4×C11C2211yescharacteristic
11C22×C22⋊D11C2212yes—
12C22×C22⋊D11C2214yes—
13C22×D44C2211yescharacteristic
14C22×D44C2212yes—
15C22×C4×D11C2211yescharacteristic
16C2×C11×C4⋊C4C2211yescharacteristic
17C2×C11×C22⋊C4C2214yes—
18C2×C42×C11C2211yescharacteristic
19C2×C22⋊Dic11C2212yes—
20#66993—24no—
21#66991—24no—
22C2×(C2×D11)⋊C4C2212yes—
23C2×(C2×D11)⋊C4C2212yes—
24C2×(C2×D11)⋊C4C2214yes—
25C2×C4⋊Dic11C2211yescharacteristic
26C2×Dic11⋊C4C2212yes—
27C2×C4×Dic11C2211yescharacteristic
28#66948—24no—
29#66925—24no—
30C11⋊#95C2214yes—
31C24×C11C2312yes—
32C23×D11C2312yes—
33C23×D11C2311yescharacteristic
34C2×D4×C11—22no—
35C2×D4×C11—22no—
36C2×D4×C11—22no—
37C22×C4×C11C2312yes—
38C22×C4×C11C2312yes—
39C22×C4×C11C2311yescharacteristic
40C2×C22⋊D11—28no—
41C2×C22⋊D11—24no—
42C2×C22⋊D11—44no—
43C2×C22⋊D11—44no—
44C2×C22⋊D11—44no—
45C22×Dic11C2312yes—
46C22×Dic11C2312yes—
47C22×Dic11C2311yescharacteristic
48C2×D44—24no—
49C2×D44—22no—
50C2×D44—22no—
51C2×D44—22no—
52C2×D44—44no—
53C2×C4×D11—22no—
54C2×C4×D11—22no—
55C2×C4×D11D414yescomplemented
56C11×C4⋊C4—22no—
57C11×C22⋊C4—44no—
58C42×C11—22no—
59C22⋊Dic11—24no—
60(C2×D11)⋊C4—24no—
61(C2×D11)⋊C4—24no—
62(C2×D11)⋊C4—44no—
63C4⋊Dic11—22no—
64Dic11⋊C4—24no—
65C4×Dic11D414yescomplemented
66#2619—111nomaximal, Sylow
67C23×C11—22no—
68C23×C11—22no—
69C23×C11—22no—
70C23×C11—24no—
71C23×C11—24no—
72C23×C11C2411yescharacteristic
73C22×D11—21no—
74C22×D11—21no—
75C22×D11—22no—
76C22×D11—22no—
77C22×D11—22no—
78C22×D11—22no—
79C22×D11—22no—
80C22×D11—24no—
81C22×D11—24no—
82C22×D11C2×D412yes—
83D4×C11—44no—
84C2×C4×C11—22no—
85C2×C4×C11—22no—
86C2×C4×C11—22no—
87C2×C4×C11—22no—
88C2×C4×C11—24no—
89C2×C4×C11C2×D412yes—
90C2×C4×C11C2×D412yes—
91C2×C4×C11#4112yes—
92C22⋊D11—416no—
93C22⋊D11—48no—
94C2×Dic11—21no—
95C2×Dic11—21no—
96C2×Dic11—22no—
97C2×Dic11—22no—
98C2×Dic11—22no—
99C2×Dic11—24no—
100C2×Dic11C2×D414yes—
101C2×Dic11C2×D412yes—
102D44—48no—
103D44—44no—
104C4×D11—28no—
105C2×#127—111no—
106C2×#121—111no—
107C2×#121—111no—
108C2×#117—111no—
109#394—114no—
110#392—114no—
111#390—111no—
112#390D1111yescharacteristic, complemented
113#382—111no—
114C22×C11—42no—
115C22×C11—42no—
116C22×C11—44no—
117C22×C11—42no—
118C22×C11—44no—
119C22×C11C22×D411yescharacteristic
120C22×C11C22×D411yescharacteristic
121C22×C11C2×#4111yescharacteristic
122C22×C11H(5,𝔽2)12yes—
123C22×C11H(5,𝔽2)12yes—
124C2×D11—22no—
125C2×D11—22no—
126C2×D11—22no—
127C2×D11—42no—
128C2×D11—42no—
129C2×D11—42no—
130C2×D11—42no—
131C2×D11—42no—
132C2×D11—44no—
133C2×D11—44no—
134C4×C11—42no—
135C4×C11—42no—
136C4×C11#12114yes—
137Dic11—22no—
138Dic11—24no—
139Dic11—42no—
140C22×D4—112no—
141C22×D4—114no—
142C22×D4—111no—
143C22×D4—112no—
144C22×D4C2×D1111yescharacteristic
145C23×C4—111no—
146#127—224no—
147#121—224no—
148#121—224no—
149#117—224no—
150C2×C4⋊C4—112no—
151C2×C4⋊C4—111no—
