Finite Group Databasebeta

← #14684456

Subgroups — #14684456

Order 2816.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622286C211yesmaximal, characteristic
2C2×#10622542C211yesmaximal, characteristic, complemented
3C2×#10622553C211yesmaximal, characteristic
4#13066886C211yesmaximal, characteristic, complemented
5#13066881C211yesmaximal, characteristic, complemented
6#13066872C212yesmaximal, complemented
7#13066871C212yesmaximal, complemented
8#13066864C212yesmaximal
9#13066862C211yesmaximal, characteristic
10#13066862C212yesmaximal
11C11×#1434C211yesmaximal, characteristic, complemented
12C23×Dic22C2211yescharacteristic
13C22×C42×C11C2211yescharacteristic
14C22×(C2×D11)⋊C4C2211yescharacteristic
15C22×(C2×D11)⋊C4C2211yescharacteristic
16C22×(C2×D11)⋊C4C2211yescharacteristic
17C22×(C2×D11)⋊C4C2212yes—
18C22×(C2×D11)⋊C4C2212yes—
19C22×C4⋊Dic11C2211yescharacteristic
20C22×C4⋊Dic11C2212yes—
21C22×Dic11⋊C4C2211yescharacteristic
22C2×C11×#95C2211yescharacteristic
23C2×C11×#95C2212yes—
24C2×C11×#95C2212yes—
25C2×C11×#95C2211yescharacteristic
26C2×C11⋊#95C2212yes—
27C2×C11⋊#95C2212yes—
28C2×C11⋊#95C2211yescharacteristic
29C2×C11⋊#95C2211yescharacteristic
30C2×C11⋊#95C2212yes—
31C2×C11⋊#95C2212yes—
32C2×C11⋊#95C2212yes—
33C2×C11⋊#95C2212yes—
34C2×C11⋊#95C2212yes—
35#10622553—24no—
36#10622542—24no—
37#10622286C418yescomplemented
38C24×D11C2311yescharacteristic
39C23×C4×C11C2312yes—
40C23×C4×C11C2312yes—
41C23×C4×C11C2311yescharacteristic
42C23×C4×C11C2311yescharacteristic
43C23×C4×C11C2311yescharacteristic
44C23×Dic11C2311yescharacteristic
45C23×Dic11C2312yes—
46C23×Dic11C2311yescharacteristic
47C23×Dic11C2311yescharacteristic
48C23×Dic11C2312yes—
49C22×Dic22—22no—
50C22×Dic22—22no—
51C22×Dic22—42no—
52C22×Dic22—42no—
53C22×Dic22C2×C414yes—
54C2×C42×C11—42no—
55C2×C42×C11D414yes—
56C2×(C2×D11)⋊C4—22no—
57C2×(C2×D11)⋊C4—22no—
58C2×(C2×D11)⋊C4—22no—
59C2×(C2×D11)⋊C4—22no—
60C2×(C2×D11)⋊C4—42no—
61C2×(C2×D11)⋊C4—42no—
62C2×(C2×D11)⋊C4—42no—
63C2×(C2×D11)⋊C4—42no—
64C2×(C2×D11)⋊C4—42no—
65C2×(C2×D11)⋊C4—42no—
66C2×(C2×D11)⋊C4—42no—
67C2×(C2×D11)⋊C4C2×C414yes—
68C2×(C2×D11)⋊C4C2×C418yes—
69C2×C4⋊Dic11—22no—
70C2×C4⋊Dic11—42no—
71C2×C4⋊Dic11—42no—
72C2×C4⋊Dic11—42no—
73C2×C4⋊Dic11—42no—
74C2×Dic11⋊C4—22no—
75C2×Dic11⋊C4—42no—
76C11×#95—22no—
77C11×#95—42no—
78C11×#95—42no—
79C11×#95C2×C414yes—
80C11⋊#95—24no—
81C11⋊#95—24no—
82C11⋊#95—22no—
83C11⋊#95—22no—
84C11⋊#95—42no—
85C11⋊#95—42no—
86C11⋊#95—42no—
87C11⋊#95—42no—
88C11⋊#95C2×C418yes—
89#14625—111nomaximal, Sylow
90C24×C11C2411yescharacteristic
91C23×D11—21no—
92C23×D11—21no—
93C23×D11—22no—
