Finite Group Databasebeta

← #13077912

Subgroups — #13077912

Order 1408.

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

normal subgroups only

This group has 700 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×#67040C211yesmaximal, characteristic
2C22×#4326C211yesmaximal, characteristic, complemented
3C2×#66962C211yesmaximal, characteristic, complemented
4#10623057C211yesmaximal, characteristic, complemented
5C11×#419C211yesmaximal, characteristic, complemented
6#10622742C211yesmaximal, characteristic, complemented
7#10622615C211yesmaximal, characteristic, complemented
8#10622423C214yesmaximal, complemented
9#10622421C214yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11#67040—24no—
12#67040—24no—
13C2×#4343—21no—
14C2×#4343C2212yes—
15C2×Q8×D11—21no—
16C2×Q8×D11C2212yes—
17C2×#4339—22no—
18C2×#4339C2211yescharacteristic
19C2×#4339C2211yescharacteristic
20C22×Dic22C2211yescharacteristic
21C2×C11×M16C2211yescharacteristic
22C11×#117C2211yescharacteristic
23C2×#4326—24no—
24C2×#4326C2214yes—
25C22×C11⋊C8C2211yescharacteristic
26#66962—24no—
27C2×Dic44C2212yes—
28C2×#4314C2212yes—
29#66950C2214yes—
30#66939—22no—
31#66928—21no—
32#66922—21no—
33C11×#103C2214yes—
34(C2×D11)⋊C8C2214yes—
35#66874C2212yes—
36#66873C2212yes—
37C2×Q8×C11—21no—
38C2×Q8×C11—22no—
39C2×Q8×C11D414yes—
40C2×Q8×C11C2312yes—
41C22×C4×C11—21no—
42C22×C4×C11C2311yescharacteristic
43C2×C22⋊D11—21no—
44C2×C22⋊D11D412yes—
45C22×Dic11—21no—
46#4343—28no—
47#4343—24no—
48#4343—24no—
49Q8×D11—28no—
50Q8×D11—24no—
51Q8×D11—24no—
52#4339—216no—
53#4339—22no—
54#4339—22no—
55#4339—22no—
56#4339D414yes—
57C2×D44—22no—
58C2×D44D411yescharacteristic
59C2×D44C2311yescharacteristic
60C2×C4×D11—22no—
61C2×C4×D11—22no—
62C2×C4×D11—21no—
63C2×C4×D11C2312yes—
64C2×Dic22—22no—
65C2×Dic22—22no—
66C2×Dic22—22no—
67C2×Dic22—21no—
68C2×Dic22D411yescharacteristic
69C2×Dic22C2312yes—
70C2×Dic22C2311yescharacteristic
71C11×M16—22no—
72C2×C8×C11C2312yes—
73C11×C4⋊C4C2312yes—
74C11×C22⋊C4—21no—
75C42×C11—21no—
76#4326—28no—
77(C2×D11)⋊C4—24no—
78C4⋊Dic11—24no—
79C2×C11⋊C8—22no—
80C2×C11⋊C8C2312yes—
81Dic44—24no—
82#4314—24no—
83#3011—111nomaximal, Sylow
84C23×C11C2×D411yescharacteristic
85C22×D11—21no—
86C22×D11C2×D412yes—
87Q8×C11—24no—
88Q8×C11—22no—
89Q8×C11C2×D414yes—
90C2×C4×C11—24no—
91C2×C4×C11—22no—
92C2×C4×C11—22no—
93C2×C4×C11C2×D412yes—
94C2×C4×C11C2×D411yescharacteristic
95C2×C4×C11C2×D412yes—
96C2×C4×C11C2411yescharacteristic
97C22⋊D11—24no—
98C22⋊D11—24no—
99C2×Dic11—21no—
100C2×Dic11—22no—
101C2×Dic11—42no—
102C2×Dic11C2×D412yes—
103D44—28no—
104D44—22no—
105D44—21no—
106D44C2×D412yes—
107C4×D11—28no—
108C4×D11—28no—
109C4×D11—24no—
110C4×D11—22no—
111C4×D11—22no—
112Dic22—28no—
113Dic22—22no—
114Dic22—21no—
115Dic22—22no—
116Dic22—22no—
117Dic22—42no—
118Dic22C2×D412yes—
119C8×C11—22no—
120C11⋊C8—22no—
121C2×2−1+4—111no—
122C2×#137—111no—
123C22×Q16—111no—
124#536—111no—
125#452—114no—
126#451—114no—
127#419D1111yescharacteristic, complemented
128#416—111no—
129#408—111no—
130C22×C11—22no—
131C22×C11#12012yes—
132C22×C11C22×D411yescharacteristic
133C2×D11—22no—
134C2×D11—22no—
135C2×D11—24no—
136C2×D11—42no—