152C2×C4⋊C4C2×D1111yescharacteristic
153C2×C22⋊C4—112no—
154C2×C22⋊C4—112no—
155C2×C22⋊C4—112no—
156C2×C22⋊C4—114no—
157C2×C22⋊C4C2×D1114yes—
158C2×C42—111no—
159C2×C42C2×D1111yescharacteristic
160#95—114no—
161C2×C11—82no—
162C2×C11C2×#12111yescharacteristic
163C2×C11#53511yescharacteristic
164C2×C11#53512yes—
165C2×C11#53511yescharacteristic
166C2×C11#54012yes—
167D11—42no—
168D11—82no—
169C24—111no—
170C24—112no—
171C24C22×D1112yes—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—224no—
176C2×D4—228no—
177C2×D4—224no—
178C2×D4—222no—
179C2×D4—222no—
180C2×D4—222no—
181C2×D4—444no—
182C2×D4—444no—
183C2×D4—444no—
184C2×D4—444no—
185C22×C4—112no—
186C22×C4—111no—
187C22×C4—114no—
188C22×C4—112no—
189C22×C4—222no—
190C22×C4—222no—
191C22×C4C22×D1112yes—
192C22×C4C22×D1112yes—
193C22×C4C22×D1111yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—224no—
196C4⋊C4—222no—
197C22⋊C4—44no—
198C22⋊C4—224no—
199C22⋊C4—224no—
200C22⋊C4—224no—
201C22⋊C4—444no—
202C42—22no—
203C42—114no—
204C11#261911yescharacteristic, complemented, Sylow
205C23—22no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—112no—
211C23—221no—
212C23—221no—
213C23—222no—
214C23—222no—
215C23—222no—
216C23—222no—
217C23—222no—
218C23—224no—
219C23—224no—
220C23C23×D1111yescharacteristic
221D4—44no—
222D4—448no—
223D4—4416no—
224D4—448no—
225D4—444no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—114no—
232C2×C4—112no—
233C2×C4—221no—
234C2×C4—221no—
235C2×C4—222no—
236C2×C4—222no—
237C2×C4—228no—
238C2×C4—222no—
239C2×C4—224no—
240C2×C4#433912yes—
241C2×C4D4×D1112yes—
242C2×C4D4×D1112yes—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—44no—
247C22—42no—
248C22—222no—
249C22—222no—
250C22—222no—
251C22—442no—
252C22—442no—
253C22—442no—
254C22—442no—
255C22—444no—
256C22—444no—
257C22—442no—
258C22C2×#433911yescharacteristic
259C22C2×D4×D1111yescharacteristic
260C22C2×D4×D1111yescharacteristic
261C22#6703712yes—
262C22#6704212yes—
263C4—42no—
264C4—42no—
265C4—222no—
266C4—224no—
267C4—442no—
268C4#6693714yes—
269C2—82no—
270C2—442no—
271C2—882no—
272C2C2×#6693711yescharacteristic
273C2#1062309312yes—
274C2#1062313812yes—
275C2#1062320511yescharacteristic
276C2#1062324611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#390C2
hypercenter#390D11
hypocenterC11#2619
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#66993C2semidirect
2. C2×#66991C2semidirect
3. C2×#66948C2semidirect
4. C2×#66925C2semidirect
5. C11×#390C2semidirect
6. #10622590C2semidirect
7. #10622573C2semidirect
8. #10622572C2semidirect
9. #10622552C2semidirect
55. C2×C4×D11D4semidirect
65. C4×Dic11D4semidirect
112. #390D11semidirect
204. C11#2619semidirect

Maximal subgroups

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

subgroupindex
C2×#669932
C2×#669912
C2×#669482
C2×#669252
C11×#3902
#106225902
#106225732
#106225722
#106225522
#261911