Finite Group Databasebeta

← #14756840

Subgroups — #14756840

Order 2816.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622631C211yesmaximal, characteristic
2#13071081C211yesmaximal, characteristic
3C4×#66940C211yesmaximal, characteristic, complemented
4#13069246C211yesmaximal, characteristic
5#13069239C211yesmaximal, characteristic
6#13069211C211yesmaximal, characteristic, complemented
7#13069190C211yesmaximal, characteristic, complemented
8#13068968C211yesmaximal, characteristic
9#13068968C211yesmaximal, characteristic
10#13068885C211yesmaximal, characteristic, complemented
11#13068871C211yesmaximal, characteristic, complemented
12#13068514C211yesmaximal, characteristic, complemented
13#13068447C211yesmaximal, characteristic, complemented
14#13068364C211yesmaximal, characteristic, complemented
15C11×#1762C211yesmaximal, characteristic
16C2×C4×D4×C11C2211yescharacteristic
17C22×#4324C2211yescharacteristic
18C2×#66979—21no—
19C2×#66940C2211yescharacteristic
20C2×#66940C2211yescharacteristic
21C2×C4×Dic22C2211yescharacteristic
22C2×C11×#102C2211yescharacteristic
23C2×C11×#102C2211yescharacteristic
24C2×C11×#97C2211yescharacteristic
25C4×C11×C4⋊C4C2211yescharacteristic
26#10622834C2211yescharacteristic
27#10622834C2211yescharacteristic
28C2×#66896C2211yescharacteristic
29C4×C22⋊Dic11—21no—
30C2×#66880C2211yescharacteristic
31C2×#66880C2211yescharacteristic
32#10622631C418yescomplemented
33#10622598C2211yescharacteristic
34#10622598C2211yescharacteristic
35C4⋊C4×Dic11C2211yescharacteristic
36C4⋊C4×Dic11C2211yescharacteristic
37C2×#66874C2211yescharacteristic
38C2×#66872C2211yescharacteristic
39C2×#66871C2211yescharacteristic
40C2×#66871C2211yescharacteristic
41#10622564—21no—
42#10622564—21no—
43C4×C4⋊Dic11C2211yescharacteristic
44C4×Dic11⋊C4—21no—
45C42×Dic11—21no—
46C42×Dic11C2211yescharacteristic
47C2×#66867C2211yescharacteristic
48#10622262—21no—
49#10622262—21no—
50C11×#340C2211yescharacteristic
51C11×#340C2211yescharacteristic
52#10622189C2211yescharacteristic
53#10622189C2211yescharacteristic
54#10622188C2211yescharacteristic
55#10622188C2211yescharacteristic
56#10622188C2211yescharacteristic
57#10622188C2211yescharacteristic
58#10622167C2211yescharacteristic
59#10622167C2211yescharacteristic
60C22×D4×C11C2311yescharacteristic
61C23×C4×C11—21no—
62C22×Dic22C2311yescharacteristic
63C22×C8×C11C2311yescharacteristic
64C22×C8×C11C2311yescharacteristic
65C4×D4×C11—22no—
66C4×D4×C11C2×C414yes—
67C2×C11×C4⋊C4C2311yescharacteristic
68C2×C11×C4⋊C4C2311yescharacteristic
69C2×C11×C4⋊C4C2311yescharacteristic
70C2×C11×C22⋊C4—21no—
71C2×C42×C11—21no—
72C2×C42×C11C2311yescharacteristic
73C2×C22⋊Dic11—21no—
74C2×C22⋊Dic11—21no—
75C2×C22⋊Dic11—21no—
76C2×C22⋊Dic11—21no—
77C2×#4324—22no—
78C2×#4324—22no—
79C2×#4324C2×C414yes—
80#66979—44no—
