Finite Group Databasebeta

← #13072248

Subgroups — #13072248

Order 1408.

1348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×D4×D11C211yesmaximal, characteristic, complemented
2C2×#66993C212yesmaximal, complemented
3C2×#66990C212yesmaximal, complemented
4C2×#66937C211yesmaximal, characteristic, complemented
5C2×#66935C212yesmaximal, complemented
6C11×#392C211yesmaximal, characteristic, complemented
7#10622836C211yesmaximal, characteristic, complemented
8#10622573C211yesmaximal, characteristic, complemented
9#10622570C212yesmaximal, complemented
10#10622284C211yesmaximal, characteristic, complemented
11#10622279C211yesmaximal, characteristic, complemented
12C24×D11C2212yes—
13C22×D4×C11C2211yescharacteristic
14C22×D4×C11C2212yescomplemented
15C22×C22⋊D11C2211yescharacteristic
16C22×C22⋊D11C2212yes—
17C22×C22⋊D11C2211yescharacteristic, complemented
18C22×C22⋊D11C2212yescomplemented
19C22×C22⋊D11C2212yescomplemented
20C2×D4×D11—28no—
21C2×D4×D11—24no—
22C22×D44C2211yescharacteristic
23C22×D44C2211yescharacteristic
24C22×D44C2212yescomplemented
25C22×C4×D11C2211yescharacteristic
26C2×C11×C22⋊C4C2212yescomplemented
27C2×C11×C22⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic11C2212yescomplemented
29#66993—28no—
30#66990—28no—
31C2×(C2×D11)⋊C4C2212yes—
32C2×(C2×D11)⋊C4C2212yescomplemented
33C2×(C2×D11)⋊C4C2211yescharacteristic, complemented
34C2×(C2×D11)⋊C4C2212yescomplemented
35C2×Dic11⋊C4C2211yescharacteristic, complemented
36C2×C4×Dic11C2212yescomplemented
37#66937—24no—
38#66935—28no—
39C11×#95C2211yescharacteristic, complemented
40C11⋊#95C2211yescharacteristic, complemented
41C24×C11C2311yescharacteristic
42C24×C11C2312yes—
43C23×D11—24no—
44C23×D11—24no—
45C23×D11—24no—
46C23×D11—24no—
47C23×D11—24no—
48C23×D11—42no—
49C23×D11—42no—
50C23×D11C2311yescharacteristic
51C23×D11C2312yes—
52C23×D11C2311yescharacteristic
53C2×D4×C11—24no—
54C2×D4×C11—24no—
55C2×D4×C11—22no—
56C2×D4×C11—24no—
57C2×D4×C11—42no—
58C22×C4×C11C2311yescharacteristic
59C22×C4×C11C2311yescharacteristic
60C22×C4×C11C2312yes—
61C2×C22⋊D11—24no—
62C2×C22⋊D11—24no—
63C2×C22⋊D11—22no—
64C2×C22⋊D11—24no—
65C2×C22⋊D11—24no—
66C2×C22⋊D11—24no—
67C2×C22⋊D11—24no—
68C2×C22⋊D11—22no—
69C2×C22⋊D11—24no—
70C2×C22⋊D11—24no—
71C2×C22⋊D11—42no—
72C2×C22⋊D11—42no—
73C2×C22⋊D11D418yescomplemented
74C22×Dic11C2311yescharacteristic
75C22×Dic11C2312yes—
76C22×Dic11C2311yescharacteristic
77D4×D11—232no—
78C2×D44—24no—
79C2×D44—24no—
80C2×D44—22no—
81C2×D44—24no—
82C2×D44—22no—
83C2×D44—24no—
84C2×D44—42no—
85C2×C4×D11—24no—
86C2×C4×D11D414yescomplemented
87C11×C22⋊C4—22no—
88C11×C22⋊C4—24no—
89C22⋊Dic11—24no—
90(C2×D11)⋊C4—24no—
91(C2×D11)⋊C4—22no—
92(C2×D11)⋊C4—24no—
93(C2×D11)⋊C4D418yescomplemented
94Dic11⋊C4—22no—
95C4×Dic11—24no—
96#2433—111nomaximal, Sylow
97C23×C11—22no—
98C23×C11—22no—
99C23×C11—22no—
100C23×C11—22no—
101C23×C11—22no—
102C23×C11—22no—
103C23×C11—21no—
104C23×C11—22no—
105C23×C11—22no—
106C23×C11—21no—
107C23×C11—21no—
108C23×C11C2×D414yes—
109C23×C11C2411yescharacteristic
110C22×D11—22no—
111C22×D11—21no—
112C22×D11—21no—
113C22×D11—216no—
114C22×D11—22no—
115C22×D11—22no—
116C22×D11—22no—
117C22×D11—22no—
118C22×D11—22no—
119C22×D11—22no—
120C22×D11—21no—
121C22×D11—22no—
