Finite Group Databasebeta

← #14681705

Subgroups — #14681705

Order 2816.

1750 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622598C212yesmaximal, complemented
2C2×#10622564C212yesmaximal
3C2×#10622825C212yesmaximal
4#13066881C212yesmaximal, complemented
5#13066869C212yesmaximal, complemented
6#13066861C211yesmaximal, characteristic
7#13066861C211yesmaximal, characteristic
8#13066850C212yesmaximal, complemented
9C2×C11×#385C211yesmaximal, characteristic
10C24×C4×C11C2211yescharacteristic
11C22×C11×C4⋊C4C2212yes—
12C22×C11×C22⋊C4C2211yescharacteristic
13C22×C11×C22⋊C4C2211yescharacteristic
14C22×C22⋊Dic11C2212yes—
15C22×C22⋊Dic11C2212yes—
16C22×C4⋊Dic11C2212yes—
17C22×C4⋊Dic11C2211yescharacteristic
18C22×C4⋊Dic11C2212yes—
19C22×C4⋊Dic11C2212yes—
20C22×C4⋊Dic11C2211yescharacteristic
21C22×C4×Dic11C2212yes—
22C11×#385C418yescomplemented
23C2×C11×#95C2212yes—
24#10622825—28no—
25C2×C11⋊#95C2212yes—
26C2×C11⋊#95C2212yes—
27C2×C11⋊#95C2212yes—
28C2×C11⋊#95C2212yes—
29C2×C11⋊#95C2211yescharacteristic
30C2×C11⋊#95C2212yes—
31C2×C11⋊#95C2211yescharacteristic
32C2×C11⋊#95C2212yes—
33#10622598—28no—
34#10622564—28no—
35C25×C11C2311yescharacteristic
36C23×C4×C11—24no—
37C23×C4×C11—24no—
38C23×C4×C11—41no—
39C23×C4×C11—41no—
40C23×C4×C11C2×C414yes—
41C23×C4×C11C2311yescharacteristic
42C23×C4×C11C2311yescharacteristic
43C23×C4×C11C2312yes—
44C23×C4×C11C2312yes—
45C23×Dic11C2312yes—
46C23×Dic11C2312yes—
47C23×Dic11C2312yes—
48C23×Dic11C2312yes—
49C2×C11×C4⋊C4—24no—
50C2×C11×C4⋊C4—24no—
51C2×C11×C4⋊C4C2×C418yes—
52C2×C11×C22⋊C4—41no—
53C2×C11×C22⋊C4—41no—
54C2×C11×C22⋊C4C2×C414yes—
55C2×C11×C22⋊C4C2×C414yes—
56C2×C11×C22⋊C4D414yes—
57C2×C11×C22⋊C4Q814yes—
58C2×C22⋊Dic11—24no—
59C2×C22⋊Dic11—24no—
60C2×C22⋊Dic11—24no—
61C2×C22⋊Dic11—24no—
62C2×C22⋊Dic11—42no—
63C2×C22⋊Dic11—42no—
64C2×C4⋊Dic11—24no—
65C2×C4⋊Dic11—24no—
66C2×C4⋊Dic11—24no—
67C2×C4⋊Dic11—24no—
68C2×C4⋊Dic11—24no—
69C2×C4⋊Dic11—24no—
70C2×C4⋊Dic11—24no—
71C2×C4⋊Dic11—42no—
72C2×C4⋊Dic11—42no—
73C2×C4⋊Dic11—42no—
74C2×C4⋊Dic11—41no—
75C2×C4⋊Dic11—41no—
76C2×C4⋊Dic11—42no—
77C2×C4×Dic11—24no—
78C2×C4×Dic11—24no—
79C2×C4×Dic11—42no—
80C11×#95C2×C418yes—
81C11⋊#95—24no—
82C11⋊#95—24no—
83C11⋊#95—24no—
84C11⋊#95—24no—
85C11⋊#95—42no—
86C11⋊#95—41no—
87C11⋊#95—41no—
88C11⋊#95D418yes—
89#14402—111nomaximal, Sylow
90C24×C11—22no—
91C24×C11—22no—
92C24×C11—22no—
93C24×C11—21no—
94C24×C11—21no—
95C24×C11—21no—
96C24×C11—21no—
97C24×C11—22no—
98C24×C11C22×C412yes—
99C24×C11C2×D412yes—
100C24×C11C2×Q812yes—
101C24×C11C2411yescharacteristic
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—22no—
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—22no—
112C22×C4×C11—22no—
113C22×C4×C11—21no—
114C22×C4×C11—21no—
115C22×C4×C11—21no—
116C22×C4×C11—21no—
117C22×C4×C11—22no—
118C22×C4×C11—216no—
119C22×C4×C11—44no—
120C22×C4×C11—44no—
121C22×C4×C11—48no—
122C22×C4×C11—44no—
