Finite Group Databasebeta

← #35371990

Subgroups — #35371990

Order 21248.

938 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
1#24916552C211yesmaximal, characteristic, complemented
2#24910114C211yesmaximal, characteristic, complemented
3#24906040C212yesmaximal, complemented
4#24906023C212yesmaximal, complemented
5#24906010C212yesmaximal, complemented
6#24906001C212yesmaximal, complemented
7#24905988C211yesmaximal, characteristic, complemented
8#24905971C211yesmaximal, characteristic, complemented
9#24905958C211yesmaximal, characteristic, complemented
10#24905949C211yesmaximal, characteristic, complemented
11C83×#3522C211yesmaximal, characteristic
12#19127832—21no—
13#19127832—22no—
14Q8×C83⋊C8—21no—
15#19127758C2211yescharacteristic
16#19127758C2212yes—
17#19127754C2212yes—
18#19127754C2211yescharacteristic
19#19127608—21no—
20#19127600—21no—
21#19127587—22no—
22#19127569C2212yes—
23#19127567C2212yes—
24C4×C4⋊Dic83C2211yescharacteristic
25#19127547C2211yescharacteristic
26#19127438C2212yes—
27#19127438C2211yescharacteristic
28#19127438C2211yescharacteristic
29#19127435C2212yes—
30#19127435C2212yes—
31#19127435C2212yes—
32#19127435C2211yescharacteristic
33#19127435C2212yes—
34#19127435C2211yescharacteristic
35#19127435C2212yes—
36#559×C83C411yescharacteristic, complemented
37#559×C83C412yescomplemented
38Q82×C83C411yescharacteristic, complemented
39#554×C83C411yescharacteristic, complemented
40#554×C83C412yescomplemented
41#550×C83C411yescharacteristic, complemented
42#548×C83—21no—
43#547×C83—22no—
44#547×C83—21no—
45#542×C83—21no—
46#542×C83—22no—
47#532×C83C2212yes—
48#532×C83C2211yescharacteristic
49#529×C83C2211yescharacteristic
50#529×C83C2212yes—
51#520×C83—21no—
52C2×C4×Q8×C83C2211yescharacteristic
53C2×C4⋊Dic83—21no—
54C2×C4⋊Dic83—22no—
55C2×C4×Dic83—21no—
56C2×C4×Dic83—21no—
57C4⋊(C83⋊C8)—21no—
58C4⋊(C83⋊C8)—22no—
59C4⋊(C83⋊C8)—22no—
60C4⋊(C83⋊C8)—21no—
61C22⋊(C83⋊C8)C2311yescharacteristic
62C22⋊(C83⋊C8)C2312yes—
63C22⋊(C83⋊C8)C2311yescharacteristic
64#14581703—22no—
65#14581703—21no—
66C4×C83⋊C8—21no—
67C4×C83⋊C8—22no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—22no—
71C83⋊#95—21no—
72C83⋊#95C2312yes—
73C83⋊#95C2311yescharacteristic
74C83⋊#95C2311yescharacteristic
75C2×#41×C83—21no—
76C2×#41×C83—21no—
77C2×#41×C83—22no—
78C22×Q8×C83D411yescharacteristic
79C4⋊Q8×C83—22no—
80C4⋊Q8×C83—21no—
81C4⋊Q8×C83C2×C412yes—
82C4⋊Q8×C83C2×C411yescharacteristic
83#127×C83C2×C412yes—
84#127×C83C2×C411yescharacteristic
85#126×C83—22no—
86#126×C83—21no—
87#125×C83C2×C411yescharacteristic
88#125×C83C2×C412yes—
89#124×C83—21no—
90#124×C83—22no—
91#124×C83—22no—
92#124×C83—21no—
93#124×C83C2×C411yescharacteristic
94#124×C83C2×C412yes—
95#123×C83—21no—
96#123×C83—22no—
97#123×C83—22no—
98#123×C83—22no—
99#123×C83—22no—
100C22⋊Q8×C83—21no—
101C22⋊Q8×C83—22no—
