Finite Group Databasebeta

← #14682230

Subgroups — #14682230

Order 2816.

1658 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×#10622564C212yesmaximal, complemented
2#13066865C212yesmaximal, complemented
3#13066861C212yesmaximal, complemented
4#13066857C212yesmaximal
5#13066853C212yesmaximal, complemented
6#13066848C212yesmaximal, complemented
7#13066846C211yesmaximal, characteristic, complemented
8#13066843C211yesmaximal, characteristic, complemented
9C2×C11×#390C211yesmaximal, characteristic
10C23×D4×C11C2211yescharacteristic
11C22×C11×C4⋊C4C2211yescharacteristic
12C22×C11×C22⋊C4C2212yes—
13C22×C11×C22⋊C4C2212yes—
14C22×C42×C11C2211yescharacteristic
15C22×C22⋊Dic11C2212yes—
16C22×C22⋊Dic11C2212yes—
17C22×C22⋊Dic11C2212yes—
18C22×C22⋊Dic11C2212yes—
19C22×C4⋊Dic11C2212yes—
20C22×C4×Dic11C2212yes—
21C11×#390C418yescomplemented
22C2×C11⋊#95C2212yes—
23C2×C11⋊#95C2211yescharacteristic
24C2×C11⋊#95C2212yes—
25C2×C11⋊#95C2212yes—
26C2×C11⋊#95C2212yes—
27C2×C11⋊#95C2211yescharacteristic
28C2×C11⋊#95C2211yescharacteristic
29C2×C11⋊#95C2212yes—
30C2×C11⋊#95C2212yes—
31C2×C11⋊#95C2211yescharacteristic
32#10622564—28no—
33C25×C11C2312yes—
34C22×D4×C11—22no—
35C22×D4×C11—22no—
36C22×D4×C11—42no—
37C22×D4×C11—42no—
38C22×D4×C11C2×C414yes—
39C23×C4×C11C2312yes—
40C23×C4×C11C2311yescharacteristic
41C23×C4×C11C2311yescharacteristic
42C23×C4×C11C2311yescharacteristic
43C23×Dic11C2312yes—
44C23×Dic11C2312yes—
45C23×Dic11C2312yes—
46C23×Dic11C2312yes—
47C2×C11×C4⋊C4—42no—
48C2×C11×C4⋊C4C2×C414yes—
49C2×C11×C22⋊C4—24no—
50C2×C11×C22⋊C4—24no—
51C2×C11×C22⋊C4—42no—
52C2×C11×C22⋊C4—42no—
53C2×C11×C22⋊C4C2×C418yes—
54C2×C11×C22⋊C4C2×C418yes—
55C2×C42×C11—42no—
56C2×C42×C11C2×C414yes—
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×C22⋊Dic11—42no—
68C2×C22⋊Dic11—42no—
69C2×C4⋊Dic11—24no—
70C2×C4⋊Dic11—24no—
71C2×C4⋊Dic11—42no—
72C2×C4×Dic11—24no—
73C2×C4×Dic11—42no—
74C2×C4×Dic11—42no—
75C11⋊#95—24no—
76C11⋊#95—24no—
77C11⋊#95—22no—
78C11⋊#95—24no—
79C11⋊#95—42no—
80C11⋊#95—41no—
81C11⋊#95—42no—
82C11⋊#95—41no—
83C11⋊#95—42no—
84C11⋊#95D414yescomplemented
85#14444—111nomaximal, Sylow
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×C11—22no—
99C24×C11—22no—
100C24×C11C22×C414yes—
101C24×C11C2411yescharacteristic
102C2×D4×C11—28no—
103C2×D4×C11—44no—
104C2×D4×C11—44no—
105C2×D4×C11—48no—
106C2×D4×C11—48no—
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—22no—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—22no—
118C22×C4×C11—22no—
119C22×C4×C11—22no—
120C22×C4×C11—21no—
121C22×C4×C11C22×C412yes—
122C22×C4×C11C22×C414yes—
123C22×C4×C11C22×C412yes—
124C22×C4×C11C22×C412yes—
125C22×C4×C11C2×D412yes—
126C22×C4×C11#4112yes—
127C22×C4×C11#4112yes—
128C22×C4×C11#4112yes—
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—22no—
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11C2×D414yes—
156C22×Dic11#4114yes—
157C11×C4⋊C4—44no—
158C11×C22⋊C4—216no—
159C11×C22⋊C4—48no—
160C42×C11—44no—
161C22⋊Dic11—48no—
162C22⋊Dic11—48no—
163C22⋊Dic11—48no—
164C22⋊Dic11—48no—
165C4⋊Dic11—48no—
166C4×Dic11—48no—
167C2×#390D1111yescharacteristic
168C2×#388—112no—
169#1440—112no—
170#1439—112no—
171#1438—112no—
172#1437—112no—
173#1436—112no—
174C4⋊#95—111no—
175#1434—111no—
176C23×C11—24no—
177C23×C11—24no—
178C23×C11—24no—
179C23×C11—24no—
180C23×C11—24no—
181C23×C11—24no—
182C23×C11—24no—
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×C11—42no—
209C23×C11—42no—
210C23×C11—42no—
211C23×C11C23×C411yescharacteristic
212C23×C11C22×D411yescharacteristic
213C23×C11C2×#4111yescharacteristic
214C23×C11C2×#4111yescharacteristic
