Finite Group Databasebeta

← #14684377

Subgroups — #14684377

Order 2816.

1550 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13066884C211yesmaximal, characteristic, complemented
2#13066865C212yesmaximal, complemented
3#13066857C212yesmaximal
4#13066853C212yesmaximal, complemented
5#13066853C212yesmaximal, complemented
6#13066853C212yesmaximal, complemented
7#13066848C211yesmaximal, characteristic, complemented
8#13066841C212yesmaximal, complemented
9C2×C11×#394C211yesmaximal, characteristic
10C23×D4×C11C2211yescharacteristic
11C22×C11×C4⋊C4C2211yescharacteristic
12C22×C11×C22⋊C4C2212yes—
13C22×C11×C22⋊C4C2212yes—
14C22×C22⋊Dic11C2212yes—
15C22×C22⋊Dic11C2212yes—
16C22×C22⋊Dic11C2212yes—
17C22×C22⋊Dic11C2212yes—
18C22×C4×Dic11C2212yes—
19C22×C4×Dic11C2211yescharacteristic
20C11×#394C418yescomplemented
21C2×C11×#95C2211yescharacteristic
22C2×C11⋊#95C2212yes—
23C2×C11⋊#95C2212yes—
24C2×C11⋊#95C2212yes—
25C2×C11⋊#95C2212yes—
26C2×C11⋊#95C2212yes—
27C2×C11⋊#95C2212yes—
28C2×C11⋊#95C2211yescharacteristic
29C2×C11⋊#95C2212yes—
30C2×C11⋊#95C2211yescharacteristic
31C2×C11⋊#95C2211yescharacteristic
32C25×C11C2312yes—
33C22×D4×C11—42no—
34C22×D4×C11—42no—
35C22×D4×C11—41no—
36C22×D4×C11—41no—
37C22×D4×C11C2×C414yes—
38C23×C4×C11C2312yes—
39C23×C4×C11C2311yescharacteristic
40C23×C4×C11C2311yescharacteristic
41C23×C4×C11C2311yescharacteristic
42C23×Dic11C2312yes—
43C23×Dic11C2312yes—
44C23×Dic11C2312yes—
45C23×Dic11C2312yes—
46C2×C11×C4⋊C4—41no—
47C2×C11×C4⋊C4—41no—
48C2×C11×C4⋊C4C2×C414yes—
49C2×C11×C22⋊C4—42no—
50C2×C11×C22⋊C4—42no—
51C2×C11×C22⋊C4—42no—
52C2×C11×C22⋊C4—42no—
53C2×C11×C22⋊C4C2×C418yes—
54C2×C11×C22⋊C4C2×C418yes—
55C2×C22⋊Dic11—24no—
56C2×C22⋊Dic11—24no—
57C2×C22⋊Dic11—24no—
58C2×C22⋊Dic11—24no—
59C2×C22⋊Dic11—24no—
60C2×C22⋊Dic11—24no—
61C2×C22⋊Dic11—42no—
62C2×C22⋊Dic11—42no—
63C2×C22⋊Dic11—42no—
64C2×C22⋊Dic11—42no—
65C2×C22⋊Dic11—42no—
66C2×C22⋊Dic11—42no—
67C2×C4×Dic11—24no—
68C2×C4×Dic11—42no—
69C2×C4×Dic11—42no—
70C2×C4×Dic11—42no—
71C2×C4×Dic11—41no—
72C11×#95C2×C414yes—
73C11⋊#95—24no—
74C11⋊#95—24no—
75C11⋊#95—42no—
76C11⋊#95—42no—
77C11⋊#95—42no—
78C11⋊#95—42no—
79C11⋊#95—41no—
80C11⋊#95—41no—
81C11⋊#95—42no—
82C11⋊#95—41no—
83#14617—111nomaximal, Sylow
84C24×C11—22no—
85C24×C11—22no—
86C24×C11—22no—
87C24×C11—22no—
88C24×C11—22no—
89C24×C11—22no—
90C24×C11—22no—
91C24×C11—22no—
92C24×C11—22no—
93C24×C11—22no—
94C24×C11—22no—
95C24×C11—22no—
96C24×C11—22no—
97C24×C11—22no—
98C24×C11C22×C414yes—
99C24×C11C2411yescharacteristic
100C2×D4×C11—44no—
101C2×D4×C11—48no—
102C2×D4×C11—44no—
103C2×D4×C11—48no—
104C2×D4×C11—44no—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11—22no—
110C22×C4×C11—22no—
111C22×C4×C11—21no—
112C22×C4×C11—21no—
113C22×C4×C11—21no—
114C22×C4×C11—22no—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—22no—
118C22×C4×C11—22no—
119C22×C4×C11—21no—
120C22×C4×C11—22no—
121C22×C4×C11—21no—
122C22×C4×C11—21no—