81C2×C4⋊Dic11—21no—
82C2×C4⋊Dic11—21no—
83C2×C4⋊Dic11C2311yescharacteristic
84C2×C4⋊Dic11C2311yescharacteristic
85C2×C4⋊Dic11C2311yescharacteristic
86C2×Dic11⋊C4—21no—
87C2×Dic11⋊C4—21no—
88C2×Dic11⋊C4—21no—
89C2×C4×Dic11—21no—
90C2×C4×Dic11—21no—
91C2×C4×Dic11—21no—
92C2×C4×Dic11—21no—
93C2×C4×Dic11—21no—
94C2×C4×Dic11—21no—
95C2×C4×Dic11D411yescharacteristic
96C2×C4×Dic11D411yescharacteristic
97C2×C4×Dic11D411yescharacteristic
98C2×C4×Dic11D411yescharacteristic
99C2×C4×Dic11C2311yescharacteristic
100C2×C4×Dic11C2311yescharacteristic
101C22×C11⋊C8C2311yescharacteristic
102C22×C11⋊C8C2311yescharacteristic
103#66940—22no—
104#66940—22no—
105#66940—22no—
106#66940—22no—
107C4×Dic22—22no—
108C4×Dic22C2×C414yes—
109C11×#102—22no—
110C11×#102—22no—
111C11×#97—22no—
112C11×#95—21no—
113#66896C2×C414yes—
114C11⋊#95—21no—
115C11⋊#95—21no—
116C11⋊#95—21no—
117C11⋊#95—21no—
118C11⋊#95—21no—
119#66880—22no—
120#66880—22no—
121#66874C2×C414yes—
122#66872—22no—
123#66871—22no—
124#66871C2×C414yes—
125#66867C2×C414yes—
126#21124—111nomaximal, Sylow
127C24×C11—21no—
128C2×D4×C11—21no—
129C2×D4×C11—21no—
130C2×D4×C11—21no—
131C2×D4×C11—21no—
132C2×D4×C11—21no—
133C2×D4×C11C22×C412yes—
134C22×C4×C11—21no—
135C22×C4×C11—21no—
136C22×C4×C11—21no—
137C22×C4×C11—24no—
138C22×C4×C11—42no—
139C22×C4×C11—42no—
140C22×C4×C11C2×D411yescharacteristic
141C22×C4×C11C2×D411yescharacteristic
142C22×C4×C11C2411yescharacteristic
143C22×Dic11—21no—
144C22×Dic11—21no—
145C22×Dic11—21no—
146C22×Dic11—21no—
147C22×Dic11C2×D411yescharacteristic
148C22×Dic11C2×D411yescharacteristic
149C22×Dic11C2×D411yescharacteristic
150C22×Dic11C2×D411yescharacteristic
151C2×Dic22—21no—
152C2×Dic22—21no—
153C2×Dic22—21no—
154C2×Dic22C22×C412yes—
155C2×Dic22#4112yes—
156C2×Dic22#4112yes—
157C2×C8×C11—21no—
158C2×C8×C11—21no—
159C2×C8×C11—21no—
160C2×C8×C11—21no—
161C2×C8×C11—21no—
162C2×C8×C11—21no—
163C11×C4⋊C4—21no—
164C11×C4⋊C4—21no—
165C11×C4⋊C4—21no—
166C11×C4⋊C4C22×C412yes—
167C11×C4⋊C4#4112yes—
168C11×C4⋊C4#4112yes—
169C11×C22⋊C4—24no—
170C42×C11—42no—
171C42×C11C22×C412yes—
172C42×C11#4112yes—
173C22⋊Dic11—42no—
174C22⋊Dic11—42no—
175C22⋊Dic11—42no—
176C22⋊Dic11—42no—
177#4324—28no—
178C4⋊Dic11—21no—
179C4⋊Dic11—21no—
180C4⋊Dic11—21no—
181C4⋊Dic11—21no—
182C4⋊Dic11—21no—
183C4⋊Dic11—42no—
184C4⋊Dic11—42no—
185C4⋊Dic11C22×C412yes—
186Dic11⋊C4—24no—
187Dic11⋊C4—24no—
188Dic11⋊C4—24no—
189C4×Dic11—24no—
190C4×Dic11—24no—
191C4×Dic11—22no—
192C4×Dic11—22no—
193C4×Dic11—24no—
194C4×Dic11—21no—
195C4×Dic11—21no—
196C4×Dic11—21no—
197C4×Dic11—21no—
198C4×Dic11—41no—
199C4×Dic11—41no—
200C4×Dic11—42no—
201C4×Dic11—42no—
202C4×Dic11—42no—
203C2×C11⋊C8—21no—
204C2×C11⋊C8—21no—
205C2×C11⋊C8—21no—
206C2×C11⋊C8—21no—
207C2×C11⋊C8C22×C412yes—
208C2×C11⋊C8C22×C412yes—
209C2×#440—111no—
210#2293—111no—
