Finite Group Databasebeta

← #14917070

Subgroups — #14917070

Order 2816.

2410 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623256C211yesmaximal, characteristic
2#13071771C214yesmaximal, complemented
3#13071692C212yesmaximal, complemented
4#13071683C212yesmaximal, complemented
5#13071555C214yesmaximal, complemented
6#13071454C212yesmaximal, complemented
7#13071445C212yesmaximal, complemented
8#13071380C214yesmaximal, complemented
9#13071341C211yesmaximal, characteristic, complemented
10#13071339C211yesmaximal, characteristic, complemented
11#13071336C211yesmaximal, characteristic, complemented
12#13071248C214yesmaximal, complemented
13#13071009C211yesmaximal, characteristic, complemented
14#13071008C211yesmaximal, characteristic, complemented
15C11×#2327C211yesmaximal, characteristic, complemented
16C22×#4343C2212yes—
17C2×C11×C4⋊Q8C2211yescharacteristic
18C2×C4×Q8×C11C2212yes—
19C2×#67000C2212yes—
20C2×Q8×Dic11C2212yes—
21#10623256C4116yescomplemented
22C2×#66948C2212yes—
23C2×#66948C2212yes—
24C2×#66946C2212yes—
25C2×#66946C2212yes—
26C2×#66946C2212yes—
27C2×#66927C2211yescharacteristic
28C2×C4×D44C2212yes—
29C2×C42×D11C2211yescharacteristic
30C11×#391C2212yes—
31C11×#387C2214yes—
32C11×#387C2214yes—
33C11×#382C2211yescharacteristic
34C11×#376C2211yescharacteristic
35#10622843C2212yes—
36#10622603C2214yes—
37#10622602C2214yes—
38#10622601C2214yes—
39#10622601C2212yes—
40#10622600C2212yes—
41#10622600C2214yes—
42#10622594C2214yes—
43#10622594C2214yes—
44C4⋊C4×Dic11C2212yes—
45#10622553C2214yes—
46#10622552C2214yes—
47#10622552C2212yes—
48C4×(C2×D11)⋊C4C2214yes—
49C4×(C2×D11)⋊C4C2211yescharacteristic
50C4×(C2×D11)⋊C4C2211yescharacteristic
51C4×(C2×D11)⋊C4C2214yes—
52#10622549C2211yescharacteristic
53#10622545C2211yescharacteristic
54C4×Dic11⋊C4C2211yescharacteristic
55C4×Dic11⋊C4C2211yescharacteristic
56#10622283C2214yes—
57#10622282C2214yes—
58#10622281C2214yes—
59#10622280C2212yes—
60#10622279C2212yes—
61#10622279C2214yes—
62#10622278C2214yes—
63#10622278C2214yes—
64#10622278C2214yes—
65#10622278C2214yes—
66#10622278C2214yes—
67#10622278C2214yes—
68#10622277C2212yes—
69#10622277C2214yes—
70#10622277C2212yes—
71D11×#95C2214yes—
72#10622264C2212yes—
73#10622264C2212yes—
74#10622263C2212yes—
75#10622261C2212yes—
76C22×Q8×C11C2312yes—
77C2×#4343—44no—
78C2×#4343—44no—
79C2×#4343C2×C4116yescomplemented
80C22×D44C2312yes—
81C22×D44C2312yes—
82C22×D44C2312yes—
83C22×D44C2314yes—
84C22×C4×D11C2311yescharacteristic
85C22×C4×D11C2314yes—
86C22×C4×D11C2314yes—
87C22×C4×D11C2311yescharacteristic
88C22×C4×D11C2311yescharacteristic
89C22×C4×D11C2314yes—
90C11×C4⋊Q8C2×C418yescomplemented
91C4×Q8×C11—44no—
92C2×C11×C4⋊C4C2312yes—
93C2×C11×C4⋊C4C2314yes—
94C2×C11×C4⋊C4C2312yes—
95C2×C11×C4⋊C4C2312yes—
96C2×C42×C11C2311yescharacteristic
97C2×C42×C11C2311yescharacteristic
98C2×C42×C11C2314yes—
99C2×C42×C11C2311yescharacteristic
100#67000C2×C4116yescomplemented
