Finite Group Databasebeta

← #14812726

Subgroups — #14812726

Order 2816.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13079146C211yesmaximal, characteristic, complemented
2#13078482C211yesmaximal, characteristic
3#13078467C211yesmaximal, characteristic
4#13077214C211yesmaximal, characteristic
5#13070129C211yesmaximal, characteristic, complemented
6#13070125C211yesmaximal, characteristic, complemented
7#13069966C211yesmaximal, characteristic, complemented
8#13069960C211yesmaximal, characteristic, complemented
9#13069400C211yesmaximal, characteristic, complemented
10#13069368C211yesmaximal, characteristic, complemented
11#13068844C211yesmaximal, characteristic
12#13068785C211yesmaximal, characteristic, complemented
13#13068673C211yesmaximal, characteristic
14#13068461C211yesmaximal, characteristic
15C11×#1860C211yesmaximal, characteristic
16C11×#520C2211yescharacteristic
17#10623358C2211yescharacteristic
18C2×#67001C2211yescharacteristic
19#10623050C2211yescharacteristic
20#10623031C2211yescharacteristic
21C11×#428C2211yescharacteristic
22C2×C11×#104C2211yescharacteristic
23C11×#417C2211yescharacteristic
24C11×#382C2211yescharacteristic
25#10622846C2211yescharacteristic
26#10622845C2211yescharacteristic
27C2×#66886C2211yescharacteristic
28#10622735C2211yescharacteristic
29#10622734C2211yescharacteristic
30#10622734C2211yescharacteristic
31#10622732C2211yescharacteristic
32#10622635—21no—
33#10622634—21no—
34#10622622—21no—
35#10622620—21no—
36#10622612C2211yescharacteristic
37#10622611—21no—
38#10622611C2211yescharacteristic
39#10622608—21no—
40#10622608C2211yescharacteristic
41#10622577—21no—
42#10622577C2211yescharacteristic
43#10622577C2211yescharacteristic
44C2×#66872C2211yescharacteristic
45C2×#66872C2211yescharacteristic
46C2×#66871—21no—
47C2×#66871C2211yescharacteristic
48C2×#66871C2211yescharacteristic
49#10622548C2211yescharacteristic
50C2×#66867C2211yescharacteristic
51#10622491—21no—
52#10622489—21no—
53#10622469—21no—
54#10622468—21no—
55#10622409—21no—
56#10622404—21no—
57#10622379—21no—
58#10622374—21no—
59C11×#343C2211yescharacteristic
60C11×#339C2211yescharacteristic
61#10622196C2211yescharacteristic
62#10622192C2211yescharacteristic
63#10622192C2211yescharacteristic
64#10622168C2211yescharacteristic
65#10622161C2211yescharacteristic
66#10622161C2211yescharacteristic
67C2×C11×#41C2311yescharacteristic
68C2×C11×M16C2311yescharacteristic
69C2×C11×M16C2311yescharacteristic
70C4×Q8×C11—21no—
71C4×D4×C11—21no—
72C4×D4×C11—41no—
73C11×#117—21no—
74C11×#117C2311yescharacteristic
75C2×C11×C4⋊C4—21no—
76C2×C11×C4⋊C4C2311yescharacteristic
77C2×C11×C4⋊C4C2311yescharacteristic
78C2×C42×C11C2311yescharacteristic
79#67001—22no—
80#67001Q814yescomplemented
81Q8×Dic11—21no—
82D4×Dic11—21no—
83D4×Dic11—41no—
84#66979—21no—
85#66979C2311yescharacteristic
86#66979C2311yescharacteristic
87C2×C4⋊Dic11—21no—
88C2×C4⋊Dic11—41no—
89C2×C4⋊Dic11C2311yescharacteristic
90C2×C4⋊Dic11C2311yescharacteristic
91#66977—41no—
92C2×#4318—21no—
93C2×#4318C2311yescharacteristic
94C2×#4318C2311yescharacteristic
95C22×C11⋊C8—21no—
96C22×C11⋊C8C2311yescharacteristic
97C22×C11⋊C8C2311yescharacteristic
98#66943—21no—
99#66943—21no—
100#66943—21no—
101#66941—21no—
102#66941—21no—
103#66941—21no—
104#66931—41no—
105#66923—21no—
106#66922—21no—
107C11×#107—21no—
108C11×#106—21no—
109C11×#104Q814yescomplemented