215C23×C11C2×#4111yescharacteristic
216C23×C11C2×#4112yes—
217C23×C11H(5,𝔽2)12yes—
218C23×C11H(5,𝔽2)12yes—
219C23×C11H(5,𝔽2)12yes—
220C23×C11H(5,𝔽2)11yescharacteristic
221C23×C112−1+411yescharacteristic
222D4×C11—416no—
223C2×C4×C11—22no—
224C2×C4×C11—22no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—24no—
228C2×C4×C11—24no—
229C2×C4×C11—24no—
230C2×C4×C11—24no—
231C2×C4×C11—24no—
232C2×C4×C11—22no—
233C2×C4×C11—22no—
234C2×C4×C11—24no—
235C2×C4×C11—24no—
236C2×C4×C11—22no—
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#11714yes—
244C2×C4×C11C4×D414yes—
245C2×Dic11—24no—
246C2×Dic11—24no—
247C2×Dic11—24no—
248C2×Dic11—24no—
249C2×Dic11—24no—
250C2×Dic11—24no—
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—
268C2×Dic11—42no—
269C2×Dic11—42no—
270C2×Dic11—42no—
271C2×Dic11—42no—
272C2×Dic11—42no—
273C23×D4C2×D1111yescharacteristic
274C22×C4⋊C4—112no—
275C22×C4⋊C4C2×D1111yescharacteristic
276C22×C22⋊C4—112no—
277C22×C22⋊C4—112no—
278C22×C22⋊C4—112no—
279C22×C22⋊C4—112no—
280C22×C22⋊C4C2×D1112yes—
281C22×C22⋊C4C2×D1112yes—
282C22×C42—112no—
283C22×C42C2×D1111yescharacteristic
284#390Dic1118yescomplemented
285#388—228no—
286C2×#95—111no—
287C2×#95—112no—
288C2×#95—111no—
289C2×#95—112no—
290C2×#95—112no—
291C2×#95—112no—
292C2×#95—111no—
293C2×#95—111no—
294C2×#95—112no—
295C2×#95—112no—
296C22×C11—44no—
297C22×C11—44no—
298C22×C11—44no—
299C22×C11—44no—
300C22×C11—44no—
301C22×C11—44no—
302C22×C11—44no—
303C22×C11—82no—
304C22×C11—82no—
305C22×C11—82no—
306C22×C11—82no—
307C22×C11—82no—
308C22×C11—82no—
309C22×C11—82no—
310C22×C11—82no—
311C22×C11C2×#11711yescharacteristic
312C22×C11C2×C4×D411yescharacteristic
313C22×C11C4×#4111yescharacteristic
314C22×C11#51811yescharacteristic
315C22×C11#51811yescharacteristic
316C22×C11#51811yescharacteristic
317C22×C11#52011yescharacteristic
318C22×C11#53512yes—
319C22×C11#53812yes—
320C22×C11#53812yes—
321C22×C11#53812yes—
322C22×C11#54012yes—
323C22×C11#54212yes—
324C22×C11#54612yes—
325C22×C11#54711yescharacteristic
326C22×C11#55011yescharacteristic
327C22×C11#55312yes—
328C22×C11#55312yes—
329C22×C11#55511yescharacteristic
330C22×C11#55611yescharacteristic
331C22×C11#55912yes—
332C22×C11#56012yes—
333C22×C11#56212yes—
334C4×C11—24no—
335C4×C11—24no—
336C4×C11—44no—
337C4×C11—42no—
338Dic11—44no—
339Dic11—44no—
340Dic11—82no—
341Dic11—82no—
342C25C22×D1112yes—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—42no—
346C22×D4—42no—
347C22×D4C2×Dic1114yes—
348C23×C4—112no—
349C23×C4—112no—
350C23×C4—112no—
351C23×C4—112no—
352C23×C4C22×D1112yes—
353C23×C4C22×D1111yescharacteristic
354C23×C4C22×D1111yescharacteristic
355C23×C4C22×D1111yescharacteristic
356C2×C4⋊C4—42no—
357C2×C4⋊C4—224no—
358C2×C4⋊C4—224no—
359C2×C4⋊C4—442no—
360C2×C4⋊C4C2×Dic1114yes—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4—42no—
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⋊C4—442no—
372C2×C22⋊C4—442no—
373C2×C22⋊C4—442no—
374C2×C22⋊C4—442no—
375C2×C22⋊C4—442no—
376C2×C22⋊C4—442no—
377C2×C22⋊C4C2×Dic1118yes—
378C2×C22⋊C4C2×Dic1118yes—
379C2×C42—42no—
380C2×C42—224no—
381C2×C42—442no—
382C2×C42—442no—
383C2×C42C2×Dic1114yes—
384#95—114no—
385#95—224no—
386#95—224no—
387#95—224no—
388#95—222no—
389#95—442no—
390#95—442no—
391#95—442no—
392#95—441no—
393#95—441no—
394C2×C11—84no—
395C2×C11#231211yescharacteristic
396C2×C11#232811yescharacteristic
397C2×C11#233711yescharacteristic
398C2×C11#235411yescharacteristic
399C2×C11#235711yescharacteristic
400C2×C11#236411yescharacteristic
401C2×C11#237711yescharacteristic
402C2×C11#271012yes—
403C2×C11#271412yes—
404C2×C11#280012yes—
