Finite Group Databasebeta

← #13078246

Subgroups — #13078246

Order 1408.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C11×M16C211yesmaximal, characteristic, complemented
2#10623322C211yesmaximal, characteristic
3C2×#66977C211yesmaximal, characteristic, complemented
4C2×#66962C211yesmaximal, characteristic, complemented
5#10622750C212yesmaximal, complemented
6#10622747C212yesmaximal, complemented
7#10622746C211yesmaximal, characteristic, complemented
8#10622734C211yesmaximal, characteristic, complemented
9#10622733C211yesmaximal, characteristic, complemented
10#10622729C212yesmaximal, complemented
11#10622727C212yesmaximal, complemented
12C23×C4×C11C2211yescharacteristic
13C2×#4339C2211yescharacteristic
14C22×Dic22C2211yescharacteristic
15C2×C11×M16—24no—
16C2×C11×M16C2211yescharacteristic
17C2×C11×M16C2212yes—
18C2×C11×M16C2211yescharacteristic
19C22×C8×C11C2212yes—
20#67006—21no—
21C2×C22⋊Dic11—21no—
22#66983C2211yescharacteristic
23#66983C2211yescharacteristic
24#66982—21no—
25C4×C22⋊D11—21no—
26C4×C22⋊D11—21no—
27#66979C2211yescharacteristic
28C2×C4⋊Dic11C2211yescharacteristic
29#66977—21no—
30#66977—21no—
31#66977C2212yes—
32#66977C2211yescharacteristic
33#66977C2212yes—
34#66977C2211yescharacteristic
35C2×Dic11⋊C4—21no—
36#66962—24no—
37C2×Dic44C2212yescomplemented
38C2×#4314C2212yescomplemented
39#66885C2212yescomplemented
40#66882C2212yescomplemented
41#66881C2212yescomplemented
42#66880C2212yescomplemented
43#66880C2212yescomplemented
44#66880C2212yescomplemented
45C24×C11D411yescharacteristic
46C22×C4×C11—22no—
47C22×C4×C11—22no—
48C22×C4×C11D411yescharacteristic
49C22×C4×C11D411yescharacteristic
50C22×C4×C11D411yescharacteristic
51C22×C4×C11C2311yescharacteristic
52C22×C4×C11C2311yescharacteristic
53C22×C4×C11C2311yescharacteristic
54C2×C22⋊D11—21no—
55C2×C22⋊D11—21no—
56C2×C22⋊D11—21no—
57C22×Dic11—21no—
58C22×Dic11—21no—
59#4339—22no—
60#4339—22no—
61C2×D44C2311yescharacteristic
62C2×C4×D11—21no—
63C2×Dic22—22no—
64C2×Dic22—21no—
65C2×Dic22—22no—
66C2×Dic22C2312yes—
67C2×Dic22C2311yescharacteristic
68C11×M16—24no—
69C11×M16—22no—
70C11×M16D414yescomplemented
71C2×C8×C11—24no—
72C2×C8×C11C2312yes—
73C2×C8×C11C2312yes—
74C22⋊Dic11—22no—
75C22⋊Dic11—22no—
76C22⋊Dic11—21no—
77C22⋊Dic11—21no—
78C22⋊Dic11—21no—
79(C2×D11)⋊C4—21no—
80(C2×D11)⋊C4—21no—
81C4⋊Dic11—21no—
82C4⋊Dic11C2311yescharacteristic
83C4⋊Dic11C2311yescharacteristic
84C4⋊Dic11C2312yes—
85Dic11⋊C4—22no—
86Dic11⋊C4—22no—
87Dic11⋊C4—21no—
88Dic11⋊C4—21no—
89C4×Dic11—21no—
90Dic44—24no—
91#4314—24no—
92#3056—111nomaximal, Sylow
93C23×C11—22no—
94C23×C11—22no—
95C23×C11—41no—
96C23×C11C2×D411yescharacteristic
97C23×C11C2×D411yescharacteristic