102C22⋊Q8×C83—22no—
103C22⋊Q8×C83—22no—
104C22⋊Q8×C83—22no—
105C22⋊Q8×C83C2311yescharacteristic
106#121×C83—21no—
107#121×C83—22no—
108#121×C83—22no—
109#121×C83—21no—
110#121×C83C2311yescharacteristic
111#121×C83C2312yes—
112C4×Q8×C83—21no—
113C4×Q8×C83C2×C411yescharacteristic
114C4×Q8×C83C2×C412yes—
115C4×Q8×C83C2×C411yescharacteristic
116C4×Q8×C83C2×C411yescharacteristic
117C4×Q8×C83C2×C412yes—
118C4×Q8×C83D411yescharacteristic
119C4×Q8×C83D411yescharacteristic
120C4×Q8×C83D412yes—
121C4×D4×C83—21no—
122C4×D4×C83—22no—
123C4×D4×C83—22no—
124C4×D4×C83—21no—
125C4×D4×C83—21no—
126C4×D4×C83—22no—
127C4×D4×C83C2×C411yescharacteristic
128C4×D4×C83C2×C412yes—
129C4×D4×C83C2×C412yes—
130C4×D4×C83C2×C412yes—
131C4×D4×C83C2×C412yes—
132#117×C83—21no—
133#117×C83—22no—
134C2×C4⋊C4×C83—22no—
135C2×C4⋊C4×C83—21no—
136C2×C4⋊C4×C83D412yes—
137C2×C4⋊C4×C83D411yescharacteristic
138C2×C42×C83C2311yescharacteristic
139C2×C42×C83C2312yes—
140C22×Dic83—21no—
141C22×Dic83—21no—
142C22×Dic83—22no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—22no—
146C4⋊Dic83—42no—
147C4⋊Dic83—82no—
148C4×Dic83—42no—
149C4×Dic83—81no—
150#41×C83—44no—
151#41×C83—81no—
152#41×C83—82no—
153#41×C83—81no—
154#41×C83—82no—
155C2×Q8×C83—22no—
156C2×Q8×C83—21no—
157C2×Q8×C83—42no—
158C2×Q8×C83C22⋊C411yescharacteristic
159C2×Q8×C83C22⋊C411yescharacteristic
160C2×Q8×C83C22⋊C412yes—
161C2×Q8×C83C22×C411yescharacteristic
162C2×D4×C83—21no—
163C2×D4×C83—22no—
164C2×D4×C83—22no—
165C2×D4×C83—21no—
166C2×D4×C83—22no—
167C2×D4×C83—21no—
168C2×D4×C83C22×C411yescharacteristic
169C2×D4×C83C22×C412yes—
170C22×C4×C83—22no—
171C22×C4×C83—21no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D412yes—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D412yes—
179C22×C4×C83C2411yescharacteristic
180C4⋊C4×C83—21no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83—22no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—22no—
186C4⋊C4×C83—22no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83C22⋊C412yes—
190C4⋊C4×C83C22⋊C411yescharacteristic
191C4⋊C4×C83C22⋊C412yes—
192C4⋊C4×C83C22⋊C411yescharacteristic
193C4⋊C4×C83C22⋊C412yes—
194C4⋊C4×C83C22⋊C412yes—
195C4⋊C4×C83C22⋊C412yes—
196C4⋊C4×C83C22×C411yescharacteristic
197C4⋊C4×C83C22×C411yescharacteristic
198C4⋊C4×C83C22×C412yes—
199C22⋊C4×C83—21no—
200C22⋊C4×C83—21no—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—21no—
203C22⋊C4×C83—22no—
204C22⋊C4×C83—22no—
205C22⋊C4×C83—21no—
206C22⋊C4×C83—22no—
207C22⋊C4×C83—22no—
208C22⋊C4×C83—22no—
209C22⋊C4×C83—21no—
210C22⋊C4×C83—22no—
211C42×C83—22no—
212C42×C83—21no—
213C42×C83C22×C411yescharacteristic
214C42×C83C22×C412yes—
215C42×C83C22×C412yes—
216C42×C83C22×C411yescharacteristic
217C42×C83C2×D411yescharacteristic
218C42×C83C2×D412yes—
219C42×C83C2×D412yes—
220C42×C83C2×D411yescharacteristic
221C23×C83—22no—