123C22×C4×C11C22×C414yes—
124C22×C4×C11C22×C412yes—
125C22×C4×C11C22×C412yes—
126C22×C4×C11C22×C412yes—
127C22×Dic11—22no—
128C22×Dic11—22no—
129C22×Dic11—22no—
130C22×Dic11—22no—
131C22×Dic11—22no—
132C22×Dic11—22no—
133C22×Dic11—22no—
134C22×Dic11—22no—
135C22×Dic11—22no—
136C22×Dic11—22no—
137C22×Dic11—22no—
138C22×Dic11—22no—
139C22×Dic11—22no—
140C22×Dic11—22no—
141C22×Dic11—22no—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11—22no—
145C22×Dic11—22no—
146C22×Dic11—22no—
147C22×Dic11—22no—
148C22×Dic11—22no—
149C22×Dic11—22no—
150C22×Dic11—22no—
151C22×Dic11—22no—
152C22×Dic11#4114yes—
153C22×Dic11#4114yes—
154C22×Dic11#4114yes—
155C11×C4⋊C4—44no—
156C11×C22⋊C4—48no—
157C11×C22⋊C4—48no—
158C22⋊Dic11—48no—
159C22⋊Dic11—48no—
160C22⋊Dic11—48no—
161C22⋊Dic11—48no—
162C4×Dic11—48no—
163C4×Dic11—44no—
164C2×#394D1111yescharacteristic
165#1445—111no—
166#1440—112no—
167#1438—112no—
168#1437—112no—
169#1437—112no—
170#1437—112no—
171#1436—111no—
172#1434—112no—
173C23×C11—24no—
174C23×C11—24no—
175C23×C11—24no—
176C23×C11—24no—
177C23×C11—24no—
178C23×C11—24no—
179C23×C11—24no—
180C23×C11—42no—
181C23×C11—42no—
182C23×C11—42no—
183C23×C11—42no—
184C23×C11—42no—
185C23×C11—42no—
186C23×C11—42no—
187C23×C11—42no—
188C23×C11—42no—
189C23×C11—42no—
190C23×C11—42no—
191C23×C11—42no—
192C23×C11—42no—
193C23×C11—42no—
194C23×C11—42no—
195C23×C11—42no—
196C23×C11—42no—
197C23×C11—42no—
198C23×C11—42no—
199C23×C11—42no—
200C23×C11—42no—
201C23×C11—42no—
202C23×C11—42no—
203C23×C11—42no—
204C23×C11—42no—
205C23×C11—42no—
206C23×C11—42no—
207C23×C11—42no—
208C23×C11C23×C411yescharacteristic
209C23×C11C2×#4112yes—
210C23×C11C2×#4112yes—
211C23×C11C2×#4112yes—
212C23×C11H(5,𝔽2)12yes—
213C23×C11H(5,𝔽2)11yescharacteristic
214C23×C11H(5,𝔽2)12yes—
215C23×C11H(5,𝔽2)12yes—
216C23×C11H(5,𝔽2)11yescharacteristic
217D4×C11—416no—
218C2×C4×C11—24no—
219C2×C4×C11—22no—
220C2×C4×C11—22no—
221C2×C4×C11—24no—
222C2×C4×C11—24no—
223C2×C4×C11—24no—
224C2×C4×C11—22no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—24no—
228C2×C4×C11—42no—
229C2×C4×C11—42no—
230C2×C4×C11—42no—
231C2×C4×C11—42no—
232C2×C4×C11—42no—
233C2×C4×C11—42no—
234C2×C4×C11—41no—
235C2×C4×C11—42no—
236C2×C4×C11—41no—
237C2×C4×C11—42no—
238C2×C4×C11—41no—
239C2×C4×C11—41no—
240C2×Dic11—24no—
241C2×Dic11—24no—
242C2×Dic11—24no—
243C2×Dic11—24no—
244C2×Dic11—24no—
245C2×Dic11—24no—
246C2×Dic11—24no—
247C2×Dic11—24no—
248C2×Dic11—24no—
249C2×Dic11—42no—
250C2×Dic11—42no—
251C2×Dic11—42no—
252C2×Dic11—42no—
253C2×Dic11—42no—
254C2×Dic11—42no—
255C2×Dic11—42no—
256C2×Dic11—42no—
257C2×Dic11—42no—
258C2×Dic11—42no—
259C2×Dic11—42no—
260C2×Dic11—42no—
261C2×Dic11—42no—
262C2×Dic11—42no—
263C2×Dic11—42no—
264C2×Dic11—42no—
265C2×Dic11—42no—
266C2×Dic11—42no—
267C2×Dic11—42no—
268C23×D4C2×D1111yescharacteristic
269C22×C4⋊C4C2×D1111yescharacteristic
270C22×C22⋊C4—112no—
271C22×C22⋊C4—112no—
272C22×C22⋊C4—112no—