101Q8×Dic11—44no—
102C2×(C2×D11)⋊C4C2314yes—
103C2×(C2×D11)⋊C4C2314yes—
104C2×(C2×D11)⋊C4C2312yes—
105C2×(C2×D11)⋊C4C2314yes—
106C2×(C2×D11)⋊C4C2312yes—
107C2×(C2×D11)⋊C4C2314yes—
108C2×(C2×D11)⋊C4C2314yes—
109C2×(C2×D11)⋊C4C2314yes—
110C2×(C2×D11)⋊C4C2314yes—
111C2×(C2×D11)⋊C4C2314yes—
112C2×(C2×D11)⋊C4C2314yes—
113C2×C4⋊Dic11C2312yes—
114C2×C4⋊Dic11C2314yes—
115C2×Dic11⋊C4C2312yes—
116C2×Dic11⋊C4C2312yes—
117C2×C4×Dic11C2311yescharacteristic
118C2×C4×Dic11C2311yescharacteristic
119C2×C4×Dic11C2314yes—
120C2×C4×Dic11C2314yes—
121C2×C4×Dic11C2311yescharacteristic
122C2×C4×Dic11C2311yescharacteristic
123C2×C4×Dic11C2311yescharacteristic
124#66948C2×C4116yescomplemented
125#66948C2×C4116yescomplemented
126#66946—44no—
127#66946C2×C4116yescomplemented
128#66946C2×C4116yescomplemented
129#66927C2×C418yes—
130C4×D44—44no—
131C42×D11C2×C418yescomplemented
132C11×#95C2312yes—
133C11×#95C2312yes—
134C11×#95C2312yes—
135C11×#95C2312yes—
136C11×#95C2314yes—
137C11×#95C2314yes—
138C11⋊#95C2314yes—
139C11⋊#95C2312yes—
140C11⋊#95C2312yes—
141C11⋊#95C2312yes—
142C11⋊#95C2312yes—
143C11⋊#95C2312yes—
144C11⋊#95C2314yes—
145C11⋊#95C2312yes—
146C11⋊#95C2312yes—
147C11⋊#95C2312yes—
148C11⋊#95C2314yes—
149C11⋊#95C2314yes—
150#34402—111nomaximal, Sylow
151C23×D11C2414yes—
152C23×D11C2411yescharacteristic
153C2×Q8×C11—24no—
154C2×Q8×C11—24no—
155C2×Q8×C11C22×C418yes—
156C22×C4×C11C2412yes—
157C22×C4×C11C2414yes—
158C22×C4×C11C2412yes—
159C22×C4×C11C2411yescharacteristic
160C22×C4×C11C2411yescharacteristic
161C22×C4×C11C2414yes—
162C22×C4×C11C2411yescharacteristic
163C22×Dic11C2412yes—
164C22×Dic11C2414yes—
165C22×Dic11C2412yes—
166C22×Dic11C2411yescharacteristic
167C22×Dic11C2411yescharacteristic
168C22×Dic11C2411yescharacteristic
169#4343—432no—
170C2×D44—24no—
171C2×D44—24no—
172C2×D44—44no—
173C2×D44—44no—
174C2×D44—44no—
175C2×D44—44no—
176C2×D44C22×C418yes—
177C2×D44C22×C418yes—
178C2×D44C22×C418yes—
179C2×D44C22×C4116yes—
180C2×C4×D11—22no—
181C2×C4×D11—22no—
182C2×C4×D11—42no—
183C2×C4×D11—42no—
184C2×C4×D11—44no—
185C2×C4×D11—44no—
186C2×C4×D11—44no—
187C2×C4×D11—44no—
188C2×C4×D11—44no—
189C2×C4×D11—44no—
190C2×C4×D11—44no—
191C2×C4×D11—44no—
192C2×C4×D11C22×C414yes—
193C2×C4×D11C22×C414yes—
194C2×C4×D11C22×C414yes—
195C2×C4×D11C22×C4116yes—
196C11×C4⋊C4—24no—
197C11×C4⋊C4—44no—
198C11×C4⋊C4C22×C418yes—
199C11×C4⋊C4C22×C418yes—
200C42×C11—22no—
201C42×C11—22no—
202C42×C11—44no—
203C42×C11C22×C414yes—
204(C2×D11)⋊C4—44no—
205(C2×D11)⋊C4—44no—
206(C2×D11)⋊C4—44no—
207(C2×D11)⋊C4—44no—
208(C2×D11)⋊C4—42no—
209(C2×D11)⋊C4—44no—
210(C2×D11)⋊C4—44no—
211(C2×D11)⋊C4—44no—
212(C2×D11)⋊C4—42no—
213(C2×D11)⋊C4C22×C4116yes—
214(C2×D11)⋊C4C22×C4116yes—
215C4⋊Dic11—24no—
216C4⋊Dic11—44no—
217Dic11⋊C4—42no—
218Dic11⋊C4—42no—
219C4×Dic11—22no—
220C4×Dic11—22no—
221C4×Dic11—44no—
222C4×Dic11—44no—
223C4×Dic11C22×C414yes—
224C4×Dic11C22×C414yes—