123C22×C4×C11C22×C412yes—
124C22×C4×C11C22×C412yes—
125C22×C4×C11C22×C414yes—
126C22×C4×C11C22×C414yes—
127C22×C4×C11C2×D414yes—
128C22×C4×C11C2×D414yes—
129C22×C4×C11C2×Q814yes—
130C22×C4×C11#4114yes—
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—22no—
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11—22no—
156C22×Dic11—22no—
157C22×Dic11C2×D414yes—
158C22×Dic11C2×D414yes—
159C11×C4⋊C4—216no—
160C11×C22⋊C4—44no—
161C11×C22⋊C4C4⋊C4116yescomplemented
162C22⋊Dic11—48no—
163C22⋊Dic11—48no—
164C4⋊Dic11—48no—
165C4⋊Dic11—48no—
166C4⋊Dic11—48no—
167C4⋊Dic11—44no—
168C4⋊Dic11—44no—
169C4×Dic11—48no—
170C2×#389—112no—
171C2×#388—112no—
172C2×#385D1111yescharacteristic
173C2×#380—112no—
174#1444—112no—
175#1441—112no—
176#1439—111no—
177#1439—111no—
178#1436—112no—
179C23×C11—24no—
180C23×C11—22no—
181C23×C11—22no—
182C23×C11—24no—
183C23×C11—22no—
184C23×C11—22no—
185C23×C11—22no—
186C23×C11—22no—
187C23×C11—24no—
188C23×C11—24no—
189C23×C11—24no—
190C23×C11—24no—
191C23×C11—42no—
192C23×C11—42no—
193C23×C11—42no—
194C23×C11—42no—
195C23×C11—41no—
196C23×C11—42no—
197C23×C11—41no—
198C23×C11—41no—
199C23×C11—42no—
200C23×C11—41no—
201C23×C11C2×C4⋊C414yes—
202C23×C11C23×C411yescharacteristic
203C23×C11C22×D411yescharacteristic
204C23×C11C22×D412yes—
205C23×C11C22×Q811yescharacteristic
206C23×C11C2×#4112yes—
207C23×C11H(5,𝔽2)11yescharacteristic
208C23×C11H(5,𝔽2)11yescharacteristic
209C23×C11H(5,𝔽2)12yes—
210C23×C112−1+411yescharacteristic
211C23×C112−1+411yescharacteristic
212C23×C112−1+412yes—
213C2×C4×C11—24no—
214C2×C4×C11—24no—
215C2×C4×C11—24no—
216C2×C4×C11—24no—
217C2×C4×C11—24no—
218C2×C4×C11—24no—
219C2×C4×C11—24no—
220C2×C4×C11—24no—
221C2×C4×C11—24no—
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—24no—
229C2×C4×C11—416no—
230C2×C4×C11—41no—
231C2×C4×C11—42no—
232C2×C4×C11—41no—
233C2×C4×C11—41no—
234C2×C4×C11—42no—
235C2×C4×C11—41no—
236C2×C4×C11C2×C4⋊C418yes—
237C2×C4×C11C4×D418yes—
238C2×Dic11—24no—
239C2×Dic11—24no—
240C2×Dic11—24no—
241C2×Dic11—24no—
242C2×Dic11—24no—
243C2×Dic11—24no—
244C2×Dic11—42no—
245C2×Dic11—42no—
246C2×Dic11—42no—
247C2×Dic11—42no—
248C2×Dic11—42no—
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—
266C24×C4C2×D1111yescharacteristic
267C22×C4⋊C4—112no—
268C22×C4⋊C4—111no—
269C22×C4⋊C4—112no—
270C22×C4⋊C4—112no—
271C22×C4⋊C4—111no—
272C22×C4⋊C4C2×D1112yes—
273C22×C22⋊C4—112no—
274C22×C22⋊C4—112no—
275C22×C22⋊C4C2×D1111yescharacteristic
276C22×C22⋊C4C2×D1111yescharacteristic
277C22×C42—112no—
278#389—228no—
279#388—228no—
280#385Dic1118yescomplemented
281#380—228no—
282C2×#95—112no—
283C2×#95—112no—
284C2×#95—112no—
285C2×#95—112no—
286C2×#95—112no—
287C2×#95—111no—
288C2×#95—112no—
289C2×#95—111no—
290C2×#95C2×D1112yes—
291C22×C11—24no—
292C22×C11—28no—
293C22×C11—44no—
294C22×C11—42no—
295C22×C11—42no—
296C22×C11—44no—
297C22×C11—42no—
298C22×C11—42no—
299C22×C11—44no—
