Finite Group Databasebeta

← #14721591

Subgroups — #14721591

Order 2816.

1254 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13082378C211yesmaximal, characteristic
2#13077656C211yesmaximal, characteristic, complemented
3#13071587C211yesmaximal, characteristic, complemented
4#13067177C211yesmaximal, characteristic, complemented
5#13067176C211yesmaximal, characteristic, complemented
6#13067143C214yesmaximal, complemented
7#13067130C214yesmaximal, complemented
8#13067110C211yesmaximal, characteristic, complemented
9C11×#1502C211yesmaximal, characteristic, complemented
10C2×#67042—21no—
11C11×#532C2211yescharacteristic
12#10623364—21no—
13#10623257C412yescomplemented
14#10623248C412yescomplemented
15#10623207—22no—
16#10623136—22no—
17D11×#121C2214yes—
18#10623082C412yescomplemented
19#10623081—22no—
20D4×D44C412yescomplemented
21#10623008C2211yescharacteristic
22C2×C4×D44C2211yescharacteristic
23C11×#431C2211yescharacteristic, complemented
24C4×C11×C4⋊C4C2211yescharacteristic, complemented
25#10622603—22no—
26#10622552—21no—
27#10622549C2211yescharacteristic, complemented
28C4×C4⋊Dic11—21no—
29#10622538C2211yescharacteristic, complemented
30#10622451—22no—
31#10622339C2214yescomplemented
32#10622301—22no—
33#10622279C2212yescomplemented
34#10622277C2212yescomplemented
35C11×#327C2212yescomplemented
36C11×#327C2212yescomplemented
37#10622175C2214yescomplemented
38#10622108C2214yescomplemented
39#10622089C2212yescomplemented
40#10622089C2212yescomplemented
41C2×C11×#41—21no—
42#67042—44no—
43#67042—44no—
44C2×#4343—22no—
45C2×D4×D11—24no—
46C2×D4×D11—22no—
47C2×D4×D11C2×C414yes—
48C2×#4339—21no—
49C2×#4339—22no—
50C22×D44—22no—
51C22×D44D411yescharacteristic
52C22×C4×D11C2312yes—
53C11×C4⋊Q8C2×C411yescharacteristic
54C11×#127C2×C411yescharacteristic
55C11×#124—21no—
56C11×#121—21no—
57C11×#121C2312yes—
58C4×D4×C11—21no—
59C4×D4×C11C2×C412yes—
60C2×C11×C4⋊C4—21no—
61C2×C42×C11—21no—
62C2×C42×C11C2311yescharacteristic
63#66999—22no—
64#66992—24no—
65#66991—22no—
66#66991C2×C414yes—
67#66990—24no—
68#66983—21no—
69#66983—22no—
70#66983C2312yes—
71C2×(C2×D11)⋊C4—22no—
72C2×(C2×D11)⋊C4—22no—
73C2×(C2×D11)⋊C4D412yes—
74C2×C4⋊Dic11—21no—
75C2×C4⋊Dic11D411yescharacteristic
76#66977—22no—
77C2×C4×Dic11—22no—
78C2×#4313—24no—
79#66948C2×C414yes—
80#66947—24no—
81C4⋊C4×D11C2×C414yes—
82#66937—24no—
83#66936—24no—
84#66935—24no—
85C22⋊C4×D11—24no—
86#66928—21no—
87#66927C2×C411yescharacteristic
88C4×D44—21no—
89C4×D44C2×C412yes—
90C4×D44C2×C412yes—
91C4×D44C2×C412yes—
92C4×D44D412yes—
93C4×D44D412yes—
94#66922C2×C411yescharacteristic
95C11×C4⋊C8—21no—
96C11×C22⋊C8C2312yes—
97C4×C8×C11—21no—
98C11×#95C2311yescharacteristic
99C11×#95C2311yescharacteristic
100C11⋊#95—21no—
101C11⋊#95—22no—
102C11⋊#95C2312yes—
103C22⋊(C11⋊C8)C2312yes—
104(C2×D11)⋊C8—24no—
105C4⋊(C11⋊C8)—21no—
106C4×C11⋊C8—21no—
107#17626—111nomaximal, Sylow
108C23×D11—24no—
109C23×D11C2×D412yes—
110C11×#41—42no—
111C11×#41—42no—
112C2×Q8×C11—21no—
113C2×D4×C11—22no—
114C2×D4×C11—21no—
115C2×D4×C11—21no—
116C2×D4×C11C22×C412yes—