222C23×C83—21no—
223C23×C83C22×D411yescharacteristic
224Q8×C83—22no—
225Q8×C83—22no—
226Q8×C83—42no—
227Q8×C83—41no—
228Q8×C83—81no—
229Q8×C83—82no—
230D4×C83—41no—
231D4×C83—42no—
232D4×C83—44no—
233D4×C83—42no—
234D4×C83—41no—
235D4×C83—41no—
236D4×C83—42no—
237D4×C83—42no—
238D4×C83—81no—
239D4×C83—82no—
240C2×C4×C83—21no—
241C2×C4×C83—22no—
242C2×C4×C83—21no—
243C2×C4×C83—22no—
244C2×C4×C83—22no—
245C2×C4×C83—21no—
246C2×C4×C83—22no—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—42no—
250C2×C4×C83—41no—
251C2×C4×C83—44no—
252C2×C4×C83—42no—
253C2×C4×C83—82no—
254C2×C4×C83—82no—
255C2×C4×C83—82no—
256C2×C4×C83C2×C22⋊C412yes—
257C2×C4×C83C2×C22⋊C411yescharacteristic
258C2×C4×C83C2×C22⋊C412yes—
259C2×C4×C83C2×C22⋊C411yescharacteristic
260C2×C4×C83C2×C22⋊C412yes—
261C2×C4×C83C2×C22⋊C412yes—
262C2×C4×C83C2×C22⋊C411yescharacteristic
263C2×C4×C83C2×C22⋊C411yescharacteristic
264C2×C4×C83C23×C411yescharacteristic
265C2×C4×C83C22×D411yescharacteristic
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—42no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—42no—
272C2×Dic83—42no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C83⋊C8—41no—
276C83⋊C8—41no—
277C83⋊C8—42no—
278C22×C83—21no—
279C22×C83—21no—
280C22×C83—42no—
281C22×C83—41no—
282C22×C83—42no—
283C22×C83—41no—
284C22×C83—42no—
285C22×C83—41no—
286C22×C83C22×C22⋊C411yescharacteristic
287C22×C83#57611yescharacteristic
288C22×C83#57711yescharacteristic
289C4×C83—21no—
290C4×C83—21no—
291C4×C83—21no—
292C4×C83—22no—
293C4×C83—21no—
294C4×C83—22no—
295C4×C83—21no—
296C4×C83—22no—
297C4×C83—22no—
298C4×C83—41no—
299C4×C83—42no—
300C4×C83—41no—
301C4×C83—41no—
302C4×C83#42112yes—
303Dic83—81no—
304Dic83—82no—
305Dic83—81no—
306#17602—831nomaximal, Sylow
307C2×C83—21no—
308C2×C83—21no—
309C2×C83—82no—
310C2×C83—81no—
311C2×C83#288411yescharacteristic
312C2×C83#289711yescharacteristic
313C2×C83C2×#42111yescharacteristic
314#3522D8311yescharacteristic
315#2984—831no—
316#2320—831no—
317#1521—832no—
318#1517—832no—
319#1514—832no—
320#1512—832no—
321#1509—831no—
322#1505—831no—
323#1502—831no—
324#1500—831no—
325C83#1760211yescharacteristic, complemented, Sylow
326#559Dic8312yescomplemented
327#559Dic8311yescharacteristic, complemented
328Q82Dic8311yescharacteristic, complemented
329#554Dic8311yescharacteristic, complemented
330#554Dic8312yescomplemented
331#550Dic8311yescharacteristic, complemented
332#548—21no—
333#547—22no—
334#547—21no—
335#542—21no—
336#542—22no—
337#532C2×D8311yescharacteristic
338#532C2×D8312yes—
339#529C2×D8311yescharacteristic
340#529C2×D8312yes—
341#520—21no—
342C2×C4×Q8C2×D8311yescharacteristic
343#446—1661no—
344#446—1662no—
345C8×Q8—1661no—
346#432—831no—
347#432—832no—
348#431—831no—
349#431—832no—
350#391—1661no—
351#389—1661no—
352#387—1662no—
353#383—832no—
354#382—832no—
