Finite Group Databasebeta

← #14755300

Subgroups — #14755300

Order 2816.

1702 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66995C211yesmaximal, characteristic
2#13081817C211yesmaximal, characteristic
3C2×#10622618C211yesmaximal, characteristic
4C2×#10622838C211yesmaximal, characteristic, complemented
5#13068988C218yesmaximal, complemented
6#13068355C211yesmaximal, characteristic, complemented
7#13068326C211yesmaximal, characteristic, complemented
8C11×#2310C211yesmaximal, characteristic, complemented
9C23×Q8×C11C2211yescharacteristic
10C22×#4339C2211yescharacteristic
11C2×C11×#117C2211yescharacteristic
12C2×C11×#117C2211yescharacteristic
13C22×#4326—21no—
14C2×#66995—24no—
15C2×#66995—22no—
16C2×#66995C418yescomplemented
17C2×#66995C2214yes—
18C22×#4325—21no—
19C2×#66979C2211yescharacteristic
20C22×#4318C2211yescharacteristic
21C22×#4318C2211yescharacteristic
22#10623054—22no—
23#10623053—22no—
24#10623052—21no—
25D11×#117—22no—
26#10623047—21no—
27#10623035C2214yes—
28#10623017—22no—
29C4×#4339—22no—
30C11×#391—22no—
31C11×#391C2214yescomplemented
32C11×#381—22no—
33#10622838—22no—
34#10622838C2214yescomplemented
35C2×#66899—22no—
36#10622618—24no—
37C2×#66869C2214yes—
38#10622168C2214yescomplemented
39#10622161C2214yescomplemented
40C22×Q8×C11—21no—
41C22×Q8×C11—22no—
42C22×Q8×C11—21no—
43C22×Q8×C11—21no—
44C22×Q8×C11—22no—
45C22×Q8×C11—21no—
46C22×Q8×C11—21no—
47C22×Q8×C11—22no—
48C22×Q8×C11C2×C414yescomplemented
49C22×Q8×C11C2312yes—
50C23×C4×C11—21no—
51C23×C4×C11C2311yescharacteristic
52C22×C22⋊D11—21no—
53C2×#4339—21no—
54C2×#4339—21no—
55C2×#4339—22no—
56C2×#4339—22no—
57C2×#4339C2×C414yes—
58C2×#4339D414yes—
59C2×#4339D412yes—
60C2×#4339C2312yes—
61C22×D44D411yescharacteristic
62C22×C4×D11—21no—
63C22×Dic22D411yescharacteristic
64C11×#117—21no—
65C11×#117—21no—
66C11×#117—21no—
67C11×#117—21no—
68C11×#117—22no—
69C11×#117—22no—
70C2×C11×C4⋊C4—21no—
71C2×C11×C4⋊C4C2312yes—
72C2×C11×C22⋊C4—21no—
73C2×C11×C22⋊C4D412yes—
74C2×C42×C11D412yes—
75C2×C42×C11C2312yes—
76C2×C22⋊Dic11—21no—
77C2×#4326—28no—
78C2×#4326—24no—
79#66995—216no—
80#66995—28no—
81#66995C2×C4116yescomplemented
82C2×#4325—28no—
83C2×#4325—24no—
84C4×C22⋊D11—28no—
85#66979—21no—
86#66979—21no—
87#66979D412yes—
88#66979C2312yes—
89C2×C4⋊Dic11D412yes—
90C2×C4×Dic11—21no—
91C2×#4318—22no—
92C2×#4318—21no—
93C2×#4318—21no—
94C2×#4318—22no—
95C2×#4318—22no—
96C2×#4318C2×C414yescomplemented
97C2×#4318C2312yes—
98C2×#4318C2312yes—
99C22×C11⋊C8—21no—
100C22×C11⋊C8—21no—
101#66946—24no—
102#66945—24no—
103C4⋊C4×D11—24no—
104#66940—24no—
105(C22⋊D11)⋊C4—28no—
