Finite Group Databasebeta

← #13079397

Subgroups — #13079397

Order 1408.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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×C11×C22⋊Q8C211yesmaximal, characteristic, complemented
2#10623322C211yesmaximal, characteristic
3#10622841C211yesmaximal, characteristic, complemented
4#10622840C211yesmaximal, characteristic, complemented
5C2×C22⋊(C11⋊C8)C211yesmaximal, characteristic, complemented
6#10622664C212yesmaximal, complemented
7#10622661C212yesmaximal, complemented
8#10622660C212yesmaximal, complemented
9#10622658C212yesmaximal, complemented
10#10622584C211yesmaximal, characteristic, complemented
11#10622580C211yesmaximal, characteristic, complemented
12C22×Q8×C11C2211yescharacteristic
13C23×C4×C11C2211yescharacteristic
14C2×#4339C2211yescharacteristic
15C11×C22⋊Q8—21no—
16C11×C22⋊Q8—21no—
17C11×C22⋊Q8C2212yes—
18C11×C22⋊Q8C2212yes—
19C2×C11×C4⋊C4—21no—
20C2×C11×C4⋊C4C2211yescharacteristic
21C2×C11×C22⋊C4—21no—
22#67006—21no—
23C2×#4326C2212yescomplemented
24C2×#4325C2212yescomplemented
25#66983C2211yescharacteristic
26#66983C2211yescharacteristic
27#66982—21no—
28C4×C22⋊D11—21no—
29C4×C22⋊D11—21no—
30#66979C2211yescharacteristic
31#66977C2211yescharacteristic
32#66977C2211yescharacteristic
33C22×C11⋊C8C2212yes—
34#66899C2212yescomplemented
35#66899C2212yescomplemented
36C22⋊(C11⋊C8)C2211yescharacteristic
37C22⋊(C11⋊C8)C2211yescharacteristic
38C22⋊(C11⋊C8)C2212yes—
39#66874C2212yescomplemented
40#66873C2212yescomplemented
41#66872C2212yescomplemented
42#66871C2212yescomplemented
43C24×C11D411yescharacteristic
44C2×Q8×C11—21no—
45C2×Q8×C11—42no—
46C2×Q8×C11C2312yes—
47C22×C4×C11—21no—
48C22×C4×C11—21no—
49C22×C4×C11—42no—
50C22×C4×C11D411yescharacteristic
51C22×C4×C11D411yescharacteristic
52C22×C4×C11D411yescharacteristic
53C22×C4×C11C2311yescharacteristic
54C22×C4×C11C2311yescharacteristic
55C22×C4×C11C2311yescharacteristic
56C2×C22⋊D11—21no—
57C2×C22⋊D11—21no—
58C2×C22⋊D11—21no—
59#4339—42no—
60C2×D44C2311yescharacteristic
61C2×C4×D11—21no—
62C2×Dic22C2311yescharacteristic
63C11×C4⋊C4—22no—
64C11×C4⋊C4—22no—
65C11×C4⋊C4—21no—
66C11×C4⋊C4C2312yes—
67C11×C22⋊C4—22no—
68C11×C22⋊C4—22no—
69C22⋊Dic11—21no—
70C22⋊Dic11—21no—
71C22⋊Dic11—21no—
72#4326—42no—
73#4325—42no—
74(C2×D11)⋊C4—21no—
75(C2×D11)⋊C4—21no—
76C4⋊Dic11C2311yescharacteristic
77C4⋊Dic11C2311yescharacteristic
78Dic11⋊C4—21no—
79Dic11⋊C4—21no—
80C4×Dic11—21no—
81C2×C11⋊C8—42no—
82C2×C11⋊C8C2312yes—
83C2×C11⋊C8C2312yes—
84#3190—111nomaximal, Sylow
85C23×C11—22no—
86C23×C11—22no—
87C23×C11—41no—
88C23×C11C2×D411yescharacteristic
89C23×C11C2×D411yescharacteristic
90C23×C11C2×D411yescharacteristic
91C22×D11—21no—
92Q8×C11—22no—
93Q8×C11—22no—
94Q8×C11—42no—
95C2×C4×C11—22no—
96C2×C4×C11—22no—
97C2×C4×C11—22no—
98C2×C4×C11—22no—
99C2×C4×C11—22no—
100C2×C4×C11—41no—
101C2×C4×C11—41no—
102C2×C4×C11—41no—
103C2×C4×C11—42no—
104C2×C4×C11—42no—
105C2×C4×C11C2×D411yescharacteristic
106C2×C4×C11C2×D411yescharacteristic
107C2×C4×C11C2×D411yescharacteristic
108C2×C4×C11C2411yescharacteristic
109C22⋊D11—42no—
110C22⋊D11—42no—
111C22⋊D11—42no—
112C2×Dic11—21no—
113C2×Dic11—21no—
114C2×Dic11—21no—
115D44—22no—
116C4×D11—42no—
117Dic22—22no—
118C11⋊C8—22no—
119C11⋊C8—22no—
120#525—111no—
121C2×C22⋊Q8D1111yescharacteristic, complemented
122#485—112no—
123#483—111no—
124#480—111no—
125#467—112no—
126#465—112no—
127#452—112no—
128#451—111no—
129#448—111no—
130C2×C22⋊C8—111no—
131C22×C11—21no—
132C22×C11—21no—
133C22×C11—21no—
134C22×C11—21no—
135C22×C11—21no—
136C22×C11—21no—
137C22×C11—21no—
138C22×C11—44no—
139C22×C11#13512yes—
140C22×C11#13712yes—
141C22×C11C22×D411yescharacteristic
142C2×D11—41no—
143C2×D11—41no—
144C2×D11—41no—