110C11×#103—21no—
111C11×#102—21no—
112C11×#95—41no—
113#66899—21no—
114#66899D411yescharacteristic
115#66899D411yescharacteristic
116#66896—21no—
117#66896D411yescharacteristic
118#66896D411yescharacteristic
119#66886—22no—
120#66882—21no—
121#66882—21no—
122#66881—21no—
123#66881—21no—
124#66879—21no—
125Dic11⋊C8—21no—
126Dic11⋊C8—21no—
127C8×Dic11—21no—
128#66872—21no—
129#66872—21no—
130#66872—21no—
131#66872—21no—
132#66872—21no—
133#66872—21no—
134#66872—21no—
135#66872D411yescharacteristic
136#66872D411yescharacteristic
137#66871—21no—
138#66871—21no—
139#66871—21no—
140#66871—21no—
141#66871—21no—
142#66871—21no—
143#66871—21no—
144#66871D411yescharacteristic
145#66871D411yescharacteristic
146C4⋊(C11⋊C8)—21no—
147C4⋊(C11⋊C8)—21no—
148#66867—22no—
149#26085—111nomaximal, Sylow
150C11×#41—21no—
151C11×#41—21no—
152C11×#41C2×Q812yes—
153C11×#41C2×Q812yes—
154C2×Q8×C11C2×D411yescharacteristic
155C2×D4×C11—21no—
156C2×D4×C11C2×D411yescharacteristic
157C22×C4×C11—21no—
158C22×C4×C11—21no—
159C22×C4×C11—21no—
160C22×C4×C11—21no—
161C22×C4×C11—41no—
162C22×C4×C11C2411yescharacteristic
163C22×Dic11—21no—
164C22×Dic11—21no—
165C22×Dic11—41no—
166C2×Dic22—41no—
167C2×Dic22—41no—
168C11×M16—21no—
169C11×M16—21no—
170C11×M16—21no—
171C11×M16C2×Q812yes—
172C2×C8×C11—21no—
173C2×C8×C11—21no—
174C11×C4⋊C4—21no—
175C11×C4⋊C4—21no—
176C11×C4⋊C4—21no—
177C11×C4⋊C4—21no—
178C11×C4⋊C4—41no—
179C11×C4⋊C4C2×D411yescharacteristic
180C11×C4⋊C4C2×D411yescharacteristic
181C11×C22⋊C4—21no—
182C11×C22⋊C4—41no—
183C42×C11—21no—
184C42×C11—21no—
185C42×C11—41no—
186C42×C11C2×Q812yes—
187C22⋊Dic11—21no—
188C22⋊Dic11—21no—
189C22⋊Dic11—41no—
190C4⋊Dic11—21no—
191C4⋊Dic11—21no—
192C4⋊Dic11—21no—
193C4⋊Dic11—21no—
194C4⋊Dic11—41no—
195C4⋊Dic11—41no—
196C4⋊Dic11—41no—
197C4⋊Dic11—82no—
198C4⋊Dic11C2×D411yescharacteristic
199C4⋊Dic11C2×D411yescharacteristic
200C4⋊Dic11C2×D411yescharacteristic
201C4⋊Dic11C2×D411yescharacteristic
202Dic11⋊C4—41no—
203Dic11⋊C4—41no—
204Dic11⋊C4—41no—
205Dic11⋊C4—41no—
206C4×Dic11—21no—
207C4×Dic11—21no—
208C4×Dic11—41no—
209#4318—24no—
210#4318—21no—
211#4318—21no—
212#4318—21no—
213#4318C2×Q812yes—
214C2×C11⋊C8—24no—
215C2×C11⋊C8—21no—
216C2×C11⋊C8—21no—
217C2×C11⋊C8—21no—
218C2×C11⋊C8—21no—
219C2×C11⋊C8C2×D411yescharacteristic
220C2×C11⋊C8C2×D411yescharacteristic
221C2×C11⋊C8C2×D411yescharacteristic
222C2×C11⋊C8C2×D411yescharacteristic
223C2×C11⋊C8C2×Q812yes—
224C2×C11⋊C8#4112yes—
225#3163—111no—
226#3084—111no—
227#3083—111no—
228#2913—111no—
229#2090—111no—
230#2089—111no—
231#2061—111no—
232#2060—111no—
233#1952—111no—
234#1946—111no—
235#1860D1111yescharacteristic
236#1859—111no—
237#1847—111no—
238#1815—111no—
239#1765—111no—
240C23×C11—21no—
241C23×C11H(5,𝔽2)11yescharacteristic
242Q8×C11—21no—
243Q8×C11C4⋊Q812yes—
244D4×C11—22no—
245D4×C11—22no—
246D4×C11—21no—
247D4×C11C4⋊Q812yes—
248C2×C4×C11—21no—
249C2×C4×C11—22no—
250C2×C4×C11—22no—
251C2×C4×C11—21no—
252C2×C4×C11—21no—
253C2×C4×C11—22no—
254C2×C4×C11—41no—
255C2×C4×C11—41no—
256C2×C4×C11—41no—
257C2×C4×C11—41no—
258C2×C4×C11—81no—
259C2×C4×C11—81no—
260C2×C4×C11—161no—
261C2×C4×C11C22×D411yescharacteristic
262C2×C4×C11C22×D411yescharacteristic
263C2×C4×C11C22×Q811yescharacteristic
264C2×C4×C11C22×Q811yescharacteristic
265C2×C4×C11C2×#4111yescharacteristic
266C2×C4×C112−1+411yescharacteristic
267C2×Dic11—21no—