117C22×C4×C11—21no—
118C22×C4×C11—22no—
119C22×C4×C11—21no—
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2411yescharacteristic
123C2×C22⋊D11—24no—
124C2×C22⋊D11—22no—
125C2×C22⋊D11—24no—
126C22×Dic11—22no—
127C22×Dic11C2×D412yes—
128#4343—48no—
129D4×D11—48no—
130D4×D11—48no—
131D4×D11—48no—
132D4×D11—84no—
133#4339—48no—
134#4339—42no—
135#4339—42no—
136C2×D44—24no—
137C2×D44—21no—
138C2×D44—21no—
139C2×D44—22no—
140C2×D44—22no—
141C2×D44—48no—
142C2×D44C22⋊C412yes—
143C2×D44C22⋊C412yes—
144C2×D44C22×C412yes—
145C2×C4×D11—24no—
146C2×C4×D11—22no—
147C2×C4×D11—82no—
148C2×C4×D11C22×C414yes—
149C2×C4×D11C2×D414yes—
150C2×Dic22—21no—
151C2×C8×C11—22no—
152C11×C4⋊C4—21no—
153C11×C4⋊C4—42no—
154C11×C4⋊C4C22×C412yes—
155C11×C22⋊C4—22no—
156C11×C22⋊C4—22no—
157C42×C11—42no—
158C42×C11C22×C411yescharacteristic
159C42×C11C22×C411yescharacteristic
160C42×C11C2×D411yescharacteristic
161C42×C11C2×D411yescharacteristic
162C22⋊Dic11—24no—
163(C2×D11)⋊C4—24no—
164(C2×D11)⋊C4—22no—
165(C2×D11)⋊C4—22no—
166(C2×D11)⋊C4—44no—
167(C2×D11)⋊C4—82no—
168(C2×D11)⋊C4C22⋊C414yes—
169(C2×D11)⋊C4C22⋊C414yes—
170C4⋊Dic11—21no—
171C4⋊Dic11—42no—
172C4⋊Dic11C22⋊C412yes—
173C4⋊Dic11C22⋊C412yes—
174C4⋊Dic11C22×C412yes—
175Dic11⋊C4—24no—
176C4×Dic11—82no—
177C2×C11⋊C8—22no—
178#4313—84no—
179#3496—111no—
180#2975—111no—
181#2357—111no—
182#1517—111no—
183#1517—111no—
184#1510—114no—
185#1507—114no—
186#1502—111no—
187#1502D1111yescharacteristic, complemented
188C23×C11—22no—
189C23×C11C22×D411yescharacteristic
190C22×D11—22no—
191C22×D11—22no—
192C22×D11—22no—
193C22×D11—42no—
194C22×D11—42no—
195C22×D11—48no—
196C22×D11—84no—
197C22×D11—84no—
198C22×D11C2×C22⋊C414yes—
199Q8×C11—42no—
200D4×C11—42no—
201D4×C11—42no—
202D4×C11—42no—
203D4×C11—42no—
204D4×C11—81no—
205C2×C4×C11—22no—
206C2×C4×C11—22no—
207C2×C4×C11—41no—
208C2×C4×C11—41no—
209C2×C4×C11—41no—
210C2×C4×C11—41no—
211C2×C4×C11—44no—
212C2×C4×C11—41no—
213C2×C4×C11—41no—
214C2×C4×C11—41no—
215C2×C4×C11C2×C22⋊C411yescharacteristic
216C2×C4×C11C2×C22⋊C411yescharacteristic
217C2×C4×C11C2×C22⋊C411yescharacteristic
218C2×C4×C11C2×C22⋊C411yescharacteristic
219C2×C4×C11AGL(1,ℤ/8)12yes—
220C2×C4×C11AGL(1,ℤ/8)12yes—
221C2×C4×C11C23×C411yescharacteristic
222C2×C4×C11C22×D411yescharacteristic
223C22⋊D11—48no—
224C22⋊D11—42no—
225C22⋊D11—42no—
226C22⋊D11—84no—
227C2×Dic11—22no—
228C2×Dic11—42no—
229C2×Dic11—42no—
230C2×Dic11—42no—
231C2×Dic11—42no—
232C2×Dic11C2×C22⋊C414yes—
233D44—24no—
234D44—24no—
235D44—42no—
236D44—48no—
237D44—42no—
238D44—81no—
239C4×D11—44no—
240C4×D11—44no—
241C4×D11—48no—
242C4×D11—44no—
243Dic22—42no—
244C8×C11—42no—
245C11⋊C8—42no—
246C2×H(5,𝔽2)—221no—
247#547—112no—
248#547—112no—
249#546—222no—
250#546—222no—
251D42—112no—
252D42—112no—
253#537—221no—
254#534—222no—
255#532—111no—
256#532C2×D1111yescharacteristic
257C2×#121—114no—
258C2×C4×D4—111no—
259#436—222no—
260#436—222no—
261#431—111no—
262#431C2×D1111yescharacteristic, complemented
263#407—114no—
264#390—221no—
265#388—222no—
266#382—111no—
267#381—112no—
268C4×C4⋊C4—221no—