122C22×D11—22no—
123C22×D11—21no—
124C22×D11—21no—
125C22×D11—48no—
126C22×D11—48no—
127C22×D11—48no—
128C22×D11—48no—
129C22×D11—48no—
130C22×D11—48no—
131C22×D11C2×D414yes—
132C22×D11C2×D414yes—
133C22×D11C2×D412yes—
134D4×C11—28no—
135D4×C11—48no—
136C2×C4×C11—22no—
137C2×C4×C11—22no—
138C2×C4×C11—22no—
139C2×C4×C11—21no—
140C2×C4×C11—22no—
141C2×C4×C11C2×D412yes—
142C2×C4×C11C2×D414yescomplemented
143C22⋊D11—216no—
144C22⋊D11—48no—
145C22⋊D11—48no—
146C22⋊D11—44no—
147C22⋊D11—44no—
148C2×Dic11—22no—
149C2×Dic11—22no—
150C2×Dic11—21no—
151C2×Dic11—22no—
152C2×Dic11C2×D414yes—
153C2×Dic11C2×D412yes—
154C2×Dic11C2×D414yescomplemented
155D44—28no—
156D44—48no—
157D44—44no—
158C4×D11—28no—
159C23×D4—111no—
160C2×#127—112no—
161C2×#121—111no—
162C2×#120—112no—
163C2×#120—112no—
164#392—111no—
165#392—111no—
166#392—111no—
167#392D1111yescharacteristic, complemented
168#390—112no—
169C4×C22⋊C4—111no—
170C22×C11—24no—
171C22×C11—24no—
172C22×C11—24no—
173C22×C11—42no—
174C22×C11—42no—
175C22×C11—42no—
176C22×C11—42no—
177C22×C11—41no—
178C22×C11—41no—
179C22×C11—41no—
180C22×C11—42no—
181C22×C11—42no—
182C22×C11C22×D411yescharacteristic
183C22×C11C22×D412yes—
184C22×C11C22×D412yes—
185C22×C11H(5,𝔽2)11yescharacteristic
186C22×C11H(5,𝔽2)11yescharacteristic
187C2×D11—22no—
188C2×D11—22no—
189C2×D11—24no—
190C2×D11—24no—
191C2×D11—24no—
192C2×D11—24no—
193C2×D11—24no—
194C2×D11—432no—
195C2×D11—42no—
196C2×D11—42no—
197C2×D11—42no—
198C2×D11—42no—
199C2×D11—41no—
200C2×D11—41no—
201C2×D11—41no—
202C2×D11—42no—
203C2×D11—42no—
204C2×D11#12714yescomplemented
205C4×C11—22no—
206C4×C11—24no—
207C4×C11—41no—
208Dic11—24no—
209Dic11—41no—
210Dic11#12014yescomplemented
211C25—112no—
212C22×D4—112no—
213C22×D4—111no—
214C22×D4—111no—
215C22×D4—112no—
216C22×D4—112no—
217C22×D4—111no—
218C22×D4—111no—
219C22×D4—112no—
220C22×D4—224no—
221C22×D4—228no—
222C22×D4C2×D1111yescharacteristic
223C22×D4C2×D1112yescomplemented
224C23×C4—111no—
225#127—228no—
226#121—224no—
227#120—228no—
228#120—228no—
229C2×C4⋊C4—111no—
230C2×C22⋊C4—112no—
231C2×C22⋊C4—112no—
232C2×C22⋊C4—112no—
233C2×C22⋊C4—112no—
234C2×C22⋊C4—111no—
235C2×C22⋊C4C2×D1112yescomplemented
236C2×C22⋊C4C2×D1111yescharacteristic, complemented
237C2×C42—112no—
238#95—111no—
239#95C2×D1111yescharacteristic, complemented
240C2×C11—44no—
241C2×C11—81no—
242C2×C11C2×#12011yescharacteristic
243C2×C11C2×#12711yescharacteristic
244C2×C11#53511yescharacteristic
245C2×C11D4212yes—
246C2×C11D4212yes—
247D11—24no—
248D11—44no—
249D11—81no—
250C24—111no—
251C24—112no—
252C24—111no—
253C24—224no—
254C24—224no—
255C24—224no—
256C24—224no—
257C24—224no—
258C24—442no—
259C24—442no—
260C24C22×D1111yescharacteristic
261C24C22×D1112yes—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—42no—
267C2×D4—118no—
268C2×D4—224no—
269C2×D4—224no—
270C2×D4—222no—
271C2×D4—224no—
272C2×D4—224no—
273C2×D4—224no—
274C2×D4—224no—
275C2×D4—224no—
276C2×D4—222no—
277C2×D4—224no—
278C2×D4—2232no—
279C2×D4—224no—
280C2×D4—222no—
281C2×D4—224no—
282C2×D4—224no—
283C2×D4—222no—
284C2×D4—224no—
285C2×D4—442no—
286C2×D4—442no—
287C2×D4—442no—
288C22×C4—111no—
289C22×C4—112no—
290C22×C4—114no—
291C22×C4—111no—
292C22×C4—224no—
293C22×C4C22×D1111yescharacteristic