225C4×Dic11C22×C414yes—
226C2×#532—111no—
227#2377—114no—
228#2367—112no—
229#2367—112no—
230#2353—114no—
231#2341—112no—
232#2339—112no—
233#2333—114no—
234#2328—111no—
235#2328—111no—
236#2327—111no—
237#2327D1111yescharacteristic, complemented
238#2318—114no—
239#2283—111no—
240#2283—111no—
241C23×C11C2511yescharacteristic
242C22×D11—24no—
243C22×D11—24no—
244C22×D11—21no—
245C22×D11—21no—
246C22×D11—22no—
247C22×D11—22no—
248C22×D11—24no—
249C22×D11—24no—
250C22×D11—24no—
251C22×D11—24no—
252C22×D11C23×C418yes—
253C22×D11C23×C412yes—
254Q8×C11—216no—
255C2×C4×C11—24no—
256C2×C4×C11—24no—
257C2×C4×C11—22no—
258C2×C4×C11—22no—
259C2×C4×C11—22no—
260C2×C4×C11—24no—
261C2×C4×C11—22no—
262C2×C4×C11—24no—
263C2×C4×C11—22no—
264C2×C4×C11—22no—
265C2×C4×C11—22no—
266C2×C4×C11—24no—
267C2×C4×C11—22no—
268C2×C4×C11C23×C412yes—
269C2×C4×C11C23×C412yes—
270C2×C4×C11C23×C418yes—
271C2×C4×C11C23×C412yes—
272C2×C4×C11H(5,𝔽2)12yes—
273C2×C4×C11H(5,𝔽2)12yes—
274C2×Dic11—21no—
275C2×Dic11—21no—
276C2×Dic11—22no—
277C2×Dic11—22no—
278C2×Dic11—22no—
279C2×Dic11—22no—
280C2×Dic11—24no—
281C2×Dic11—24no—
282C2×Dic11—22no—
283C2×Dic11—22no—
284C2×Dic11—22no—
285C2×Dic11—24no—
286C2×Dic11—22no—
287C2×Dic11C23×C412yes—
288C2×Dic11C23×C412yes—
289C2×Dic11C23×C412yes—
290D44—216no—
291D44—48no—
292D44—48no—
293D44—416no—
294C4×D11—28no—
295C4×D11—44no—
296C4×D11—44no—
297C4×D11—416no—
298C4×D11—88no—
299C4×D11—88no—
300C22×#41—112no—
301#532—1116no—
302C2×C4⋊Q8C2×D1111yescharacteristic
303C2×#127—111no—
304C2×#124—112no—
305C2×#121—112no—
306C2×#121—112no—
307C2×C4×Q8—112no—
308C2×C4×Q8C2×D1112yes—
309C2×C4×D4—112no—
310C2×C4×D4—112no—
311C2×C4×D4—112no—
312C2×C4×D4—112no—
313C22×C42—111no—
314#391—112no—
315#391C2×D1112yes—
316#390—112no—
317#390—114no—
318#390—112no—
319#390—114no—
320#388—114no—
321#388—114no—
322#388—114no—
323#388—114no—
324#387—114no—
325#387—114no—
326#387C2×D1114yes—
327#387C2×D1114yes—
328#386—114no—
329#386—114no—
330#386—114no—
331#386—114no—
332#386—114no—
333#386—114no—
334#385—114no—
335#385—114no—
336#383—112no—
337#383—112no—
338#382—111no—
339#382—112no—
340#382C2×D1111yescharacteristic
341#381—114no—
342#381—112no—
343#381—112no—
344#380—114no—
345#380—112no—
346#380—112no—
347C4×C4⋊C4—112no—
348C4×C4⋊C4—111no—
349C4×C4⋊C4—111no—
350C4×C22⋊C4—114no—
351C4×C22⋊C4—114no—
352C4×C22⋊C4—111no—
353C4×C22⋊C4—111no—
354C4×C22⋊C4—112no—
355C4×C22⋊C4—114no—
356#376—111no—
357#376—112no—
358#376C2×D1111yescharacteristic
359C2×#95—114no—
360C22×C11C24×C411yescharacteristic
361C22×C11C2×H(5,𝔽2)11yescharacteristic
362C22×C11C2×H(5,𝔽2)11yescharacteristic
363C22×C11#58512yes—
364C22×C11#58512yes—
365C2×D11—22no—
366C2×D11—28no—
367C2×D11—22no—
368C2×D11—22no—
369C2×D11—28no—
370C2×D11—28no—
371C2×D11—44no—
372C2×D11—42no—
373C2×D11—44no—
374C2×D11—42no—
375C2×D11—44no—
376C2×D11—44no—
377C4×C11—28no—
378C4×C11—22no—
379C4×C11—22no—
380C4×C11—42no—