355C4×C4⋊C4—831no—
356#376—831no—
357#328—831no—
358#328—831no—
359#328—832no—
360#327—831no—
361#327—831no—
362#327—832no—
363#327—832no—
364#327—832no—
365#327—832no—
366#327—832no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—22no—
370C22×Q8C22⋊D8311yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8C2×Dic8311yescharacteristic
374C4⋊Q8C2×Dic8312yes—
375#127C2×Dic8311yescharacteristic
376#127C2×Dic8312yes—
377#126—22no—
378#126—21no—
379#125C2×Dic8311yescharacteristic
380#125C2×Dic8312yes—
381#124—21no—
382#124—22no—
383#124—22no—
384#124—21no—
385#124C2×Dic8311yescharacteristic
386#124C2×Dic8312yes—
387#123—21no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8C22×D8311yescharacteristic
398#121—21no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121C22×D8311yescharacteristic
403#121C22×D8312yes—
404C4×Q8—21no—
405C4×Q8C2×Dic8311yescharacteristic
406C4×Q8C2×Dic8312yes—
407C4×Q8C2×Dic8311yescharacteristic
408C4×Q8C2×Dic8311yescharacteristic
409C4×Q8C2×Dic8312yes—
410C4×Q8C22⋊D8311yescharacteristic
411C4×Q8C22⋊D8311yescharacteristic
412C4×Q8C22⋊D8312yes—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4C2×Dic8311yescharacteristic
420C4×D4C2×Dic8312yes—
421C4×D4C2×Dic8312yes—
422C4×D4C2×Dic8312yes—
423C4×D4C2×Dic8312yes—
424#117—21no—
425#117—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—1661no—
429C2×C4⋊C4—1662no—
430C2×C4⋊C4C22⋊D8312yes—
431C2×C4⋊C4C22⋊D8311yescharacteristic
432C2×C42—1661no—
433C2×C42—1661no—
434C2×C42C22×D8311yescharacteristic
435C2×C42C22×D8312yes—
436C4⋊C8—1662no—
437C4⋊C8—1661no—
438C4⋊C8—1661no—
439C4⋊C8—1662no—
440C22⋊C8—831no—
441C22⋊C8—831no—
442C22⋊C8—832no—
443#97—1662no—
444#97—1661no—
445C4×C8—1661no—
446C4×C8—1662no—
447#95—831no—
448#95—831no—
449#95—832no—
450#95—1661no—
451#95—1662no—
452#95—1662no—
453#95—1662no—
454#41—44no—
455#41—81no—
456#41—82no—
457#41—81no—
458#41—82no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8C22⋊Dic8311yescharacteristic
463C2×Q8C22⋊Dic8311yescharacteristic
464C2×Q8C22⋊Dic8312yes—
465C2×Q8C22×Dic8311yescharacteristic
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4C22×Dic8311yescharacteristic
473C2×D4C22×Dic8312yes—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—1661no—
480C22×C4—1661no—
481C22×C4—1662no—
482C22×C4C2×C22⋊D8311yescharacteristic
483C22×C4C2×C22⋊D8312yes—
484C22×C4C2×C22⋊D8311yescharacteristic
485C22×C4C2×C22⋊D8312yes—
486C22×C4C23×D8311yescharacteristic
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1662no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—3322no—
500C4⋊C4—6642no—
501C4⋊C4C22⋊Dic8312yes—
502C4⋊C4C22⋊Dic8311yescharacteristic
503C4⋊C4C22⋊Dic8312yes—
504C4⋊C4C22⋊Dic8311yescharacteristic
505C4⋊C4C22⋊Dic8312yes—
506C4⋊C4C22⋊Dic8312yes—
507C4⋊C4C22⋊Dic8312yes—
508C4⋊C4C22×Dic8311yescharacteristic
509C4⋊C4C22×Dic8312yes—