98C23×C11C2×D411yescharacteristic
99C22×D11—21no—
100C2×C4×C11—22no—
101C2×C4×C11—22no—
102C2×C4×C11—22no—
103C2×C4×C11—21no—
104C2×C4×C11—21no—
105C2×C4×C11—21no—
106C2×C4×C11—21no—
107C2×C4×C11C2×D411yescharacteristic
108C2×C4×C11C2×D411yescharacteristic
109C2×C4×C11C2×D411yescharacteristic
110C2×C4×C11C2×D412yes—
111C2×C4×C11#4112yes—
112C2×C4×C11C2411yescharacteristic
113C22⋊D11—42no—
114C22⋊D11—42no—
115C22⋊D11—42no—
116C2×Dic11—21no—
117C2×Dic11—21no—
118C2×Dic11—22no—
119C2×Dic11—22no—
120C2×Dic11—21no—
121C2×Dic11—42no—
122C2×Dic11—42no—
123D44—21no—
124D44—21no—
125C4×D11—42no—
126Dic22—22no—
127Dic22—21no—
128Dic22—22no—
129Dic22—21no—
130Dic22—42no—
131C8×C11—22no—
132C8×C11C2×D414yes—
133C2×#137—111no—
134C22×M16D1111yescharacteristic, complemented
135#525—111no—
136C2×C22⋊Q8—111no—
137#470—112no—
138#468—112no—
139#467—112no—
140#465—112no—
141#428—111no—
142#419—111no—
143#417—111no—
144C22×C11—21no—
145C22×C11—21no—
146C22×C11—21no—
147C22×C11—21no—
148C22×C11—21no—
149C22×C11—21no—
150C22×C11—21no—
151C22×C11—44no—
152C22×C11#12112yes—
153C22×C11#13712yescomplemented
154C22×C11C22×D411yescharacteristic
155C2×D11—41no—
156C2×D11—41no—
157C2×D11—41no—
158C4×C11—21no—
159C4×C11—21no—
160C4×C11#12112yes—
161C4×C11C22×D411yescharacteristic
162C4×C11C2×#4111yescharacteristic
163Dic11—42no—
164Dic11—41no—
165Dic11—41no—
166Dic11—42no—
167Dic11—41no—
168C2×#41—111no—
169C22×Q8—111no—
170C23×C4C2×D1111yescharacteristic
171#137—224no—
172C2×Q16—112no—
173C2×SD16—112no—
174C2×M16—24no—
175C2×M16C2×D1111yescharacteristic
176C2×M16C2×D1112yes—
177C2×M16C2×D1111yescharacteristic
178C22×C8C2×D1112yes—
179#123—221no—
180C22⋊Q8—112no—
181C22⋊Q8—112no—
182C22⋊Q8—111no—
183C22⋊Q8—111no—
184C22⋊Q8—221no—
185C22⋊Q8—221no—
186#121—111no—
187#121—111no—
188#120—221no—
189C4×D4—221no—
190C4×D4—221no—
191#117—111no—
192C2×C4⋊C4—111no—
193C2×C4⋊C4—221no—
194C2×C22⋊C4—221no—
195#107—112no—
196#106—112no—
197#103—112no—
198#103—112no—
199#103—112no—
200#102—112no—
201C2×C11—22no—
202C2×C11—22no—
203C2×C11—41no—
204C2×C11C2×#12111yescharacteristic
205C2×C11C2×#13711yescharacteristic
206C2×C11#57511yescharacteristic
207D11—81no—
208C24D4411yescharacteristic
209#41—222no—
210#41—222no—
211C2×Q8—112no—
212C2×Q8—111no—
213C2×Q8—221no—
214C2×Q8—222no—
215C2×Q8—222no—
216C2×D4—111no—
217C2×D4—221no—
218C2×D4—221no—
219C2×D4—221no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—221no—
223C22×C4—221no—
224C22×C4—221no—
225C22×C4D4411yescharacteristic
226C22×C4D4411yescharacteristic
227C22×C4D4411yescharacteristic
228C22×C4C22×D1111yescharacteristic
229C22×C4C22×D1111yescharacteristic
230C22×C4C22×D1111yescharacteristic
231Q16—224no—