300C22×C11—42no—
301C22×C11—42no—
302C22×C11C22×C4⋊C411yescharacteristic
303C22×C11C2×C4×D412yes—
304C22×C11#51811yescharacteristic
305C22×C11#51911yescharacteristic
306C22×C11#52012yes—
307C22×C11#53612yes—
308C22×C11#53711yescharacteristic
309C22×C11#53711yescharacteristic
310C22×C11#53712yes—
311C22×C11#54112yes—
312C22×C11#54212yes—
313C22×C11#54411yescharacteristic
314C22×C11#54411yescharacteristic
315C22×C11D4212yes—
316C22×C11D4211yescharacteristic
317C22×C11#54612yes—
318C22×C11#54712yes—
319C22×C11D4×Q812yes—
320C22×C11#55111yescharacteristic
321C22×C11#55412yes—
322C22×C11#56011yescharacteristic
323C22×C11#56212yes—
324C22×C11#56311yescharacteristic
325C4×C11—28no—
326C4×C11—28no—
327C4×C11—42no—
328C4×C11—42no—
329Dic11—44no—
330Dic11—44no—
331Dic11—82no—
332Dic11—82no—
333C25C22×D1111yescharacteristic
334C23×C4—24no—
335C23×C4—24no—
336C23×C4—41no—
337C23×C4—41no—
338C23×C4—112no—
339C23×C4—112no—
340C23×C4—112no—
341C23×C4—112no—
342C23×C4C2×Dic1114yes—
343C23×C4C22×D1111yescharacteristic
344C23×C4C22×D1111yescharacteristic
345C23×C4C22×D1112yes—
346C23×C4C22×D1112yes—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—224no—
350C2×C4⋊C4—224no—
351C2×C4⋊C4—224no—
352C2×C4⋊C4—224no—
353C2×C4⋊C4—224no—
354C2×C4⋊C4—224no—
355C2×C4⋊C4—224no—
356C2×C4⋊C4—442no—
357C2×C4⋊C4—442no—
358C2×C4⋊C4—442no—
359C2×C4⋊C4—441no—
360C2×C4⋊C4—441no—
361C2×C4⋊C4—442no—
362C2×C4⋊C4C2×Dic1118yes—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—224no—
366C2×C22⋊C4—224no—
367C2×C22⋊C4—224no—
368C2×C22⋊C4—224no—
369C2×C22⋊C4—442no—
370C2×C22⋊C4—442no—
371C2×C22⋊C4Dic2214yes—
372C2×C22⋊C4D4414yes—
373C2×C22⋊C4C2×Dic1114yes—
374C2×C22⋊C4C2×Dic1114yes—
375C2×C42—224no—
376C2×C42—224no—
377C2×C42—442no—
378#95—118no—
379#95—224no—
380#95—224no—
381#95—224no—
382#95—224no—
383#95—441no—
384#95—441no—
385#95—442no—
386#95C2×Dic1118yes—
387C2×C11—44no—
388C2×C11#231611yescharacteristic
389C2×C11#232312yes—
390C2×C11D4×C4⋊C412yes—
391C2×C11#235811yescharacteristic
392C2×C11#237611yescharacteristic
393C2×C11#267212yes—
394C2×C11#267511yescharacteristic
395C2×C11#267711yescharacteristic
396C2×C11#270412yes—
397C2×C11#272211yescharacteristic
398C2×C11#274211yescharacteristic
399C24—22no—
400C24—22no—
401C24—21no—
402C24—21no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—22no—
407C24C2×Dic2212yes—
408C24C2×D4412yes—
409C24C22×Dic1112yes—
410C24C23×D1111yescharacteristic
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—216no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—114no—
433C22×C4—114no—
434C22×C4—222no—
435C22×C4—222no—
436C22×C4—222no—
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×C4C2×Dic2214yes—
461C22×C4C2×D4414yes—
462C22×C4D4×D1114yes—
463C22×C4#434114yes—
464C22×C4C22×Dic1112yes—
465C22×C4C22×Dic1112yes—
466C22×C4C22×Dic1114yes—
467C22×C4C22×Dic1114yes—
468C4⋊C4—216no—
469C4⋊C4—448no—
470C4⋊C4—448no—