211C42×Q8—111no—
212#1926—111no—
213#1925—111no—
214#1921—111no—
215#1918—111no—
216#1880—111no—
217#1880—111no—
218#1865—111no—
219#1863—111no—
220#1776—111no—
221#1763—111no—
222#1762D1111yescharacteristic
223#1745—111no—
224C23×C11—22no—
225C23×C11—41no—
226C23×C11—41no—
227C23×C11—41no—
228C23×C11—41no—
229C23×C11—41no—
230C23×C11—41no—
231C23×C11C22×D411yescharacteristic
232D4×C11—22no—
233D4×C11—22no—
234D4×C11—41no—
235D4×C11—41no—
236C2×C4×C11—21no—
237C2×C4×C11—21no—
238C2×C4×C11—21no—
239C2×C4×C11—21no—
240C2×C4×C11—22no—
241C2×C4×C11—22no—
242C2×C4×C11—22no—
243C2×C4×C11—48no—
244C2×C4×C11—41no—
245C2×C4×C11—41no—
246C2×C4×C11—41no—
247C2×C4×C11—41no—
248C2×C4×C11—41no—
249C2×C4×C11—41no—
250C2×C4×C11C4×D412yes—
251C2×C4×C11C4×D412yes—
252C2×C4×C11C23×C411yescharacteristic
253C2×C4×C11C2×#4111yescharacteristic
254C2×C4×C11C2×#4111yescharacteristic
255C2×C4×C11C2×#4111yescharacteristic
256C2×C4×C11C2×#4111yescharacteristic
257C2×C4×C11C2×#4111yescharacteristic
258C2×Dic11—22no—
259C2×Dic11—21no—
260C2×Dic11—21no—
261C2×Dic11—22no—
262C2×Dic11—22no—
263C2×Dic11—21no—
264C2×Dic11—21no—
265C2×Dic11—22no—
266C2×Dic11—21no—
267C2×Dic11—21no—
268C2×Dic11—21no—
269C2×Dic11—21no—
270C2×Dic11—21no—
271C2×Dic11—21no—
272C2×Dic11—41no—
273C2×Dic11—41no—
274C2×Dic11—41no—
275C2×Dic11—41no—
276C2×Dic11—41no—
277C2×Dic11—41no—
278C2×Dic11—41no—
279C2×Dic11—41no—
280C2×Dic11AGL(1,ℤ/8)12yes—
281C2×Dic11#13712yes—
282Dic22—22no—
283Dic22—22no—
284Dic22—22no—
285Dic22#11714yescomplemented
286C8×C11—41no—
287C8×C11—41no—
288C11⋊C8—22no—
289C11⋊C8—22no—
290C22×SD16—111no—
291C2×C4×Q8—111no—
292C2×C4×Q8—111no—
293C2×C4×Q8—111no—
294C2×C4×D4C2×D1111yescharacteristic
295C2×#117—221no—
296#440—118no—
297C2×#107—111no—
298C2×#107—111no—
299C2×#106—111no—
300C2×#103—111no—
301C2×#103—111no—
302C2×#103—111no—
303C2×#102—111no—
304C2×#102C2×D1111yescharacteristic
305C2×#102C2×D1111yescharacteristic
306C2×#97—111no—
307C2×#97C2×D1111yescharacteristic
308#390—111no—
309#390—111no—
310#389—111no—
311#389—111no—
312#388—221no—
313#388—221no—
314C4×C4⋊C4—111no—
315C4×C4⋊C4—111no—
316C4×C4⋊C4—111no—
317C4×C4⋊C4—221no—
318C4×C4⋊C4—221no—
319C4×C4⋊C4—221no—
320C4×C4⋊C4C2×D1111yescharacteristic
321C4×C22⋊C4—221no—
322C43—111no—
323C43—221no—
324#340—111no—
325#340—111no—
326#340—111no—
327#340—111no—
328#340C2×D1111yescharacteristic
329#340C2×D1111yescharacteristic
330#336—111no—
331#336—111no—
332#336—111no—
333#336—111no—
334C22×C11—42no—
335C22×C11—42no—
336C22×C11—42no—
337C22×C11—81no—
338C22×C11—81no—
339C22×C11—81no—
340C22×C11—81no—
341C22×C11C2×C4×D411yescharacteristic
342C22×C11#52511yescharacteristic
343C22×C11#52511yescharacteristic
344C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C11C2×#13711yescharacteristic
346C22×C11#57511yescharacteristic