232SD16—224no—
233M16—22no—
234M16—24no—
235M16C22⋊D1114yescomplemented
236C2×C8—24no—
237C2×C8C22×D1112yes—
238C2×C8C22×D1112yes—
239C4⋊C4—112no—
240C4⋊C4—111no—
241C4⋊C4—111no—
242C4⋊C4—221no—
243C4⋊C4—222no—
244C4⋊C4—222no—
245C4⋊C4—221no—
246C4⋊C4—221no—
247C22⋊C4—222no—
248C22⋊C4—222no—
249C22⋊C4—221no—
250C22⋊C4—221no—
251C22⋊C4—221no—
252C22⋊C4—221no—
253C22⋊C4—221no—
254C42—221no—
255C11#305611yescharacteristic, complemented, Sylow
256C23—22no—
257C23—22no—
258C23—41no—
259C23—221no—
260C23C2×D4411yescharacteristic
261C23C2×D4411yescharacteristic
262C23C2×D4411yescharacteristic
263Q8—222no—
264Q8—221no—
265Q8—222no—
266Q8—221no—
267Q8—442no—
268D4—221no—
269D4—221no—
270D4—442no—
271D4—442no—
272D4—442no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—221no—
281C2×C4—221no—
282C2×C4—222no—
283C2×C4—221no—
284C2×C4—222no—
285C2×C4—442no—
286C2×C4—442no—
287C2×C4—442no—
288C2×C4C2×D4411yescharacteristic
289C2×C4C2×D4411yescharacteristic
290C2×C4C2×D4411yescharacteristic
291C2×C4#433912yes—
292C2×C4C2×C22⋊D1112yes—
293C2×C4C23×D1111yescharacteristic
294C8—22no—
295C8C2×C22⋊D1114yes—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—441no—
305C22—441no—
306C22—441no—
307C22#6696212yescomplemented
308C22#6698312yes—
309C22C22×D4411yescharacteristic
310C4—21no—
311C4—21no—
312C4—441no—
313C4—442no—
314C4—441no—
315C4—441no—
316C4—442no—
317C4#6698312yes—
318C4C2×#433911yescharacteristic
319C4C22×C22⋊D1111yescharacteristic
320C2—22no—
321C2—22no—
322C2—41no—
323C2—881no—
324C2C2×#6696211yescharacteristic
325C2#1062327911yescharacteristic
326C2C2×#6698311yescharacteristic
subgroupG/H
centerC22C22×D44
commutator, cosocleC2×C4×C11C24
FrattiniC2×C4C23×D11
FittingC22×C11×M16C2
socleC22×C11C22×D4
prime radical#10623322C2
hypercenterC22×M16D11
hypocenterC11#3056
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×C11×M16C2semidirect
3. C2×#66977C2semidirect
4. C2×#66962C2semidirect
5. #10622750C2semidirect
6. #10622747C2semidirect
7. #10622746C2semidirect
8. #10622734C2semidirect
9. #10622733C2semidirect
10. #10622729C2semidirect
11. #10622727C2semidirect
37. C2×Dic44C22semidirect
38. C2×#4314C22semidirect
39. #66885C22semidirect
40. #66882C22semidirect
41. #66881C22semidirect
42. #66880C22semidirect
43. #66880C22semidirect
44. #66880C22semidirect
70. C11×M16D4semidirect
134. C22×M16D11semidirect
153. C22×C11#137semidirect
235. M16C22⋊D11semidirect
255. C11#3056semidirect
307. C22#66962semidirect

Maximal subgroups

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

subgroupindex
C22×C11×M162
#106233222
C2×#669772
C2×#669622
#106227502
#106227472
#106227462
#106227342
#106227332
#106227292
#106227272
#305611