145C4×C11—21no—
146C4×C11—21no—
147C4×C11—21no—
148C4×C11—42no—
149C4×C11—42no—
150C4×C11H(5,𝔽2)12yes—
151Dic11—41no—
152Dic11—41no—
153Dic11—41no—
154C2×#41—111no—
155C22×Q8C2×D1111yescharacteristic
156C23×C4C2×D1111yescharacteristic
157C2×Q16—112no—
158C2×SD16—112no—
159C22×C8—112no—
160#123—221no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—111no—
164C22⋊Q8—111no—
165C22⋊Q8C2×D1112yes—
166C22⋊Q8C2×D1112yes—
167#121—111no—
168#121—111no—
169#120—221no—
170C4×D4—221no—
171C4×D4—221no—
172#117—111no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×D1111yescharacteristic
175C2×C22⋊C4—21no—
176#107—112no—
177#106—112no—
178#103—112no—
179#103—112no—
180#103—112no—
181#102—112no—
182C22⋊C8—111no—
183C22⋊C8—111no—
184C22⋊C8—112no—
185C2×C11—22no—
186C2×C11—22no—
187C2×C11—41no—
188C2×C11#53411yescharacteristic
189C2×C11C2×#13511yescharacteristic
190C2×C11C2×#13711yescharacteristic
191D11—81no—
192C24C22⋊D1111yescharacteristic
193#41—442no—
194C2×Q8—21no—
195C2×Q8—42no—
196C2×Q8—111no—
197C2×Q8C22×D1112yes—
198C2×D4—111no—
199C2×D4—221no—
200C2×D4—221no—
201C2×D4—221no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—221no—
206C22×C4C22⋊D1111yescharacteristic
207C22×C4C22⋊D1111yescharacteristic
208C22×C4C22⋊D1111yescharacteristic
209C22×C4C22×D1111yescharacteristic
210C22×C4C22×D1111yescharacteristic
211C22×C4C22×D1111yescharacteristic
212Q16—442no—
213SD16—442no—
214C2×C8—112no—
215C2×C8—112no—
216C2×C8—442no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—111no—
221C4⋊C4—111no—
222C4⋊C4—221no—
223C4⋊C4—221no—
224C4⋊C4C22×D1112yes—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—221no—
228C22⋊C4—221no—
229C22⋊C4—221no—
230C22⋊C4—221no—
231C22⋊C4—221no—
232C42—221no—
233C11#319011yescharacteristic, complemented, Sylow
234C23—22no—
235C23—22no—
236C23—41no—
237C23—221no—
238C23C2×C22⋊D1111yescharacteristic
239C23C2×C22⋊D1111yescharacteristic
240C23C2×C22⋊D1111yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—222no—
245D4—222no—
246D4—442no—
247D4—442no—
248D4—442no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—221no—
260C2×C4—221no—
261C2×C4—221no—
262C2×C4—442no—
263C2×C4C2×C22⋊D1111yescharacteristic
264C2×C4C2×C22⋊D1111yescharacteristic
265C2×C4C2×C22⋊D1111yescharacteristic
266C2×C4C23×D1111yescharacteristic
267C8—222no—
268C8—222no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—441no—
278C22—441no—
279C22—441no—
280C22#6699512yes—
281C22#6700312yes—
282C22C22×C22⋊D1111yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—441no—
289C4—441no—
290C4—441no—
291C4#6703712yes—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—881no—
296C2#1062333511yescharacteristic
297C2C2×#6699511yescharacteristic
298C2C2×#6700311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D11
commutator, cosocleC2×C4×C11C24
FrattiniC2×C4C23×D11
FittingC2×C11×C22⋊Q8C2
socleC22×C11C22×D4
prime radical#10623322C2
hypercenterC2×C22⋊Q8D11
hypocenterC11#3190
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C11×C22⋊Q8C2semidirect
3. #10622841C2semidirect
4. #10622840C2semidirect
5. C2×C22⋊(C11⋊C8)C2semidirect
6. #10622664C2semidirect
7. #10622661C2semidirect
8. #10622660C2semidirect
9. #10622658C2semidirect
10. #10622584C2semidirect
11. #10622580C2semidirect
23. C2×#4326C22semidirect
24. C2×#4325C22semidirect
34. #66899C22semidirect
35. #66899C22semidirect
39. #66874C22semidirect
40. #66873C22semidirect
41. #66872C22semidirect
42. #66871C22semidirect
121. C2×C22⋊Q8D11semidirect
233. C11#3190semidirect

Maximal subgroups

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

subgroupindex
C2×C11×C22⋊Q82
#106233222
#106228412
#106228402
C2×C22⋊(C11⋊C8)2
#106226642
#106226612
#106226602
#106226582
#106225842
#106225802
#319011