471C4⋊C4—448no—
472C4⋊C4—444no—
473C4⋊C4—444no—
474C22⋊C4—44no—
475C22⋊C4—448no—
476C22⋊C4—448no—
477C22⋊C4C4⋊Dic11116yescomplemented
478C42—448no—
479C11#1440211yescharacteristic, complemented, Sylow
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23C2×C4⋊Dic1114yes—
503C23C22×Dic2211yescharacteristic
504C23C22×D4411yescharacteristic
505C23C2×D4×D1112yes—
506C23C2×#434112yes—
507C23#6703711yescharacteristic
508C23#6703711yescharacteristic
509C23#6704011yescharacteristic
510C23#6704011yescharacteristic
511C23#6704212yes—
512C23#6704312yes—
513C23C23×Dic1111yescharacteristic
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—416no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—224no—
538C2×C4—224no—
539C2×C4—224no—
540C2×C4—224no—
541C2×C4—224no—
542C2×C4—224no—
543C2×C4—442no—
544C2×C4—442no—
545C2×C4—442no—
546C2×C4—442no—
547C2×C4—442no—
548C2×C4—442no—
549C2×C4—442no—
550C2×C4—442no—
551C2×C4—442no—
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×C4C2×C4⋊Dic1118yes—
566C2×C4D4×Dic1118yes—
567C22—24no—
568C22—28no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—42no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22#1062303111yescharacteristic
579C22#1062303111yescharacteristic
580C22#1062304111yescharacteristic
581C22#1062304111yescharacteristic
582C22D4×Dic2212yes—
583C22#1062307412yes—
584C22D4×D4412yes—
585C22#1062308612yes—
586C22#1062311811yescharacteristic
587C22#1062312111yescharacteristic
588C22#1062312212yes—
589C22#1062315612yes—
590C22#1062315912yes—
591C22#1062316012yes—
592C22#1062317412yes—
593C22#1062318211yescharacteristic
594C22#1062319711yescharacteristic
595C22#1062319812yes—
596C22C22×C4⋊Dic1111yescharacteristic
597C22C2×D4×Dic1112yes—
598C22#1062332911yescharacteristic
599C22#1062334311yescharacteristic
600C22#1062335812yes—
601C4—28no—
602C4—28no—
603C4—42no—
604C4—42no—
605C4—444no—
606C4—444no—
607C4—882no—
608C4—882no—
609C2—44no—
610C2#1307123411yescharacteristic
611C2#1307130012yes—
612C2D4×C4⋊Dic1112yes—
613C2#1307160011yescharacteristic
614C2#1307176411yescharacteristic
615C2#1307487012yes—
616C2#1307490711yescharacteristic
617C2#1307493111yescharacteristic
618C2#1307527812yes—
619C2#1307552211yescharacteristic
620C2#1307574511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC2×C11×#385C2
socle, cosocleC24×C11C24
prime radicalC25×C11C23
hypercenterC2×#385D11
hypocenterC11#14402
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C2×#10622598C2semidirect
4. #13066881C2semidirect
5. #13066869C2semidirect
8. #13066850C2semidirect
22. C11×#385C4semidirect
161. C11×C22⋊C4C4⋊C4semidirect
280. #385Dic11semidirect
477. C22⋊C4C4⋊Dic11semidirect
479. C11#14402semidirect

Maximal subgroups

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

subgroupindex
C2×#106225982
C2×#106225642
C2×#106228252
#130668812
#130668692
#130668612
#130668502
C2×C11×#3852
#1440211