106C22⋊C4×D11—24no—
107#66930—24no—
108C4×D44—24no—
109#66925—24no—
110C42×D11—24no—
111C4×Dic22—24no—
112C11×#95—28no—
113#66899—28no—
114#66869—28no—
115#20944—111nomaximal, Sylow
116C24×C11C2×D411yescharacteristic
117C23×D11—21no—
118C2×Q8×C11—24no—
119C2×Q8×C11—22no—
120C2×Q8×C11—24no—
121C2×Q8×C11—24no—
122C2×Q8×C11—24no—
123C2×Q8×C11—24no—
124C2×Q8×C11—22no—
125C2×Q8×C11—22no—
126C2×Q8×C11—22no—
127C2×Q8×C11—22no—
128C2×Q8×C11—41no—
129C2×Q8×C11—41no—
130C2×Q8×C11—42no—
131C2×Q8×C11—42no—
132C2×Q8×C11—42no—
133C2×Q8×C11—42no—
134C2×Q8×C11—42no—
135C2×Q8×C11C22×C414yes—
136C22×C4×C11—24no—
137C22×C4×C11—22no—
138C22×C4×C11—22no—
139C22×C4×C11—41no—
140C22×C4×C11—41no—
141C22×C4×C11—42no—
142C22×C4×C11C22×C412yes—
143C22×C4×C11C2×D414yes—
144C22×C4×C11C2×D411yescharacteristic
145C22×C4×C11C2×D411yescharacteristic
146C22×C4×C11C2×D412yes—
147C22×C4×C11C2×D411yescharacteristic
148C22×C4×C11C2×D411yescharacteristic
149C22×C4×C11C2×D411yescharacteristic
150C22×C4×C11#4112yes—
151C22×C4×C11#4112yes—
152C22×C4×C11C2411yescharacteristic
153C2×C22⋊D11—24no—
154C2×C22⋊D11—22no—
155C2×C22⋊D11—24no—
156C2×C22⋊D11—22no—
157C22×Dic11—21no—
158C22×Dic11—22no—
159#4339—24no—
160#4339—24no—
161#4339—24no—
162#4339—24no—
163#4339—24no—
164#4339—22no—
165#4339—22no—
166#4339C22⋊C418yes—
167#4339C22×C414yes—
168#4339C2×D414yes—
169C2×D44—21no—
170C2×D44—22no—
171C2×D44C22⋊C414yes—
172C2×D44C2×D412yes—
173C2×C4×D11—28no—
174C2×C4×D11—24no—
175C2×C4×D11—22no—
176C2×C4×D11—24no—
177C2×C4×D11—22no—
178C2×Dic22—21no—
179C2×Dic22—22no—
180C2×Dic22C22⋊C414yes—
181C2×Dic22C2×D412yes—
182C11×C4⋊C4—22no—
183C11×C4⋊C4—22no—
184C11×C4⋊C4—42no—
185C11×C22⋊C4—22no—
186C11×C22⋊C4—22no—
187C11×C22⋊C4—42no—
188C42×C11—22no—
189C42×C11—22no—
190C42×C11—22no—
191C42×C11—22no—
192C22⋊Dic11—22no—
193C22⋊Dic11—22no—
194#4326—216no—
195#4325—216no—
196(C2×D11)⋊C4—28no—
197C4⋊Dic11—22no—
198C4⋊Dic11C2×D414yes—
199Dic11⋊C4—28no—
200C4×Dic11—28no—
201C4×Dic11—22no—
202C4×Dic11—22no—
203#4318—24no—
204#4318—22no—
205#4318—22no—
206#4318—22no—
207#4318—42no—
208#4318C22×C414yes—
209C2×C11⋊C8—24no—
210C2×C11⋊C8—22no—
211C2×C11⋊C8—22no—
212C2×C11⋊C8—42no—
213C22×#137—111no—
214#3428—111no—
215C2×#421—111no—
216C2×#419—111no—
217#2310D1111yescharacteristic, complemented
218#1882—118no—
219#1742—111no—
220#1733—111no—
221C23×C11—42no—
222C23×C11C4×D412yes—
223C23×C11#12011yescharacteristic
224C23×C11#12011yescharacteristic
225C23×C11#12112yes—
226C23×C11C22×D411yescharacteristic
227C22×D11—22no—
228C22×D11—24no—
229C22×D11—82no—
230Q8×C11—24no—
231Q8×C11—24no—
232Q8×C11—42no—
233Q8×C11—42no—