94C23×D11—22no—
95C23×D11—22no—
96C23×D11—21no—
97C23×D11—21no—
98C23×D11—21no—
99C23×D11—21no—
100C23×D11—22no—
101C23×D11C22×C412yes—
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—22no—
105C22×C4×C11—21no—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11—22no—
110C22×C4×C11—22no—
111C22×C4×C11—22no—
112C22×C4×C11—22no—
113C22×C4×C11—22no—
114C22×C4×C11—21no—
115C22×C4×C11—21no—
116C22×C4×C11—22no—
117C22×C4×C11—21no—
118C22×C4×C11—22no—
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11—22no—
122C22×C4×C11—22no—
123C22×C4×C11—22no—
124C22×C4×C11—22no—
125C22×C4×C11C22×C414yes—
126C22×C4×C11C22×C412yes—
127C22×C4×C11C2×D412yes—
128C22×C4×C11C2×D412yes—
129C22×C4×C11C2×D412yes—
130C22×C4×C11#4112yes—
131C22×Dic11—22no—
132C22×Dic11—21no—
133C22×Dic11—22no—
134C22×Dic11—21no—
135C22×Dic11—21no—
136C22×Dic11—21no—
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—21no—
149C22×Dic11—22no—
150C22×Dic11—22no—
151C22×Dic11—22no—
152C22×Dic11—22no—
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11—21no—
156C22×Dic11—21no—
157C22×Dic11—21no—
158C22×Dic11C22×C412yes—
159C22×Dic11C22×C414yes—
160C2×Dic22—28no—
161C2×Dic22—48no—
162C2×Dic22—44no—
163C2×Dic22—44no—
164C2×Dic22—48no—
165C42×C11—44no—
166(C2×D11)⋊C4—28no—
167(C2×D11)⋊C4—48no—
168(C2×D11)⋊C4—44no—
169(C2×D11)⋊C4—44no—
170(C2×D11)⋊C4—48no—
171C4⋊Dic11—44no—
172C4⋊Dic11—48no—
173Dic11⋊C4—44no—
174C2×#393—111no—
175C2×#391—111no—
176C2×#388—111no—
177#1446—111no—
178#1444—111no—
179#1442—112no—
180#1442—112no—
181#1440—112no—
182#1439—111no—
183#1439—112no—
184#1434D1111yescharacteristic, complemented
185C23×C11C23×C411yescharacteristic
186C23×C11C22×D411yescharacteristic
187C23×C11C2×#4111yescharacteristic
188C23×C11H(5,𝔽2)12yes—
189C23×C11H(5,𝔽2)11yescharacteristic
190C23×C11H(5,𝔽2)12yes—
191C23×C11H(5,𝔽2)12yes—
192C23×C112−1+411yescharacteristic
193C23×C112−1+411yescharacteristic
194C23×C112−1+411yescharacteristic
195C23×C112−1+412yes—
196C22×D11—22no—
197C22×D11—24no—
198C22×D11—22no—
199C22×D11—24no—
200C22×D11—22no—
201C22×D11—41no—
202C22×D11—41no—
203C22×D11—41no—
204C22×D11—41no—
205C22×D11—41no—
206C22×D11—41no—
207C22×D11—42no—
208C22×D11—41no—
209C22×D11—42no—
210C22×D11—41no—
211C22×D11—42no—
212C22×D11—42no—
213C22×D11—42no—
214C22×D11—42no—
215C22×D11—42no—
216C22×D11—42no—
217C22×D11—42no—
218C22×D11—42no—
219C2×C4×C11—24no—
220C2×C4×C11—22no—
221C2×C4×C11—22no—
222C2×C4×C11—24no—
223C2×C4×C11—24no—
224C2×C4×C11—24no—
225C2×C4×C11—24no—
226C2×C4×C11—22no—
227C2×C4×C11—22no—
228C2×C4×C11—22no—
229C2×C4×C11—22no—
230C2×C4×C11—22no—
231C2×C4×C11—22no—
