Finite Group Databasebeta

← #13074758

Subgroups — #13074758

Order 1408.

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

normal subgroups only

This group has 534 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×#66989C211yesmaximal, characteristic, complemented
2C2×#66932C211yesmaximal, characteristic, complemented
3C11×#392C211yesmaximal, characteristic
4#10622835C211yesmaximal, characteristic, complemented
5#10622835C212yesmaximal, complemented
6#10622834C212yesmaximal, complemented
7#10622564C212yesmaximal, complemented
8#10622561C212yesmaximal, complemented
9#10622560C211yesmaximal, characteristic, complemented
10#10622269C211yesmaximal, characteristic, complemented
11#10622262C211yesmaximal, characteristic, complemented
12C22×D4×C11C2211yescharacteristic
13C22×D4×C11C2212yes—
14C22×Dic22C2211yescharacteristic
15C2×C11×C22⋊C4C2211yescharacteristic
16C2×C11×C22⋊C4C2212yes—
17C2×C22⋊Dic11C2211yescharacteristic
18C2×C22⋊Dic11C2211yescharacteristic
19C2×C22⋊Dic11C2212yes—
20C2×C22⋊Dic11C2212yes—
21C2×C22⋊Dic11C2212yes—
22C2×C22⋊Dic11C2212yes—
23C2×C22⋊Dic11C2212yes—
24#66989—24no—
25C2×C4⋊Dic11C2211yescharacteristic
26C2×C4⋊Dic11C2212yes—
27C2×Dic11⋊C4C2211yescharacteristic
28C2×Dic11⋊C4C2211yescharacteristic
29C2×Dic11⋊C4C2212yes—
30C2×C4×Dic11C2212yes—
31C2×C4×Dic11C2211yescharacteristic
32#66932—24no—
33C11×#95C2211yescharacteristic
34C11⋊#95C2211yescharacteristic
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C24×C11C2311yescharacteristic
38C24×C11C2312yes—
39C2×D4×C11—22no—
40C2×D4×C11—24no—
41C2×D4×C11—42no—
42C2×D4×C11—42no—
43C2×D4×C11—42no—
44C22×C4×C11C2311yescharacteristic
45C22×C4×C11C2311yescharacteristic
46C22×C4×C11C2312yes—
47C22×Dic11C2312yes—
48C22×Dic11C2311yescharacteristic
49C22×Dic11C2312yes—
50C22×Dic11C2311yescharacteristic
51C22×Dic11C2311yescharacteristic
52C22×Dic11C2311yescharacteristic
53C2×Dic22—22no—
54C2×Dic22—42no—
55C11×C22⋊C4—22no—
56C11×C22⋊C4—24no—
57C22⋊Dic11—22no—
58C22⋊Dic11—22no—
59C22⋊Dic11—24no—
60C22⋊Dic11—24no—
61C22⋊Dic11—42no—
62C22⋊Dic11—42no—
63C22⋊Dic11—42no—
64C4⋊Dic11—22no—
65C4⋊Dic11—42no—
66Dic11⋊C4—22no—
67Dic11⋊C4—22no—
68Dic11⋊C4—24no—
69C4×Dic11—42no—
70C4×Dic11D414yescomplemented
71#2662—111nomaximal, Sylow
72C23×C11—21no—
73C23×C11—22no—
74C23×C11—22no—
75C23×C11—22no—
76C23×C11—22no—
77C23×C11—22no—
78C23×C11—22no—
79C23×C11—22no—
80C23×C11—22no—
81C23×C11—21no—
82C23×C11—21no—
83C23×C11—22no—
84C23×C11C2411yescharacteristic
85D4×C11—44no—
86D4×C11—48no—
87C2×C4×C11—22no—
88C2×C4×C11—22no—
89C2×C4×C11—22no—
90C2×C4×C11—21no—
91C2×C4×C11—22no—
92C2×C4×C11C2×D412yes—
93C2×C4×C11#4114yes—
94C2×Dic11—22no—
95C2×Dic11—21no—
96C2×Dic11—21no—
97C2×Dic11—21no—
98C2×Dic11—21no—
99C2×Dic11—22no—
100C2×Dic11—22no—
101C2×Dic11—22no—
102C2×Dic11—22no—
103C2×Dic11—22no—
104C2×Dic11—21no—
105C2×Dic11—22no—
106C2×Dic11—22no—
107C2×Dic11—21no—
108C2×Dic11C2×D412yes—
109C2×Dic11C2×D412yes—
110Dic22—44no—
111C2×#126—111no—
112C2×#124—111no—
113#396—112no—
114#395—111no—
115#394—111no—
116#394—112no—
117#393—111no—
118#392D1111yescharacteristic
119#390—112no—
120#388—112no—
121C4×C4⋊C4—111no—
122C22×C11—41no—
123C22×C11—42no—
124C22×C11—42no—
125C22×C11—41no—
126C22×C11—41no—
127C22×C11—42no—
128C22×C11—42no—
129C22×C11—42no—
130C22×C11—42no—
131C22×C11—42no—
132C22×C11—42no—
133C22×C11—42no—