273C22×C22⋊C4—112no—
274C22×C22⋊C4C2×D1112yes—
275C22×C22⋊C4C2×D1112yes—
276C22×C42—112no—
277C22×C42—111no—
278#394Dic1118yescomplemented
279C2×#95—112no—
280C2×#95—112no—
281C2×#95—112no—
282C2×#95—112no—
283C2×#95—112no—
284C2×#95—112no—
285C2×#95—111no—
286C2×#95—111no—
287C2×#95—112no—
288C2×#95—111no—
289C2×#95C2×D1111yescharacteristic
290C22×C11—44no—
291C22×C11—44no—
292C22×C11—44no—
293C22×C11—44no—
294C22×C11—44no—
295C22×C11—44no—
296C22×C11—44no—
297C22×C11—82no—
298C22×C11—82no—
299C22×C11—82no—
300C22×C11—82no—
301C22×C11—82no—
302C22×C11—82no—
303C22×C11—82no—
304C22×C11—82no—
305C22×C11C4×#4112yes—
306C22×C11C4×#4111yescharacteristic
307C22×C11#51811yescharacteristic
308C22×C11#51811yescharacteristic
309C22×C11#51812yes—
310C22×C11#53812yes—
311C22×C11#53812yes—
312C22×C11#53812yes—
313C22×C11#54012yes—
314C22×C11#54012yes—
315C22×C11#54012yes—
316C22×C11#55112yes—
317C22×C11#55112yes—
318C22×C11#55312yes—
319C22×C11#55312yes—
320C22×C11#55511yescharacteristic
321C22×C11#55511yescharacteristic
322C22×C11#55912yes—
323C22×C11#56011yescharacteristic
324C22×C11#56012yes—
325C22×C11H(3,𝔽4)11yescharacteristic
326C4×C11—42no—
327C4×C11—44no—
328C4×C11—42no—
329C4×C11—42no—
330Dic11—44no—
331Dic11—44no—
332Dic11—44no—
333Dic11—82no—
334C25C22×D1112yes—
335C22×D4—42no—
336C22×D4—42no—
337C22×D4—41no—
338C22×D4—41no—
339C22×D4C2×Dic1114yes—
340C23×C4—112no—
341C23×C4—112no—
342C23×C4—112no—
343C23×C4—112no—
344C23×C4C22×D1111yescharacteristic
345C23×C4C22×D1111yescharacteristic
346C23×C4C22×D1112yes—
347C23×C4C22×D1111yescharacteristic
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4C2×Dic1114yes—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—224no—
356C2×C22⋊C4—224no—
357C2×C22⋊C4—224no—
358C2×C22⋊C4—224no—
359C2×C22⋊C4—224no—
360C2×C22⋊C4—224no—
361C2×C22⋊C4—442no—
362C2×C22⋊C4—442no—
363C2×C22⋊C4—442no—
364C2×C22⋊C4—442no—
365C2×C22⋊C4—442no—
366C2×C22⋊C4—442no—
367C2×C22⋊C4C2×Dic1118yes—
368C2×C22⋊C4C2×Dic1118yes—
369C2×C42—224no—
370C2×C42—442no—
371C2×C42—442no—
372C2×C42—442no—
373C2×C42—441no—
374#95—224no—
375#95—224no—
376#95—442no—
377#95—442no—
378#95—442no—
379#95—441no—
380#95—441no—
381#95—441no—
382#95—442no—
383#95—442no—
384#95C2×Dic1114yes—
385C2×C11—84no—
386C2×C11#233312yes—
387C2×C11#233311yescharacteristic
388C2×C11#237211yescharacteristic
389C2×C11#237412yes—
390C2×C11#237511yescharacteristic
391C2×C11#275212yes—
392C2×C11#275312yes—
393C2×C11#275612yes—
394C2×C11#280312yes—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24C22×Dic1114yes—
410C24C23×D1111yescharacteristic
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—48no—
415C2×D4—44no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—114no—
435C22×C4—114no—
436C22×C4—114no—
437C22×C4—222no—
438C22×C4—222no—
439C22×C4—222no—
440C22×C4—222no—
441C22×C4—222no—
442C22×C4—222no—
443C22×C4—222no—
444C22×C4—222no—
445C22×C4—222no—
446C22×C4—222no—