269C4×C4⋊C4C2×D1111yescharacteristic, complemented
270C4×C22⋊C4—112no—
271#327—114no—
272#327—114no—
273#327—112no—
274#327—112no—
275#327C2×D1112yescomplemented
276#327C2×D1112yescomplemented
277C22×C11—21no—
278C22×C11—21no—
279C22×C11—42no—
280C22×C11—42no—
281C22×C11—42no—
282C22×C11C22×C22⋊C411yescharacteristic
283C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
285C2×D11—24no—
286C2×D11—24no—
287C2×D11—24no—
288C2×D11—42no—
289C2×D11—42no—
290C2×D11—42no—
291C2×D11—82no—
292C2×D11—82no—
293C2×D11—168no—
294C4×C11—21no—
295C4×C11—21no—
296C4×C11—21no—
297C4×C11—21no—
298C4×C11—42no—
299C4×C11—42no—
300C4×C11—81no—
301C4×C11#41812yes—
302C4×C11#41812yes—
303Dic11—24no—
304Dic11—42no—
305Dic11—82no—
306H(5,𝔽2)—444no—
307H(5,𝔽2)—444no—
308C2×#41—21no—
309C2×#41—222no—
310C2×#41—222no—
311C2×#41—221no—
312C22×D4—114no—
313C22×D4—111no—
314C22×D4—222no—
315C22×D4—222no—
316C22×D4—224no—
317C23×C4—112no—
318C2×M16—224no—
319C4⋊Q8—111no—
320C4⋊Q8C4×D1111yescharacteristic
321#127—111no—
322#127C4×D1111yescharacteristic
323#124—21no—
324#124—221no—
325#123—224no—
326#123—224no—
327C22⋊Q8—222no—
328C22⋊Q8—222no—
329#121—21no—
330#121—114no—
331#121—112no—
332#121—114no—
333#121—224no—
334#121—222no—
335#121—222no—
336#121—224no—
337#121—221no—
338#121C22×D1112yes—
339#120—224no—
340#120—224no—
341C4×D4—21no—
342C4×D4—112no—
343C4×D4—112no—
344C4×D4—112no—
345C4×D4—112no—
346C4×D4—112no—
347C4×D4—221no—
348C4×D4C4×D1112yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—111no—
351C2×C4⋊C4—114no—
352C2×C4⋊C4—221no—
353C2×C22⋊C4—112no—
354C2×C22⋊C4—224no—
355C2×C22⋊C4—222no—
356C2×C22⋊C4—222no—
357C2×C42—21no—
358C2×C42—222no—
359C2×C42C22×D1111yescharacteristic
360C4⋊C8—21no—
361C4⋊C8—221no—
362C22⋊C8—112no—
363C22⋊C8—224no—
364C22⋊C8C22×D1112yes—
365C4×C8—21no—
366C4×C8—221no—
367#95—112no—
368#95—222no—
369#95—221no—
370#95C22×D1111yescharacteristic
371#95C22×D1111yescharacteristic
372C2×C11—21no—
373C2×C11—21no—
374C2×C11—82no—
375C2×C11#288311yescharacteristic
376C2×C11C2×#41811yescharacteristic
377C2×C11#290111yescharacteristic
378D11—44no—
379D11—82no—
380C24—112no—
381C24—224no—
382#41—42no—
383#41—42no—
384#41—448no—
385#41—448no—
386#41—442no—
387#41—442no—
388C2×Q8—21no—
389C2×Q8—221no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—112no—
394C2×D4—112no—
395C2×D4—112no—
396C2×D4—224no—
397C2×D4—224no—
398C2×D4—224no—
399C2×D4—222no—
400C2×D4—221no—
401C2×D4—221no—
402C2×D4—222no—
403C2×D4—222no—
404C2×D4—448no—
405C2×D4—448no—
406C2×D4—448no—
407C2×D4—448no—
408C2×D4—884no—
409C2×D4C2×C4×D1112yes—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—112no—
414C22×C4—114no—
415C22×C4—114no—
416C22×C4—224no—
417C22×C4—222no—
418C22×C4—222no—
419C22×C4—882no—
420C22×C4D4×D1111yescharacteristic
421C22×C4D4×D1111yescharacteristic
422C22×C4C23×D1111yescharacteristic
423M16—884no—
424C2×C8—22no—
425C2×C8—222no—
426C4⋊C4—21no—
427C4⋊C4—42no—
428C4⋊C4—112no—
429C4⋊C4—112no—
430C4⋊C4—112no—
431C4⋊C4—224no—
432C4⋊C4—221no—
433C4⋊C4—442no—
434C4⋊C4C2×C4×D1112yes—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—114no—