234Q8×C11—44no—
235Q8×C11—44no—
236C2×C4×C11—24no—
237C2×C4×C11—24no—
238C2×C4×C11—42no—
239C2×C4×C11—42no—
240C2×C4×C11—44no—
241C2×C4×C11—44no—
242C2×C4×C11—44no—
243C2×C4×C11—44no—
244C2×C4×C11—42no—
245C2×C4×C11C2×C22⋊C412yes—
246C2×C4×C11C2×C22⋊C412yes—
247C2×C4×C11C4×D412yes—
248C2×C4×C11C4×D412yes—
249C2×C4×C11C4×D412yes—
250C2×C4×C11#12011yescharacteristic
251C2×C4×C11#12011yescharacteristic
252C2×C4×C11#12112yes—
253C2×C4×C11#12112yes—
254C2×C4×C11#12112yes—
255C2×C4×C11#12312yes—
256C2×C4×C11#12312yes—
257C2×C4×C11C23×C411yescharacteristic
258C2×C4×C11C22×D411yescharacteristic
259C2×C4×C11C22×D411yescharacteristic
260C2×C4×C11C22×D411yescharacteristic
261C2×C4×C11C2×#4111yescharacteristic
262C2×C4×C11C2×#4111yescharacteristic
263C22⋊D11—24no—
264C22⋊D11—24no—
265C22⋊D11—28no—
266C2×Dic11—22no—
267C2×Dic11—24no—
268C2×Dic11—24no—
269C2×Dic11—82no—
270D44—22no—
271D44—24no—
272D44C2×C22⋊C414yes—
273C4×D11—24no—
274C4×D11—24no—
275C4×D11—28no—
276C4×D11—88no—
277Dic22—22no—
278Dic22—24no—
279Dic22C2×C22⋊C414yes—
280C11⋊C8—24no—
281C11⋊C8—42no—
282C22×#41—111no—
283C23×Q8C2×D1111yescharacteristic
284C2×#137—114no—
285C2×#137—118no—
286C2×#137—222no—
287C2×#137—224no—
288C22×Q16—221no—
289C22×SD16—221no—
290C22×M16—111no—
291C22×M16—111no—
292#520—114no—
293#520—222no—
294#519—221no—
295#518—221no—
296#518—222no—
297C4×#41—222no—
298C4×#41—222no—
299C2×#117—111no—
300C2×#117—222no—
301C2×#117C2×D1111yescharacteristic
302C2×#117C2×D1111yescharacteristic
303#421—224no—
304C2×#104—114no—
305#419—114no—
306#419—222no—
307C2×#103—222no—
308#391—22no—
309#391C2×D1114yescomplemented
310#381—22no—
311#343—114no—
312#339—114no—
313C22×C11—22no—
314C22×C11—22no—
315C22×C11—22no—
316C22×C11—42no—
317C22×C11—42no—
318C22×C11#38612yes—
319C22×C11#38612yes—
320C22×C11C2×C4×D411yescharacteristic
321C22×C11C2×#12011yescharacteristic
322C22×C11C2×#12111yescharacteristic
323C2×D11—24no—
324C2×D11—44no—
325C2×D11—84no—
326C4×C11—44no—
327C4×C11—44no—
328C4×C11—82no—
329C4×C11#38612yes—
330C4×C11#38612yes—
331C4×C11C22×C22⋊C411yescharacteristic
332C4×C11C2×C4×D411yescharacteristic
333C4×C11C2×#12111yescharacteristic
334C4×C11C2×#12311yescharacteristic
335Dic11—24no—
336Dic11—44no—
337C2×#41—114no—
338C2×#41—112no—
339C2×#41—114no—
340C2×#41—112no—
341C2×#41—222no—
342C2×#41—221no—
343C2×#41—222no—
344C2×#41—221no—
345C22×Q8—22no—
346C22×Q8—21no—
347C22×Q8—21no—
348C22×Q8—21no—
349C22×Q8—22no—
350C22×Q8—21no—
351C22×Q8—21no—
352C22×Q8—22no—
353C22×Q8—111no—
354C22×Q8C4×D1114yescomplemented
355C22×Q8C22×D1112yes—
356C22×D4—111no—
357C22×D4—221no—