268C2×Dic11—21no—
269C2×Dic11—21no—
270C2×Dic11—21no—
271C2×Dic11—41no—
272C2×Dic11—41no—
273C2×Dic11—81no—
274C2×Dic11—81no—
275C2×Dic11—161no—
276Dic22—161no—
277Dic22—161no—
278C8×C11—22no—
279C8×C11—41no—
280C11⋊C8—22no—
281C11⋊C8—21no—
282C11⋊C8—21no—
283C11⋊C8—41no—
284C11⋊C8C22⋊Q812yes—
285C11⋊C8C22⋊Q812yes—
286C2×#131—111no—
287#544—111no—
288#544—111no—
289#520—111no—
290#520C2×D1111yescharacteristic
291#501—221no—
292#501—221no—
293#499—221no—
294#498—221no—
295#479—221no—
296#479—221no—
297#478—221no—
298#478—221no—
299#477—221no—
300#476—221no—
301#475—221no—
302#474—221no—
303C2×#108—111no—
304#428—111no—
305#428—111no—
306#428—111no—
307#428—111no—
308#428—111no—
309#428—221no—
310#428—221no—
311#428C2×D1111yescharacteristic
312#427—111no—
313#427—221no—
314C2×#107—111no—
315C2×#107—111no—
316C2×#107—221no—
317C2×#106—111no—
318C2×#106—111no—
319#424—111no—
320#424—111no—
321C2×#104C2×D1111yescharacteristic
322#417—111no—
323#417C2×D1111yescharacteristic
324#416—111no—
325#405—111no—
326C2×#97—111no—
327#384—111no—
328#382C2×D1111yescharacteristic
329#343—111no—
330#343—111no—
331#343—111no—
332#343C2×D1111yescharacteristic
333#341—111no—
334#339—111no—
335#339—111no—
336#339C2×D1111yescharacteristic
337C22×C11—21no—
338C22×C11—21no—
339C22×C11—22no—
340C22×C11—41no—
341C22×C11—41no—
342C22×C11#54411yescharacteristic
343C22×C11D4211yescharacteristic
344C22×C11#55411yescharacteristic
345C4×C11—22no—
346C4×C11—42no—
347C4×C11—41no—
348C4×C11—41no—
349C4×C11—81no—
350C4×C11—81no—
351C4×C11C2×C22⋊Q811yescharacteristic
352C4×C11C2×C4⋊Q811yescharacteristic
353C4×C11#54711yescharacteristic
354C4×C11D4×Q811yescharacteristic
355Dic11—41no—
356Dic11—41no—
357Dic11—41no—
358Dic11—41no—
359Dic11—81no—
360Dic11—81no—
361C2×#41C22×D1111yescharacteristic
362#131—114no—
363#131—222no—
364C2×M16—111no—
365C2×M16—111no—
366C2×M16—221no—
367C2×M16C22×D1111yescharacteristic
368C2×M16C22×D1111yescharacteristic
369C22×C8—111no—
370C22×C8—111no—
371C22×C8—221no—
372C4⋊Q8—221no—
373C4⋊Q8—221no—
374C4⋊Q8—221no—
375C4⋊Q8—221no—
376#125—221no—
377#125—221no—
378#125—221no—
379#125—221no—
380C22⋊Q8—441no—
381C22⋊Q8—441no—
382C4×Q8—21no—
383C4×Q8—221no—
384C4×D4—21no—
385C4×D4—41no—
386C4×D4—221no—
387C4×D4—441no—
388#117—21no—
389#117—111no—
390#117—111no—
391#117—221no—
392#117C22×D1111yescharacteristic
393C2×C4⋊C4—21no—
394C2×C4⋊C4—111no—
395C2×C4⋊C4—111no—
396C2×C4⋊C4—221no—
397C2×C4⋊C4—441no—
398C2×C4⋊C4C22×D1111yescharacteristic
399C2×C4⋊C4C22×D1111yescharacteristic
400C2×C42C22×D1111yescharacteristic
401#108—222no—
402#107—21no—
403#107—111no—
404#107—111no—
405#107—221no—
406#107—221no—
407#107—221no—
408#107—221no—
409#107—221no—
410#107—221no—
411#107—221no—
412#107—221no—
413#107—221no—
414#106—21no—
415#106—111no—
416#106—111no—
417#106—221no—
418#106—221no—
419#106—221no—
420#106—221no—
421#106—221no—
422#106—221no—
423#106—221no—
424#106—221no—
425#106—221no—
426C4⋊C8—221no—
427C4⋊C8—221no—
428C4⋊C8—221no—
429C4⋊C8—221no—
430#104Dic2214yescomplemented
431#103—21no—
432#103—111no—
433#103—111no—
434#103—221no—
435#102—21no—
436#102—111no—
437#102—111no—
438#102—221no—
439#97—222no—
440#97—221no—
441C4×C8—221no—
442#95—41no—
443C2×C11—21no—
444C2×C11—21no—
445C2×C11—42no—
446C2×C11#244011yescharacteristic
447C2×C11#328811yescharacteristic
448C2×C11#328911yescharacteristic
449#41—21no—
450#41—21no—