438C22⋊C4—114no—
439C22⋊C4—224no—
440C22⋊C4—224no—
441C22⋊C4—222no—
442C22⋊C4—222no—
443C22⋊C4—444no—
444C22⋊C4—882no—
445C42—42no—
446C42—882no—
447C42C2×C4×D1111yescharacteristic
448C42C2×C4×D1111yescharacteristic
449C42D4×D1111yescharacteristic
450C42D4×D1111yescharacteristic
451C11#1762611yescharacteristic, complemented, Sylow
452C23—22no—
453C23—114no—
454C23—222no—
455C23—222no—
456C23—222no—
457C23—442no—
458C23—442no—
459C23—448no—
460C23—884no—
461C23—884no—
462C23C2×D4×D1111yescharacteristic
463Q8—42no—
464Q8—442no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—81no—
470D4—224no—
471D4—224no—
472D4—442no—
473D4—448no—
474D4—448no—
475D4—442no—
476D4—442no—
477D4—442no—
478D4—881no—
479D4—884no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—114no—
491C2×C4—222no—
492C2×C4—444no—
493C2×C4—444no—
494C2×C4—448no—
495C2×C4—442no—
496C2×C4—444no—
497C2×C4—442no—
498C2×C4—442no—
499C2×C4—442no—
500C2×C4C22⋊C4×D1111yescharacteristic
501C2×C4C22⋊C4×D1111yescharacteristic
502C2×C4C22⋊C4×D1111yescharacteristic
503C2×C4C22⋊C4×D1111yescharacteristic
504C2×C4#6696412yes—
505C2×C4#6696712yes—
506C2×C4C22×C4×D1111yescharacteristic
507C2×C4C2×D4×D1111yescharacteristic
508C8—42no—
509C8—442no—
510C22—21no—
511C22—21no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—224no—
516C22—224no—
517C22—224no—
518C22—442no—
519C22—442no—
520C22—442no—
521C22—882no—
522C22—882no—
523C22—1768no—
524C22C2×C22⋊C4×D1111yescharacteristic
525C22C2×#6696411yescharacteristic
526C22C2×#6696711yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—224no—
535C4—442no—
536C4—882no—
537C4#1062238312yes—
538C4#1062238312yes—
539C2—21no—
540C2—21no—
541C2—82no—
542C2—444no—
543C2—882no—
544C2#1307707511yescharacteristic
545C2C2×#1062238311yescharacteristic
546C2#1307716011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1502C2
socleC22×C11C22×C22⋊C4
prime radical#13082378C2
cosocleC22×C4×C11C24
hypercenter#1502D11
hypocenterC11#17626
LCSG ▷ C2×C4×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #13077656C2semidirect
3. #13071587C2semidirect
4. #13067177C2semidirect
5. #13067176C2semidirect
6. #13067143C2semidirect
7. #13067130C2semidirect
8. #13067110C2semidirect
9. C11×#1502C2semidirect
13. #10623257C4semidirect
14. #10623248C4semidirect
18. #10623082C4semidirect
20. D4×D44C4semidirect
23. C11×#431C22semidirect
24. C4×C11×C4⋊C4C22semidirect
27. #10622549C22semidirect
29. #10622538C22semidirect
31. #10622339C22semidirect
33. #10622279C22semidirect
34. #10622277C22semidirect
35. C11×#327C22semidirect
36. C11×#327C22semidirect
37. #10622175C22semidirect
38. #10622108C22semidirect
39. #10622089C22semidirect
40. #10622089C22semidirect
187. #1502D11semidirect
262. #431C2×D11semidirect
269. C4×C4⋊C4C2×D11semidirect
275. #327C2×D11semidirect
276. #327C2×D11semidirect
451. C11#17626semidirect

Maximal subgroups

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

subgroupindex
#130823782
#130776562
#130715872
#130671772
#130671762
#130671432
#130671302
#130671102
C11×#15022
#1762611