137C4×C11—24no—
138C4×C11—42no—
139C4×C11#12012yes—
140C4×C11C22×D411yescharacteristic
141C4×C11C22×D411yescharacteristic
142Dic11—22no—
143Dic11—22no—
144Dic11—42no—
1452−1+4—224no—
1462−1+4—224no—
147C2×#41—111no—
148C2×#41—112no—
149C2×#41—111no—
150C2×#41—222no—
151C2×#41—221no—
152C22×Q8—112no—
153C22×Q8—111no—
154C22×Q8—221no—
155C22×Q8C2×D1111yescharacteristic
156#137—224no—
157C2×Q16—114no—
158C2×Q16—112no—
159C2×Q16—224no—
160C2×SD16—112no—
161C2×M16C2×D1111yescharacteristic
162C22×C8—111no—
163C4⋊Q8—221no—
164#124—221no—
165#123—222no—
166C22⋊Q8—114no—
167#117C2×D1111yescharacteristic
168#103—112no—
169#103C2×D1114yes—
170#102—112no—
171C22⋊C8—114no—
172C2×C11—22no—
173C2×C11C2×#12011yescharacteristic
174C2×C11#57812yes—
175D11—42no—
176D11—42no—
177#41—114no—
178#41—2216no—
179#41—224no—
180#41—228no—
181#41—224no—
182#41—222no—
183#41—222no—
184#41—222no—
185C2×Q8—21no—
186C2×Q8—22no—
187C2×Q8—111no—
188C2×Q8—111no—
189C2×Q8—112no—
190C2×Q8—222no—
191C2×Q8—224no—
192C2×Q8—228no—
193C2×Q8—224no—
194C2×Q8—221no—
195C2×Q8—222no—
196C2×Q8—222no—
197C2×Q8D4414yes—
198C2×Q8C22×D1112yes—
199C2×D4—112no—
200C2×D4—111no—
201C2×D4—111no—
202C2×D4—222no—
203C2×D4—221no—
204C22×C4—21no—
205C22×C4—112no—
206C22×C4—222no—
207C22×C4—222no—
208C22×C4—221no—
209C22×C4—221no—
210C22×C4C22×D1111yescharacteristic
211Q16—224no—
212Q16—228no—
213SD16—224no—
214M16—22no—
215C2×C8—112no—
216C2×C8—222no—
217C2×C8C22×D1112yes—
218C4⋊C4—224no—
219C4⋊C4C22×D1112yes—
220C22⋊C4—21no—
221C22⋊C4—224no—
222C42—21no—
223C11#301111yescharacteristic, complemented, Sylow
224C23—112no—
225C23—221no—
226C23D4×D1111yescharacteristic
227Q8—24no—
228Q8—22no—
229Q8—112no—
230Q8—228no—
231Q8—222no—
232Q8—221no—
233Q8—222no—
234Q8—222no—
235Q8—442no—
236Q8C2×D4414yes—
237D4—112no—
238D4—228no—
239D4—224no—
240D4—224no—
241D4—222no—
242D4—221no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—112no—
247C2×C4—228no—
248C2×C4—228no—
249C2×C4—221no—
250C2×C4—224no—
251C2×C4—222no—
252C2×C4—222no—
253C2×C4—222no—
254C2×C4—442no—
255C2×C4C2×D4412yes—
256C2×C4D4×D1111yescharacteristic
257C2×C4D4×D1112yes—
258C2×C4C23×D1111yescharacteristic
259C8—22no—
260C8—222no—
261C22—22no—
262C22—222no—
263C22—222no—
264C22—224no—
265C22—442no—
266C22#6693512yes—
267C22C2×D4×D1111yescharacteristic
268C4—24no—
269C4—42no—
270C4—222no—
271C4—222no—
272C4—442no—
273C4#6693512yes—
274C4C22×D4411yescharacteristic
275C4C2×D4×D1111yescharacteristic
276C2—22no—
277C2—442no—
278C2—442no—
279C2C2×#6693511yescharacteristic
280C2#1062331012yes—
subgroupG/H
centerC22C2×D4×D11
commutator, cosocleC2×C4×C11C24
FrattiniC2×C4C23×D11
FittingC11×#419C2
socleC22×C11C22×D4
prime radicalC2×#67040C2
hypercenter#419D11
hypocenterC11#3011
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#4326C2semidirect
3. C2×#66962C2semidirect
4. #10623057C2semidirect
5. C11×#419C2semidirect
6. #10622742C2semidirect
7. #10622615C2semidirect
8. #10622423C2semidirect
9. #10622421C2semidirect
127. #419D11semidirect
223. C11#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#670402
C22×#43262
C2×#669622
#106230572
C11×#4192
#106227422
#106226152
#106224232
#106224212
#301111