381C4×C11—44no—
382C4×C11—42no—
383C4×C11#51814yes—
384Dic11—22no—
385Dic11—22no—
386Dic11—22no—
387Dic11—42no—
388Dic11—44no—
389Dic11—42no—
390C2×#41—1116no—
391C2×#41—444no—
392C2×#41—444no—
393C22×Q8C22×D1112yes—
394C22×D4—112no—
395C22×D4—112no—
396C22×D4—112no—
397C22×D4—114no—
398C23×C4—111no—
399C23×C4—114no—
400C23×C4—111no—
401C23×C4—111no—
402C23×C4—114no—
403C23×C4—114no—
404C4⋊Q8C4×D1118yescomplemented
405#127—118no—
406#124—1116no—
407#121—1116no—
408#121—1116no—
409C4×Q8—44no—
410C4×Q8—444no—
411C4×D4—1116no—
412C4×D4—1116no—
413C4×D4—444no—
414C4×D4—444no—
415C2×C4⋊C4—112no—
416C2×C4⋊C4—114no—
417C2×C4⋊C4—112no—
418C2×C4⋊C4—112no—
419C2×C4⋊C4C22×D1112yes—
420C2×C4⋊C4C22×D1114yes—
421C2×C4⋊C4C22×D1112yes—
422C2×C4⋊C4C22×D1112yes—
423C2×C22⋊C4—114no—
424C2×C22⋊C4—114no—
425C2×C22⋊C4—114no—
426C2×C22⋊C4—114no—
427C2×C22⋊C4—112no—
428C2×C22⋊C4—114no—
429C2×C22⋊C4—112no—
430C2×C22⋊C4—114no—
431C2×C22⋊C4—114no—
432C2×C22⋊C4—114no—
433C2×C22⋊C4—114no—
434C2×C42—111no—
435C2×C42—111no—
436C2×C42—111no—
437C2×C42—118no—
438C2×C42—111no—
439C2×C42—111no—
440C2×C42—114no—
441C2×C42—114no—
442C2×C42C22×D1111yescharacteristic
443C2×C42C22×D1111yescharacteristic
444C2×C42C22×D1114yes—
445C2×C42C22×D1111yescharacteristic
446#95—112no—
447#95—114no—
448#95—114no—
449#95—112no—
450#95—112no—
451#95—112no—
452#95—112no—
453#95—112no—
454#95—112no—
455#95—114no—
456#95—112no—
457#95—114no—
458#95C22×D1114yes—
459#95C22×D1112yes—
460#95C22×D1112yes—
461#95C22×D1112yes—
462#95C22×D1114yes—
463#95C22×D1112yes—
464C2×C11C2×#51811yescharacteristic
465C2×C11#342811yescharacteristic
466C2×C11#342811yescharacteristic
467C2×C11#356212yes—
468C2×C11#356212yes—
469D11—42no—
470D11—48no—
471C24—114no—
472C24—111no—
473#41—4432no—
474C2×Q8—24no—
475C2×Q8—24no—
476C2×Q8C2×C4×D1118yes—
477C2×D4—118no—
478C2×D4—118no—
479C2×D4—118no—
480C2×D4—1116no—
481C2×D4—224no—
482C2×D4—224no—
483C2×D4—444no—
484C2×D4—444no—
485C2×D4—444no—
486C2×D4—444no—
487C22×C4—111no—
488C22×C4—111no—
489C22×C4—111no—
490C22×C4—114no—
491C22×C4—114no—
492C22×C4—114no—
493C22×C4—1116no—
494C22×C4—112no—
495C22×C4—114no—
496C22×C4—112no—
497C22×C4—222no—
498C22×C4—222no—
499C22×C4—444no—
500C22×C4—444no—
501C22×C4—442no—
502C22×C4—442no—
503C22×C4—444no—
504C22×C4—444no—
505C22×C4—444no—
506C22×C4—444no—
507C22×C4—444no—
508C22×C4—444no—
509C22×C4C23×D1112yes—
510C22×C4C23×D1114yes—
511C22×C4C23×D1112yes—
512C22×C4C23×D1111yescharacteristic
513C22×C4C23×D1111yescharacteristic
514C22×C4C23×D1114yes—
515C22×C4C23×D1111yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—224no—
519C4⋊C4—444no—
520C4⋊C4—442no—
521C4⋊C4—442no—
522C4⋊C4C2×C4×D1118yes—
523C4⋊C4C2×C4×D1118yes—
524C22⋊C4—1116no—
525C22⋊C4—1116no—
526C22⋊C4—444no—
527C22⋊C4—444no—
528C22⋊C4—444no—
529C22⋊C4—444no—
530C22⋊C4—442no—
531C22⋊C4—444no—
532C22⋊C4—444no—
533C22⋊C4—444no—
534C22⋊C4—442no—
535C42—22no—
536C42—22no—
537C42—44no—