358C23×C4—21no—
359C23×C4—221no—
360C23×C4C22×D1111yescharacteristic
361#137—1116no—
362#137—2216no—
363#137—228no—
364C2×Q16—228no—
365C2×Q16—224no—
366C2×SD16—228no—
367C2×SD16—224no—
368C2×M16—114no—
369C2×M16—112no—
370C2×M16—112no—
371C2×M16—222no—
372C2×M16—221no—
373C2×M16—221no—
374C2×M16—222no—
375C2×M16—222no—
376C22×C8—221no—
377C22×C8—221no—
378C4×Q8—224no—
379C4×Q8—224no—
380C4×D4—224no—
381C4×D4—228no—
382C4×D4—224no—
383C4×D4—228no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—22no—
389#117—22no—
390#117—112no—
391#117—112no—
392#117—224no—
393#117—224no—
394#117—224no—
395#117—221no—
396#117—221no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—112no—
399C2×C4⋊C4—224no—
400C2×C4⋊C4C22×D1112yes—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—224no—
403C2×C22⋊C4—221no—
404C2×C22⋊C4C22⋊D1112yes—
405C2×C42—224no—
406C2×C42—221no—
407C2×C42C22⋊D1112yes—
408C2×C42C22×D1112yes—
409#104—228no—
410#103—228no—
411#95—28no—
412C2×C11—21no—
413C2×C11—22no—
414C2×C11—21no—
415C2×C11—22no—
416C2×C11C2×#38611yescharacteristic
417C2×C11#302212yes—
418D11—44no—
419C24—221no—
420C24C2×C22⋊D1111yescharacteristic
421#41—118no—
422#41—114no—
423#41—114no—
424#41—224no—
425#41—224no—
426#41—224no—
427#41—224no—
428#41—224no—
429#41—222no—
430#41—222no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—42no—
448C2×Q8—114no—
449C2×Q8—112no—
450C2×Q8—222no—
451C2×Q8—221no—
452C2×Q8C2×C4×D1114yes—
453C2×D4—114no—
454C2×D4—112no—
455C2×D4—224no—
456C2×D4—222no—
457C2×D4—224no—
458C2×D4—222no—
459C2×D4—222no—
460C2×D4—221no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—228no—
468C22×C4—221no—
469C22×C4—224no—
470C22×C4—224no—
471C22×C4—222no—
472C22×C4—222no—
473C22×C4—222no—
474C22×C4C2×C4×D1112yes—
475C22×C4#433912yes—
476C22×C4D4×D1111yescharacteristic
477C22×C4D4×D1111yescharacteristic
478C22×C4D4×D1112yes—
479C22×C4D4×D1111yescharacteristic
480C22×C4D4×D1111yescharacteristic
481C22×C4#434112yes—
482C22×C4C2×C22⋊D1114yes—
483C22×C4C2×C22⋊D1111yescharacteristic
484C22×C4C23×D1111yescharacteristic
485Q16—2216no—
486SD16—2216no—
487M16—114no—
488M16—224no—
489M16—222no—
490M16—222no—
491M16—222no—
492M16—442no—
493C2×C8—224no—
494C2×C8—222no—
495C2×C8—222no—
496C2×C8—442no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—114no—
501C4⋊C4—228no—
502C4⋊C4—222no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—228no—
507C22⋊C4—222no—
508C22⋊C4—222no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—228no—
514C42—222no—
515C42—222no—
516C11#2094411yescharacteristic, complemented, Sylow
517C23—42no—
518C23—222no—