347C22×C11#57511yescharacteristic
348C4×C11—22no—
349C4×C11—22no—
350C4×C11—42no—
351C4×C11—42no—
352C4×C11—42no—
353C4×C11C2×#11711yescharacteristic
354C4×C11C4×#4111yescharacteristic
355C4×C11#52911yescharacteristic
356C4×C11#52911yescharacteristic
357Dic11—22no—
358Dic11—22no—
359Dic11—24no—
360Dic11—41no—
361Dic11—41no—
362Dic11—42no—
363Dic11—81no—
364Dic11—81no—
365C22×Q8—111no—
366C22×D4C22×D1111yescharacteristic
367C23×C4—21no—
368C2×SD16—114no—
369C2×SD16—222no—
370C2×SD16—222no—
371C22×C8—111no—
372C22×C8—111no—
373C22×C8C22×D1111yescharacteristic
374C22×C8C22×D1111yescharacteristic
375C4×Q8—114no—
376C4×Q8—222no—
377C4×Q8—222no—
378C4×Q8—222no—
379C4×Q8—222no—
380C4×Q8—222no—
381C4×D4—22no—
382C4×D4C4×D1114yes—
383#117—444no—
384C2×C4⋊C4—111no—
385C2×C4⋊C4—111no—
386C2×C4⋊C4—111no—
387C2×C4⋊C4—221no—
388C2×C4⋊C4—221no—
389C2×C4⋊C4—221no—
390C2×C4⋊C4—221no—
391C2×C4⋊C4—221no—
392C2×C4⋊C4C22×D1111yescharacteristic
393C2×C4⋊C4C22×D1111yescharacteristic
394C2×C4⋊C4C22×D1111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—221no—
397C2×C22⋊C4—221no—
398C2×C22⋊C4—221no—
399C2×C22⋊C4—221no—
400C2×C42—21no—
401C2×C42—111no—
402C2×C42—111no—
403C2×C42—111no—
404C2×C42—111no—
405C2×C42—111no—
406C2×C42—111no—
407C2×C42—221no—
408C2×C42—221no—
409C2×C42—221no—
410C2×C42—221no—
411C2×C42—221no—
412C2×C42—221no—
413C2×C42C22×D1111yescharacteristic
414#107—114no—
415#107—222no—
416#106—222no—
417#103—114no—
418#103—222no—
419#103—222no—
420#102—22no—
421#102—22no—
422#102—114no—
423#97—22no—
424#97—114no—
425#95—21no—
426#95—221no—
427#95—221no—
428#95—221no—
429#95—221no—
430#95—221no—
431C2×C11—82no—
432C2×C11#229011yescharacteristic
433C2×C11C4×AGL(1,ℤ/8)11yescharacteristic
434C2×C11C4×#13711yescharacteristic
435C2×C11#311111yescharacteristic
436C2×C11#311211yescharacteristic
437C2×C11#311511yescharacteristic
438C2×C11#311711yescharacteristic
439C24—21no—
440C2×Q8—112no—
441C2×Q8—112no—
442C2×Q8—112no—
443C2×Q8—221no—
444C2×Q8—221no—
445C2×Q8—221no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4C2×C4×D1112yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—111no—
459C22×C4—111no—
460C22×C4—111no—
461C22×C4—111no—
462C22×C4—221no—
463C22×C4—221no—
464C22×C4—221no—
465C22×C4—221no—
466C22×C4D4×D1111yescharacteristic
467C22×C4D4×D1111yescharacteristic
468C22×C4C23×D1111yescharacteristic
469SD16—228no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—112no—
477C2×C8—112no—
478C2×C8—221no—
479C2×C8—221no—
480C2×C8—221no—
481C2×C8—221no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—112no—
486C4⋊C4—224no—
487C4⋊C4—224no—
488C4⋊C4—224no—
489C4⋊C4—221no—
490C4⋊C4—221no—
491C4⋊C4—221no—
492C4⋊C4—221no—
493C4⋊C4—221no—
494C4⋊C4—442no—
495C4⋊C4—442no—
496C4⋊C4C2×C4×D1112yes—
497C4⋊C4#433912yes—
498C4⋊C4#433912yes—
499C22⋊C4—24no—
500C22⋊C4—442no—