232C2×C4×C11—42no—
233C2×C4×C11—42no—
234C2×C4×C11—42no—
235C2×C4×C11—42no—
236C2×C4×C11—42no—
237C2×C4×C11—42no—
238C2×C4×C11—42no—
239C2×C4×C11—42no—
240C2×C4×C11—42no—
241C2×C4×C11—42no—
242C2×C4×C11—42no—
243C2×C4×C11—42no—
244C2×C4×C11—42no—
245C2×C4×C11—42no—
246C2×C4×C11—42no—
247C2×C4×C11C4×D414yes—
248C2×Dic11—22no—
249C2×Dic11—24no—
250C2×Dic11—24no—
251C2×Dic11—24no—
252C2×Dic11—24no—
253C2×Dic11—42no—
254C2×Dic11—41no—
255C2×Dic11—42no—
256C2×Dic11—41no—
257C2×Dic11—42no—
258C2×Dic11—42no—
259C2×Dic11—42no—
260C2×Dic11—41no—
261C2×Dic11—41no—
262C2×Dic11—42no—
263C2×Dic11—41no—
264C2×Dic11—42no—
265C2×Dic11—41no—
266C2×Dic11—41no—
267C2×Dic11—42no—
268C2×Dic11—41no—
269C2×Dic11—42no—
270C2×Dic11—42no—
271C2×Dic11—42no—
272C2×Dic11—42no—
273C2×Dic11—42no—
274C2×Dic11—42no—
275C2×Dic11—42no—
276C2×Dic11—42no—
277Dic22—416no—
278C23×Q8—111no—
279C22×C4⋊C4—111no—
280C22×C4⋊C4—111no—
281C22×C4⋊C4—112no—
282C22×C22⋊C4—111no—
283C22×C22⋊C4—112no—
284C22×C22⋊C4—111no—
285C22×C22⋊C4—111no—
286C22×C22⋊C4—112no—
287C22×C42C2×D1111yescharacteristic
288#393—118no—
289#391—224no—
290#388—224no—
291C2×#95—112no—
292C2×#95—112no—
293C2×#95—112no—
294C2×#95—111no—
295C2×#95—111no—
296C2×#95—112no—
297C2×#95—112no—
298C2×#95—112no—
299C2×#95—112no—
300C2×#95C2×D1111yescharacteristic
301C2×#95C2×D1112yes—
302C2×#95C2×D1112yes—
303C2×#95C2×D1111yescharacteristic
304C22×C11C2×C4×D411yescharacteristic
305C22×C11#51812yes—
306C22×C11#51911yescharacteristic
307C22×C11#52012yes—
308C22×C11#52011yescharacteristic
309C22×C11#53411yescharacteristic
310C22×C11#53511yescharacteristic
311C22×C11#53511yescharacteristic
312C22×C11#53611yescharacteristic
313C22×C11#53611yescharacteristic
314C22×C11#53711yescharacteristic
315C22×C11#53812yes—
316C22×C11#54111yescharacteristic
317C22×C11#54212yes—
318C22×C11#54211yescharacteristic
319C22×C11H(3,𝔽4)12yes—
320C22×C11#56212yes—
321C22×C11#56212yes—
322C22×C11#56212yes—
323C22×C11#56212yes—
324C22×C11#56312yes—
325C22×C11#56312yes—
326C22×C11#56312yes—
327C2×D11—42no—
328C2×D11—42no—
329C2×D11—42no—
330C2×D11—44no—
331C2×D11—44no—
332C2×D11—81no—
333C2×D11—81no—
334C2×D11—81no—
335C2×D11—82no—
336C2×D11—82no—
337C2×D11—81no—
338C4×C11—24no—
339C4×C11—42no—
340C4×C11—44no—
341C4×C11—42no—
342C4×C11—82no—
343Dic11—44no—
344Dic11—42no—
345Dic11—81no—
346Dic11—82no—
347Dic11—81no—
348C25—111no—
349C22×Q8—114no—
350C22×Q8—222no—
351C22×Q8—222no—
352C22×Q8—442no—
353C22×Q8—442no—
354C23×C4—111no—
355C23×C4—112no—
356C23×C4—111no—
357C23×C4—112no—
358C23×C4—111no—
359C23×C4C22×D1112yes—
360C23×C4C22×D1112yes—
361C23×C4C22×D1111yescharacteristic