405C2×C11#280312yes—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24C22×Dic1114yes—
421C24C23×D1111yescharacteristic
422C2×D4—28no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—48no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—114no—
442C22×C4—114no—
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×C4—222no—
463C22×C4—222no—
464C22×C4—222no—
465C22×C4—222no—
466C22×C4—222no—
467C22×C4—222no—
468C22×C4—222no—
469C22×C4#433912yes—
470C22×C4#433912yes—
471C22×C4D4×D1112yes—
472C22×C4#434112yes—
473C22×C4C22×Dic1112yes—
474C22×C4C22×Dic1114yes—
475C22×C4C22×Dic1112yes—
476C22×C4C22×Dic1112yes—
477C4⋊C4—44no—
478C4⋊C4—448no—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—448no—
482C22⋊C4—448no—
483C22⋊C4—448no—
484C22⋊C4—448no—
485C42—44no—
486C42—448no—
487C11#1444411yescharacteristic, complemented, Sylow
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
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—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C2×#433911yescharacteristic
524C23C2×#433911yescharacteristic
525C23C2×D4×D1111yescharacteristic
526C23C2×#434111yescharacteristic
527C23#6703712yes—
528C23#6703712yes—
529C23#6703712yes—
530C23#41×D1112yes—
531C23#6704211yescharacteristic
532C23#6704311yescharacteristic
533C23C23×Dic1111yescharacteristic
534D4—416no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—224no—
556C2×C4—224no—
557C2×C4—224no—
558C2×C4—224no—
559C2×C4—224no—
560C2×C4—224no—
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—
571C2×C4—442no—
572C2×C4—442no—
573C2×C4—442no—
574C2×C4—442no—
575C2×C4—442no—
576C2×C4—442no—
577C2×C4—442no—
578C2×C4—442no—
579C2×C4—442no—
580C2×C4—442no—
581C2×C4—442no—
582C2×C4—442no—
583C2×C4#6697914yes—
584C2×C4D4×Dic1114yes—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22#1062302512yes—
601C22#1062302512yes—
602C22#1062307311yescharacteristic
603C22#1062308211yescharacteristic
604C22#1062308411yescharacteristic
605C22#1062308912yes—
606C22#1062309012yes—
607C22#1062309411yescharacteristic
608C22#1062313212yes—
609C22#1062314112yes—
610C22#1062315112yes—
611C22#1062315212yes—
612C22#1062320012yes—
613C22#1062320912yes—
614C22#1062324612yes—
615C22#1062325012yes—
616C22C2×#6697911yescharacteristic
617C22C2×D4×Dic1111yescharacteristic
618C22#1062332911yescharacteristic
619C22#1062332911yescharacteristic
620C22#1062332911yescharacteristic
621C22#41×Dic1111yescharacteristic
622C22#1062335811yescharacteristic
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—444no—
628C4—444no—
629C4—882no—
630C4—882no—
631C2—84no—
632C2#1307121411yescharacteristic
633C2#1307134211yescharacteristic
634C2#1307143111yescharacteristic
635C2#1307156511yescharacteristic
636C2#1307158911yescharacteristic
637C2#1307166811yescharacteristic
638C2#1307177311yescharacteristic
639C2#1307535512yes—
640C2#1307540912yes—
641C2#1307651312yes—
642C2#1307655012yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC2×C11×#390C2
socle, cosocleC24×C11C24
prime radicalC23×D4×C11C22
hypercenterC2×#390D11
hypocenterC11#14444
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#10622564C2semidirect
2. #13066865C2semidirect
3. #13066861C2semidirect
5. #13066853C2semidirect
6. #13066848C2semidirect
7. #13066846C2semidirect
8. #13066843C2semidirect
21. C11×#390C4semidirect
84. C11⋊#95D4semidirect
284. #390Dic11semidirect
487. C11#14444semidirect

Maximal subgroups

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

subgroupindex
C2×#106225642
#130668652
#130668612
#130668572
#130668532
#130668482
#130668462
#130668432
C2×C11×#3902
#1444411