538C42—114no—
539C42—114no—
540C42—114no—
541C42—222no—
542C42—222no—
543C42—444no—
544C42—444no—
545C42C2×C4×D1114yes—
546C11#3440211yescharacteristic, complemented, Sylow
547C23—118no—
548C23—112no—
549C23—224no—
550C23—221no—
551C23—224no—
552C23—221no—
553C23—222no—
554C23—222no—
555C23—224no—
556C23—224no—
557C23—224no—
558C23—224no—
559C23C24×D1111yescharacteristic
560Q8—216no—
561D4—2216no—
562D4—4416no—
563D4—448no—
564D4—448no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—112no—
579C2×C4—112no—
580C2×C4—112no—
581C2×C4—221no—
582C2×C4—221no—
583C2×C4—222no—
584C2×C4—222no—
585C2×C4—228no—
586C2×C4—222no—
587C2×C4—222no—
588C2×C4—224no—
589C2×C4—224no—
590C2×C4—222no—
591C2×C4—222no—
592C2×C4—224no—
593C2×C4—222no—
594C2×C4—222no—
595C2×C4—4416no—
596C2×C4—444no—
597C2×C4—444no—
598C2×C4—888no—
599C2×C4—888no—
600C2×C4C22×C4×D1112yes—
601C2×C4C22×C4×D1112yes—
602C2×C4C22×C4×D1118yes—
603C2×C4C22×C4×D1112yes—
604C2×C4#6703712yes—
605C2×C4#6704212yes—
606C22—222no—
607C22—228no—
608C22—222no—
609C22—222no—
610C22—228no—
611C22—228no—
612C22—444no—
613C22—444no—
614C22—442no—
615C22—444no—
616C22—444no—
617C22—442no—
618C22C23×C4×D1111yescharacteristic
619C22C2×#6703711yescharacteristic
620C22C2×#6704211yescharacteristic
621C22#1062345712yes—
622C22#1062345712yes—
623C4—28no—
624C4—22no—
625C4—22no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—222no—
630C4—222no—
631C4—222no—
632C4—442no—
633C4—444no—
634C4—442no—
635C4#1062307614yes—
636C2—442no—
637C2—448no—
638C2C2×#1062307611yescharacteristic
639C2#1308181811yescharacteristic
640C2#1308181811yescharacteristic
641C2#1308319212yes—
642C2#1308321012yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC11×#2327C2
socle, cosocleC23×C11C25
prime radicalC2×#10623256C2
hypercenter#2327D11
hypocenterC11#34402
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ #2327

Complements

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

NH ≅ G/Ntype
2. #13071771C2semidirect
3. #13071692C2semidirect
4. #13071683C2semidirect
5. #13071555C2semidirect
6. #13071454C2semidirect
7. #13071445C2semidirect
8. #13071380C2semidirect
9. #13071341C2semidirect
10. #13071339C2semidirect
11. #13071336C2semidirect
12. #13071248C2semidirect
13. #13071009C2semidirect
14. #13071008C2semidirect
15. C11×#2327C2semidirect
21. #10623256C4semidirect
79. C2×#4343C2×C4semidirect
90. C11×C4⋊Q8C2×C4semidirect
100. #67000C2×C4semidirect
124. #66948C2×C4semidirect
125. #66948C2×C4semidirect
127. #66946C2×C4semidirect
128. #66946C2×C4semidirect
131. C42×D11C2×C4semidirect
237. #2327D11semidirect
404. C4⋊Q8C4×D11semidirect
546. C11#34402semidirect

Maximal subgroups

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

subgroupindex
C2×#106232562
#130717712
#130716922
#130716832
#130715552
#130714542
#130714452
#130713802
#130713412
#130713392
#130713362
#130712482
#130710092
#130710082
C11×#23272
#3440211