510C4⋊C4C22×Dic8311yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C42—22no—
524C42—21no—
525C42—3322no—
526C42—6641no—
527C42C22×Dic8311yescharacteristic
528C42C22×Dic8312yes—
529C42C22×Dic8312yes—
530C42C22×Dic8311yescharacteristic
531C42C2×C22⋊D8311yescharacteristic
532C42C2×C22⋊D8312yes—
533C42C2×C22⋊D8312yes—
534C42C2×C22⋊D8311yescharacteristic
535C23—22no—
536C23—21no—
537C23C22×C22⋊D8311yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—41no—
542Q8—81no—
543Q8—82no—
544D4—41no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—41no—
549D4—41no—
550D4—42no—
551D4—42no—
552D4—81no—
553D4—82no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3322no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3322no—
576C2×C4—3322no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4C2×C22⋊Dic8312yes—
580C2×C4C2×C22⋊Dic8311yescharacteristic
581C2×C4C2×C22⋊Dic8312yes—
582C2×C4C2×C22⋊Dic8311yescharacteristic
583C2×C4C2×C22⋊Dic8312yes—
584C2×C4C2×C22⋊Dic8312yes—
585C2×C4C2×C22⋊Dic8311yescharacteristic
586C2×C4C2×C22⋊Dic8311yescharacteristic
587C2×C4C23×Dic8311yescharacteristic
588C2×C4C22×C22⋊D8311yescharacteristic
589C8—3321no—
590C8—3321no—
591C8—3322no—
592C22—21no—
593C22—21no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22C22×C22⋊Dic8311yescharacteristic
601C22#1912850411yescharacteristic
602C22#1912851211yescharacteristic
603C4—21no—
604C4—22no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—21no—
611C4—21no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—6641no—
617C4—6642no—
618C4—6641no—
619C4#1912771812yes—
620C2—21no—
621C2—21no—
622C2—82no—
623C2—81no—
624C2#2491592711yescharacteristic
625C2#2491598911yescharacteristic
626C2C2×#1912771811yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
Fitting, prime radicalC83×#3522C2
socleC22×C83C22×C22⋊C4
cosocleC22×C4×C83C24
hypercenter#3522D83
hypocenterC83#17602
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3522

Complements

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

NH ≅ G/Ntype
1. #24916552C2semidirect
2. #24910114C2semidirect
3. #24906040C2semidirect
4. #24906023C2semidirect
5. #24906010C2semidirect
6. #24906001C2semidirect
7. #24905988C2semidirect
8. #24905971C2semidirect
9. #24905958C2semidirect
10. #24905949C2semidirect
36. #559×C83C4semidirect
37. #559×C83C4semidirect
38. Q82×C83C4semidirect
39. #554×C83C4semidirect
40. #554×C83C4semidirect
41. #550×C83C4semidirect
325. C83#17602semidirect
326. #559Dic83semidirect
327. #559Dic83semidirect
328. Q82Dic83semidirect
329. #554Dic83semidirect
330. #554Dic83semidirect
331. #550Dic83semidirect

Maximal subgroups

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

subgroupindex
#249165522
#249101142
#249060402
#249060232
#249060102
#249060012
#249059882
#249059712
#249059582
#249059492
C83×#35222
#1760283