294C22×C4C22×D1111yescharacteristic
295C22×C4C22×D1112yes—
296C4⋊C4—222no—
297C22⋊C4—22no—
298C22⋊C4—24no—
299C22⋊C4—118no—
300C22⋊C4—224no—
301C22⋊C4—224no—
302C22⋊C4—222no—
303C22⋊C4—224no—
304C42—224no—
305C11#243311yescharacteristic, complemented, Sylow
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—114no—
318C23—112no—
319C23—114no—
320C23—2216no—
321C23—222no—
322C23—221no—
323C23—221no—
324C23—222no—
325C23—222no—
326C23—222no—
327C23—222no—
328C23—222no—
329C23—222no—
330C23—221no—
331C23—222no—
332C23—222no—
333C23—221no—
334C23—221no—
335C23—448no—
336C23—448no—
337C23—448no—
338C23—448no—
339C23—448no—
340C23—448no—
341C23D4×D1114yes—
342C23C23×D1111yescharacteristic
343D4—28no—
344D4—48no—
345D4—228no—
346D4—2216no—
347D4—448no—
348D4—444no—
349D4—444no—
350D4—448no—
351D4—448no—
352D4—444no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—114no—
359C2×C4—112no—
360C2×C4—114no—
361C2×C4—228no—
362C2×C4—222no—
363C2×C4—222no—
364C2×C4—222no—
365C2×C4—221no—
366C2×C4D4×D1112yes—
367C2×C4D4×D1114yescomplemented
368C22—24no—
369C22—24no—
370C22—24no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22—42no—
380C22—114no—
381C22—222no—
382C22—224no—
383C22—222no—
384C22—224no—
385C22—224no—
386C22—224no—
387C22—224no—
388C22—4432no—
389C22—442no—
390C22—442no—
391C22—442no—
392C22—442no—
393C22—441no—
394C22—441no—
395C22—442no—
396C22—441no—
397C22—442no—
398C22C2×D4×D1111yescharacteristic
399C22C2×D4×D1112yes—
400C22C2×D4×D1112yes—
401C22#6703711yescharacteristic
402C22#6704211yescharacteristic
403C4—22no—
404C4—24no—
405C4—41no—
406C4—114no—
407C4—224no—
408C4—441no—
409C2—44no—
410C2—81no—
411C2—224no—
412C2—444no—
413C2—881no—
414C2D11×#12011yescharacteristic
415C2#1062312112yes—
416C2#1062313811yescharacteristic
417C2#1062314012yes—
418C2D11×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#392C2
hypercenter#392D11
hypocenterC11#2433
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D11C2semidirect
2. C2×#66993C2semidirect
3. C2×#66990C2semidirect
4. C2×#66937C2semidirect
5. C2×#66935C2semidirect
6. C11×#392C2semidirect
7. #10622836C2semidirect
8. #10622573C2semidirect
9. #10622570C2semidirect
10. #10622284C2semidirect
11. #10622279C2semidirect
14. C22×D4×C11C22semidirect
17. C22×C22⋊D11C22semidirect
18. C22×C22⋊D11C22semidirect
19. C22×C22⋊D11C22semidirect
24. C22×D44C22semidirect
26. C2×C11×C22⋊C4C22semidirect
27. C2×C11×C22⋊C4C22semidirect
28. C2×C22⋊Dic11C22semidirect
32. C2×(C2×D11)⋊C4C22semidirect
33. C2×(C2×D11)⋊C4C22semidirect
34. C2×(C2×D11)⋊C4C22semidirect
35. C2×Dic11⋊C4C22semidirect
36. C2×C4×Dic11C22semidirect
39. C11×#95C22semidirect
40. C11⋊#95C22semidirect
73. C2×C22⋊D11D4semidirect
86. C2×C4×D11D4semidirect
93. (C2×D11)⋊C4D4semidirect
142. C2×C4×C11C2×D4semidirect
154. C2×Dic11C2×D4semidirect
167. #392D11semidirect
204. C2×D11#127semidirect
210. Dic11#120semidirect
223. C22×D4C2×D11semidirect
235. C2×C22⋊C4C2×D11semidirect
236. C2×C22⋊C4C2×D11semidirect
239. #95C2×D11semidirect
305. C11#2433semidirect
367. C2×C4D4×D11semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D112
C2×#669932
C2×#669902
C2×#669372
C2×#669352
C11×#3922
#106228362
#106225732
#106225702
#106222842
#106222792
#243311