447C22×C4—222no—
448C22×C4—222no—
449C22×C4—222no—
450C22×C4—222no—
451C22×C4—222no—
452C22×C4—222no—
453C22×C4—222no—
454C22×C4—222no—
455C22×C4—222no—
456C22×C4—222no—
457C22×C4—222no—
458C22×C4—222no—
459C22×C4—222no—
460C22×C4—222no—
461C22×C4—222no—
462C22×C4C22×Dic1114yes—
463C22×C4C22×Dic1112yes—
464C22×C4C22×Dic1112yes—
465C22×C4C22×Dic1112yes—
466C4⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—448no—
470C22⋊C4—448no—
471C22⋊C4—448no—
472C22⋊C4—448no—
473C42—448no—
474C42—444no—
475C11#1461711yescharacteristic, complemented, Sylow
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23#6703712yes—
512C23#6703711yescharacteristic
513C23#6703711yescharacteristic
514C23#6703712yes—
515C23#6703712yes—
516C23#41×D1112yes—
517C23#41×D1112yes—
518C23#41×D1112yes—
519C23C23×Dic1111yescharacteristic
520D4—416no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—224no—
544C2×C4—224no—
545C2×C4—224no—
546C2×C4—224no—
547C2×C4—224no—
548C2×C4—224no—
549C2×C4—224no—
550C2×C4—224no—
551C2×C4—224no—
552C2×C4—442no—
553C2×C4—442no—
554C2×C4—442no—
555C2×C4—442no—
556C2×C4—442no—
557C2×C4—442no—
558C2×C4—442no—
559C2×C4—442no—
560C2×C4—442no—
561C2×C4—442no—
562C2×C4—442no—
563C2×C4—442no—
564C2×C4—442no—
565C2×C4—442no—
566C2×C4—442no—
567C2×C4—442no—
568C2×C4—442no—
569C2×C4—442no—
570C2×C4—442no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#1062312511yescharacteristic
587C22#1062312612yes—
588C22#1062312611yescharacteristic
589C22#1062314412yes—
590C22#1062314412yes—
591C22#1062314412yes—
592C22#1062314612yes—
593C22#1062314612yes—
594C22#1062314612yes—
595C22#1062319012yes—
596C22#1062319212yes—
597C22#1062319212yes—
598C22#1062319512yes—
599C22#1062320111yescharacteristic
600C22#1062320111yescharacteristic
601C22#1062321012yes—
602C22#1062332911yescharacteristic
603C22#1062332911yescharacteristic
604C22#1062332912yes—
605C22#41×Dic1112yes—
606C22#41×Dic1111yescharacteristic
607C4—42no—
608C4—44no—
609C4—42no—
610C4—42no—
611C4—444no—
612C4—444no—
613C4—444no—
614C4—882no—
615C2—84no—
616C2#1307138311yescharacteristic
617C2#1307138312yes—
618C2#1307172811yescharacteristic
619C2#1307175212yes—
620C2#1307175711yescharacteristic
621C2#1307588212yes—
622C2#1307589212yes—
623C2#1307594312yes—
624C2#1307655612yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC2×C11×#394C2
socle, cosocleC24×C11C24
prime radicalC23×D4×C11C22
hypercenterC2×#394D11
hypocenterC11#14617
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. #13066884C2semidirect
2. #13066865C2semidirect
4. #13066853C2semidirect
5. #13066853C2semidirect
6. #13066853C2semidirect
7. #13066848C2semidirect
8. #13066841C2semidirect
20. C11×#394C4semidirect
278. #394Dic11semidirect
475. C11#14617semidirect

Maximal subgroups

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

subgroupindex
#130668842
#130668652
#130668572
#130668532
#130668482
#130668412
C2×C11×#3942
#1461711