501C22⋊C4—442no—
502C22⋊C4—442no—
503C22⋊C4—442no—
504C42—42no—
505C42—222no—
506C42—222no—
507C42—224no—
508C42—224no—
509C42—224no—
510C42—221no—
511C42—221no—
512C42—221no—
513C42—221no—
514C42—441no—
515C42—441no—
516C42—442no—
517C42—442no—
518C42—442no—
519C42C2×C4×D1112yes—
520C42#434112yes—
521C11#2112411yescharacteristic, complemented, Sylow
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D1111yescharacteristic
530Q8—114no—
531Q8—222no—
532Q8—222no—
533Q8—222no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—112no—
553C2×C4—112no—
554C2×C4—222no—
555C2×C4—221no—
556C2×C4—221no—
557C2×C4—221no—
558C2×C4—221no—
559C2×C4—222no—
560C2×C4—222no—
561C2×C4—222no—
562C2×C4—221no—
563C2×C4—221no—
564C2×C4—221no—
565C2×C4—221no—
566C2×C4—221no—
567C2×C4—221no—
568C2×C4—441no—
569C2×C4—441no—
570C2×C4—441no—
571C2×C4—441no—
572C2×C4—441no—
573C2×C4—441no—
574C2×C4—441no—
575C2×C4—441no—
576C2×C4(C22⋊D11)⋊C412yes—
577C2×C4(C22⋊D11)⋊C412yes—
578C2×C4C22×C4×D1111yescharacteristic
579C2×C4C2×#433911yescharacteristic
580C2×C4C2×#433911yescharacteristic
581C2×C4C2×#434111yescharacteristic
582C2×C4#41×D1111yescharacteristic
583C2×C4#41×D1111yescharacteristic
584C8—41no—
585C8—41no—
586C8—222no—
587C8—222no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C2×(C22⋊D11)⋊C411yescharacteristic
596C22#1062307811yescharacteristic
597C22#1062307811yescharacteristic
598C22D11×AGL(1,ℤ/8)11yescharacteristic
599C22#1062330411yescharacteristic
600C22#1062330411yescharacteristic
601C22D11×#13711yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—222no—
608C4—222no—
609C4—224no—
610C4—442no—
611C4—441no—
612C4—441no—
613C4—881no—
614C4—881no—
615C4C4×#433911yescharacteristic
616C4D11×#11711yescharacteristic
617C4#1062308811yescharacteristic
618C4#1062308811yescharacteristic
619C2—82no—
620C2#1307106211yescharacteristic
621C2#1307742011yescharacteristic
622C2#1307744011yescharacteristic
623C2#1307872111yescharacteristic
624C2#1307873611yescharacteristic
625C2#1307876311yescharacteristic
626C2#1307878511yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1762C2
socleC23×C11C22×D4
prime radicalC22×D4×C11C23
cosocleC22×C4×C11C24
hypercenter#1762D11
hypocenterC11#21124
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. C4×#66940C2semidirect
6. #13069211C2semidirect
7. #13069190C2semidirect
10. #13068885C2semidirect
11. #13068871C2semidirect
12. #13068514C2semidirect
13. #13068447C2semidirect
14. #13068364C2semidirect
32. #10622631C4semidirect
285. Dic22#117semidirect
521. C11#21124semidirect

Maximal subgroups

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

subgroupindex
C2×#106226312
#130710812
C4×#669402
#130692462
#130692392
#130692112
#130691902
#130689682
#130688852
#130688712
#130685142
#130684472
#130683642
C11×#17622
#2112411