519C23—224no—
520C23—882no—
521C23C4×C22⋊D1112yes—
522C23#6699011yescharacteristic
523C23#6699011yescharacteristic
524C23#6699112yes—
525C23C22×C22⋊D1111yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—42no—
529Q8—42no—
530Q8—44no—
531Q8—44no—
532Q8—114no—
533Q8—222no—
534Q8—224no—
535D4—114no—
536D4—224no—
537D4—224no—
538D4—228no—
539D4—222no—
540D4—224no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—222no—
551C2×C4—224no—
552C2×C4—224no—
553C2×C4—224no—
554C2×C4—228no—
555C2×C4—224no—
556C2×C4—888no—
557C2×C4—882no—
558C2×C4C22⋊C4×D1112yes—
559C2×C4C22⋊C4×D1112yes—
560C2×C4(C22⋊D11)⋊C412yes—
561C2×C4(C22⋊D11)⋊C412yes—
562C2×C4#6693612yes—
563C2×C4#6693612yes—
564C2×C4#6693712yes—
565C2×C4#6693712yes—
566C2×C4C4×C22⋊D1112yes—
567C2×C4#6699011yescharacteristic
568C2×C4#6699011yescharacteristic
569C2×C4#6699112yes—
570C2×C4C22×C4×D1111yescharacteristic
571C2×C4C2×#433911yescharacteristic
572C2×C4C2×D4×D1111yescharacteristic
573C2×C4C2×D4×D1111yescharacteristic
574C2×C4C2×D4×D1111yescharacteristic
575C2×C4C2×#434111yescharacteristic
576C8—224no—
577C8—442no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—224no—
584C22—444no—
585C22—884no—
586C22#1062256912yes—
587C22#1062256912yes—
588C22C2×C4×C22⋊D1111yescharacteristic
589C22C2×#6699011yescharacteristic
590C22C2×#6699111yescharacteristic
591C4—44no—
592C4—44no—
593C4—82no—
594C4—224no—
595C4—444no—
596C4#1062256912yes—
597C4#1062256912yes—
598C4C2×C22⋊C4×D1111yescharacteristic
599C4C2×(C22⋊D11)⋊C411yescharacteristic
600C4C2×#6693611yescharacteristic
601C4C2×#6693711yescharacteristic
602C2—21no—
603C2—22no—
604C2—21no—
605C2—22no—
606C2—444no—
607C2C2×#1062256911yescharacteristic
608C2#1307799412yes—
subgroupG/H
centerC22C2×#66990
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#2310C2
socleC22×C11C2×#120
prime radicalC22×#4339C22
cosocleC22×C4×C11C24
hypercenter#2310D11
hypocenterC11#20944
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. C2×#10622838C2semidirect
5. #13068988C2semidirect
6. #13068355C2semidirect
7. #13068326C2semidirect
8. C11×#2310C2semidirect
16. C2×#66995C4semidirect
31. C11×#391C22semidirect
34. #10622838C22semidirect
38. #10622168C22semidirect
39. #10622161C22semidirect
48. C22×Q8×C11C2×C4semidirect
81. #66995C2×C4semidirect
96. C2×#4318C2×C4semidirect
217. #2310D11semidirect
309. #391C2×D11semidirect
354. C22×Q8C4×D11semidirect
516. C11#20944semidirect

Maximal subgroups

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

subgroupindex
C22×#669952
#130818172
C2×#106226182
C2×#106228382
#130689882
#130683552
#130683262
C11×#23102
#2094411