134C22×C11C22×D411yescharacteristic
135C22×C11C2×#4112yes—
136C22×C11H(5,𝔽2)12yes—
137C22×C11H(5,𝔽2)11yescharacteristic
138C22×C112−1+411yescharacteristic
139C4×C11—22no—
140C4×C11—24no—
141C4×C11—41no—
142Dic11—22no—
143Dic11—22no—
144Dic11—42no—
145Dic11—41no—
146Dic11—41no—
147Dic11—42no—
148C22×Q8—111no—
149C22×D4C2×D1111yescharacteristic
150C22×D4C2×D1112yes—
151#126—224no—
152#124—224no—
153C2×C4⋊C4—111no—
154C2×C4⋊C4—111no—
155C2×C4⋊C4—111no—
156C2×C4⋊C4—112no—
157C2×C4⋊C4—112no—
158C2×C22⋊C4—112no—
159C2×C22⋊C4—112no—
160C2×C22⋊C4—111no—
161C2×C22⋊C4—111no—
162C2×C22⋊C4—112no—
163C2×C22⋊C4—112no—
164C2×C22⋊C4—112no—
165C2×C22⋊C4C2×D1111yescharacteristic
166C2×C22⋊C4C2×D1112yes—
167C2×C42—111no—
168C2×C42—112no—
169#95—111no—
170#95—112no—
171#95—112no—
172#95C2×D1111yescharacteristic
173C2×C11—82no—
174C2×C11—81no—
175C2×C11#53211yescharacteristic
176C2×C11#53511yescharacteristic
177C2×C11#53711yescharacteristic
178C2×C11#53812yes—
179C2×C11#54212yes—
180C24C22×D1111yescharacteristic
181C24C22×D1112yes—
182C2×Q8—222no—
183C2×Q8—442no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C22×C4—111no—
190C22×C4—111no—
191C22×C4—112no—
192C22×C4—111no—
193C22×C4—112no—
194C22×C4—111no—
195C22×C4C22×D1111yescharacteristic
196C22×C4C22×D1111yescharacteristic
197C22×C4C22×D1112yes—
198C4⋊C4—222no—
199C4⋊C4—222no—
200C4⋊C4—222no—
201C4⋊C4—224no—
202C4⋊C4—442no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—224no—
206C22⋊C4—224no—
207C22⋊C4—222no—
208C22⋊C4—222no—
209C22⋊C4—442no—
210C22⋊C4—442no—
211C22⋊C4—442no—
212C42—114no—
213C42—442no—
214C11#266211yescharacteristic, complemented, Sylow
215C23—21no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—22no—
226C23—21no—
227C23C23×D1111yescharacteristic
228Q8—444no—
229D4—44no—
230D4—48no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—112no—
237C2×C4—112no—
238C2×C4—222no—
239C2×C4—221no—
240C2×C4—221no—
241C2×C4—222no—
242C2×C4—221no—
243C2×C4—221no—
244C2×C4—222no—
245C2×C4—222no—
246C2×C4—222no—
247C2×C4—221no—
248C2×C4—222no—
249C2×C4—222no—
250C2×C4—222no—
251C2×C4—221no—
252C2×C4D4×D1112yes—
253C2×C4#434114yes—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—41no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22C2×D4×D1111yescharacteristic
267C22C2×#434112yes—
268C22#6703712yes—
269C22#6703711yescharacteristic
270C22#6704311yescharacteristic
271C4—22no—
272C4—24no—
273C4—41no—
274C4—222no—
275C4—222no—
276C4—442no—
277C4—441no—
278C4—442no—
279C4—441no—
280C2—82no—
281C2—81no—
282C2#1062312311yescharacteristic
283C2#1062312412yes—
284C2#1062313412yes—
285C2#1062313911yescharacteristic
286C2#1062324711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
Fitting, prime radicalC11×#392C2
hypercenter#392D11
hypocenterC11#2662
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#66989C2semidirect
2. C2×#66932C2semidirect
4. #10622835C2semidirect
5. #10622835C2semidirect
6. #10622834C2semidirect
7. #10622564C2semidirect
8. #10622561C2semidirect
9. #10622560C2semidirect
10. #10622269C2semidirect
11. #10622262C2semidirect
70. C4×Dic11D4semidirect
214. C11#2662semidirect

Maximal subgroups

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

subgroupindex
C2×#669892
C2×#669322
C11×#3922
#106228352
#106228342
#106225642
#106225612
#106225602
#106222692
#106222622
#266211