451#41C2×Dic2212yes—
452#41Q8×D1112yes—
453C2×Q8—441no—
454C2×Q8—441no—
455C2×Q8D4×D1111yescharacteristic
456C2×D4—21no—
457C2×D4D4×D1111yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—221no—
464C22×C4—221no—
465C22×C4—441no—
466C22×C4C23×D1111yescharacteristic
467M16—21no—
468M16—21no—
469M16—21no—
470M16—112no—
471M16—224no—
472M16—221no—
473M16—221no—
474M16—221no—
475M16C2×Dic2212yes—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—111no—
479C2×C8—111no—
480C2×C8—112no—
481C2×C8—112no—
482C2×C8—111no—
483C2×C8—111no—
484C2×C8—224no—
485C2×C8—221no—
486C2×C8—221no—
487C2×C8—221no—
488C2×C8—221no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—111no—
495C4⋊C4—111no—
496C4⋊C4—111no—
497C4⋊C4—111no—
498C4⋊C4—221no—
499C4⋊C4—221no—
500C4⋊C4—221no—
501C4⋊C4—221no—
502C4⋊C4—441no—
503C4⋊C4—441no—
504C4⋊C4—441no—
505C4⋊C4—441no—
506C4⋊C4—441no—
507C4⋊C4—441no—
508C4⋊C4—441no—
509C4⋊C4—882no—
510C4⋊C4D4×D1111yescharacteristic
511C4⋊C4D4×D1111yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—41no—
514C22⋊C4—221no—
515C22⋊C4—221no—
516C22⋊C4—441no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42—221no—
521C42—221no—
522C42—441no—
523C42C2×Dic2212yes—
524C11#2608511yescharacteristic, complemented, Sylow
525C23—21no—
526C23#6703711yescharacteristic
527Q8—21no—
528Q8—1761no—
529Q8—1761no—
530Q8#6694112yes—
531D4—22no—
532D4—22no—
533D4—21no—
534D4#6694112yes—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—161no—
548C2×C4—221no—
549C2×C4—221no—
550C2×C4—221no—
551C2×C4—221no—
552C2×C4—441no—
553C2×C4—441no—
554C2×C4—881no—
555C2×C4—881no—
556C2×C4—1761no—
557C2×C4C22×Dic2211yescharacteristic
558C2×C4C2×D4×D1111yescharacteristic
559C2×C4C2×D4×D1111yescharacteristic
560C2×C4C2×Q8×D1111yescharacteristic
561C2×C4#41×D1111yescharacteristic
562C2×C4#6704311yescharacteristic
563C8—22no—
564C8—41no—
565C8—112no—
566C8—112no—
567C8—222no—
568C8—221no—
569C8—221no—
570C8—441no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22#1062307011yescharacteristic
577C22#1062312111yescharacteristic
578C22#1062315411yescharacteristic
579C4—22no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—441no—
586C4—441no—
587C4—441no—
588C4—441no—
589C4—881no—
590C4—881no—
591C4C2×#6694111yescharacteristic
592C4D4×Dic2211yescharacteristic
593C4D11×C22⋊Q811yescharacteristic
594C4#1062318611yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2#1307231711yescharacteristic
599C2#1308038511yescharacteristic
600C2#1308039311yescharacteristic
subgroupG/H
centerC22#10623121
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1860C2
socleC22×C11D42
prime radicalC2×C11×#41C23
hypercenter#1860D11
hypocenterC11#26085
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

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

NH ≅ G/Ntype
1. #13079146C2semidirect
5. #13070129C2semidirect
6. #13070125C2semidirect
7. #13069966C2semidirect
8. #13069960C2semidirect
9. #13069400C2semidirect
10. #13069368C2semidirect
12. #13068785C2semidirect
80. #67001Q8semidirect
109. C11×#104Q8semidirect
430. #104Dic22semidirect
524. C11#26085semidirect

Maximal subgroups

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

subgroupindex
#130791462
#130784822
#130784672
#130772142
#130701292
#130701252
#130699662
#130699602
#130694002
#130693682
#130688442
#130687852
#130686732
#130684612
C11×#18602
#2608511