362C23×C4C22×D1111yescharacteristic
363C23×C4C22×D1111yescharacteristic
364C2×C4⋊C4—222no—
365C2×C4⋊C4—222no—
366C2×C4⋊C4—442no—
367C2×C4⋊C4—442no—
368C2×C4⋊C4—442no—
369C2×C4⋊C4—442no—
370C2×C4⋊C4—442no—
371C2×C22⋊C4—114no—
372C2×C22⋊C4—118no—
373C2×C22⋊C4—222no—
374C2×C22⋊C4—222no—
375C2×C22⋊C4—222no—
376C2×C22⋊C4—222no—
377C2×C22⋊C4—442no—
378C2×C22⋊C4—442no—
379C2×C22⋊C4—442no—
380C2×C22⋊C4—442no—
381C2×C22⋊C4—442no—
382C2×C22⋊C4—442no—
383C2×C22⋊C4—442no—
384C2×C42—42no—
385C2×C42D4414yes—
386#95—22no—
387#95—42no—
388#95—42no—
389#95—118no—
390#95—222no—
391#95—222no—
392#95—224no—
393#95—224no—
394#95—442no—
395#95—442no—
396#95—442no—
397#95—442no—
398#95C4×D1114yes—
399C2×C11#232111yescharacteristic
400C2×C11#232311yescharacteristic
401C2×C11#232611yescharacteristic
402C2×C11#237812yes—
403C2×C11#237912yes—
404C2×C11#281411yescharacteristic
405C2×C11#281611yescharacteristic
406C2×C11#281911yescharacteristic
407C2×C11#282111yescharacteristic
408C2×C11#283112yes—
409C2×C11#283212yes—
410D11—82no—
411C24—112no—
412C24—221no—
413C24—221no—
414C24—222no—
415C24—221no—
416C24—221no—
417C24—222no—
418C24—222no—
419C24—222no—
420C24—221no—
421C24—221no—
422C24C23×D1111yescharacteristic
423C2×Q8—228no—
424C2×Q8—448no—
425C2×Q8—444no—
426C2×Q8—444no—
427C2×Q8—448no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—112no—
452C22×C4—114no—
453C22×C4—222no—
454C22×C4—221no—
455C22×C4—221no—
456C22×C4—222no—
457C22×C4—222no—
458C22×C4—222no—
459C22×C4—222no—
460C22×C4—222no—
461C22×C4—222no—
462C22×C4—222no—
463C22×C4—221no—
464C22×C4—222no—
465C22×C4—221no—
466C22×C4—222no—
467C22×C4—222no—
468C22×C4—221no—
469C22×C4—221no—
470C22×C4—222no—
471C22×C4—222no—
472C22×C4—222no—
473C22×C4—222no—
474C22×C4—222no—
475C22×C4—222no—
476C22×C4—222no—
477C22×C4—222no—
478C22×C4—221no—
479C22×C4—221no—
480C22×C4C2×C4×D1114yes—
481C22×C4C2×C4×D1112yes—
482C22×C4C2×D4412yes—
483C22×C4C2×D4412yes—
484C22×C4C2×D4412yes—
485C22×C4#433912yes—
486C4⋊C4—444no—
487C4⋊C4—444no—
488C4⋊C4—448no—
489C22⋊C4—228no—
490C22⋊C4—448no—
491C22⋊C4—444no—
492C22⋊C4—444no—
493C22⋊C4—448no—
494C42—44no—
495C11#1462511yescharacteristic, complemented, Sylow
496C23—222no—
497C23—224no—
498C23—224no—
499C23—222no—
500C23—222no—
501C23—442no—
502C23—442no—
503C23—441no—
504C23—441no—
505C23—441no—
506C23—441no—
507C23—441no—
508C23—441no—
509C23—442no—
510C23—442no—
511C23—441no—
512C23—441no—
513C23—442no—
514C23—442no—
515C23—442no—
516C23—442no—
517C23—442no—
518C23—442no—
519C23C22×C4×D1111yescharacteristic
520C23C22×D4411yescharacteristic
521C23C2×#433911yescharacteristic
522C23#6703712yes—
523C23#6703711yescharacteristic
524C23#6704011yescharacteristic
525C23#6704212yes—
526C23#6704212yes—
527C23#6704312yes—
528C23#6704311yescharacteristic
529C23#6704311yescharacteristic
530Q8—4416no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—222no—
560C2×C4—224no—
561C2×C4—224no—
562C2×C4—224no—
563C2×C4—224no—
564C2×C4—442no—
565C2×C4—441no—
566C2×C4—442no—
567C2×C4—441no—
568C2×C4—442no—
569C2×C4—442no—
570C2×C4—442no—
571C2×C4—441no—
572C2×C4—441no—
573C2×C4—442no—
574C2×C4—441no—
575C2×C4—441no—
576C2×C4—442no—
577C2×C4—442no—
578C2×C4—441no—
579C2×C4—441no—
580C2×C4—442no—
581C2×C4—442no—
582C2×C4—442no—
583C2×C4—442no—
584C2×C4—442no—
585C2×C4—442no—
586C2×C4—442no—
587C2×C4—442no—
588C2×C4C4×D4414yes—
589C22—444no—
590C22—444no—
591C22—442no—
592C22—442no—
593C22—442no—
594C22—882no—
595C22—882no—
596C22—881no—
597C22—881no—
598C22—881no—
599C22—881no—
600C22C2×C4×D4411yescharacteristic
601C22#1062302411yescharacteristic
602C22#1062304111yescharacteristic
603C22#1062305412yes—
604C22#1062305611yescharacteristic
605C22#1062305611yescharacteristic
606C22#1062305711yescharacteristic
607C22#1062305711yescharacteristic
608C22#1062306512yes—
609C22#1062307612yes—
610C22#1062307711yescharacteristic
611C22#1062309112yes—
612C22#1062309211yescharacteristic
613C22#1062310311yescharacteristic
614C22#1062311411yescharacteristic
615C22#1062315212yes—
616C22#1062317712yes—
617C22#1062319812yes—
618C22#1062319812yes—
619C22#1062321412yes—
620C22#1062323112yes—
621C22#1062324312yes—
622C22#1062324312yes—
623C4—24no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—444no—
629C4—442no—
630C4—881no—
631C4—881no—
632C4—882no—
633C2—882no—
634C2#1307127611yescharacteristic
635C2#1307129911yescharacteristic
636C2#1307132711yescharacteristic
637C2#1307178112yes—
638C2#1307178512yes—
639C2#1307663111yescharacteristic
640C2#1307664611yescharacteristic
641C2#1307666211yescharacteristic
642C2#1307667711yescharacteristic
643C2#1307677012yes—
644C2#1307678112yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC11×#1434C2
socle, cosocleC24×C11C24
prime radicalC24×D11C23
hypercenter#1434D11
hypocenterC11#14625
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#10622542C2semidirect
4. #13066886C2semidirect
5. #13066881C2semidirect
6. #13066872C2semidirect
7. #13066871C2semidirect
11. C11×#1434C2semidirect
37. #10622286C4semidirect
184. #1434D11semidirect
495. C11#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#106222862
C2×#106225422
C2×#106225532
#130668862
#130668812
#130668722
#130668712
#130668642
#130668622
C11×#14342
#1462511