Finite Group Databasebeta

← C83×#18349 #35246173

Subgroups — C83×#18349 #35246173

Order 21248.

954 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
1C83×#3492C211yesmaximal, characteristic
2C83×#3196C212yesmaximal, complemented
3C83×#2627C211yesmaximal, characteristic, complemented
4C83×#1617C211yesmaximal, characteristic, complemented
5C83×#1616C211yesmaximal, characteristic, complemented
6C83×#1615C212yesmaximal, complemented
7C83×#1605C211yesmaximal, characteristic, complemented
8C83×#1604C211yesmaximal, characteristic, complemented
9C83×#1603C212yesmaximal, complemented
10C83×#1513C212yesmaximal, complemented
11C83×#1463C211yesmaximal, characteristic, complemented
12#554×C83—22no—
13#553×C83—22no—
14#552×C83—22no—
15#547×C83—22no—
16#544×C83C2211yescharacteristic
17#539×C83—21no—
18#539×C83C2212yes—
19#537×C83C2211yescharacteristic
20#525×C83—21no—
21#525×C83—21no—
22#525×C83C2212yes—
23C2×C22⋊Q8×C83C2211yescharacteristic
24#520×C83—21no—
25C22×C4⋊C4×C83C2211yescharacteristic
26#485×C83—22no—
27#484×C83C2212yes—
28#481×C83C2212yes—
29#470×C83—22no—
30#468×C83—22no—
31#452×C83—22no—
32#451×C83C2212yes—
33#448×C83C2212yes—
34#407×C83C2212yes—
35#400×C83C2211yescharacteristic
36#400×C83C2211yescharacteristic
37#398×C83—21no—
38#398×C83—21no—
39#395×C83C2211yescharacteristic
40#395×C83C2211yescharacteristic
41#385×C83—21no—
42#381×C83—21no—
43#380×C83C2211yescharacteristic
44#328×C83C2212yes—
45#328×C83C2212yes—
46#327×C83C2212yes—
47#327×C83C2212yes—
48#324×C83C2212yes—
49#324×C83C2211yescharacteristic
50#324×C83C2211yescharacteristic
51C2×#41×C83—21no—
52C2×#41×C83—21no—
53C2×#41×C83D412yes—
54C22×Q8×C83C2311yescharacteristic
55C23×C4×C83D411yescharacteristic
56C23×C4×C83D411yescharacteristic
57C23×C4×C83C2311yescharacteristic
58C2×Q16×C83—22no—
59C2×SD16×C83—22no—
60C2×M16×C83—22no—
61C4⋊Q8×C83—22no—
62#126×C83—22no—
63#126×C83—22no—
64#125×C83—22no—
65#125×C83—22no—
66#125×C83—22no—
67#123×C83—22no—
68#123×C83—22no—
69#123×C83—22no—
70#123×C83—22no—
71#123×C83—22no—
72C22⋊Q8×C83—21no—
73C22⋊Q8×C83—21no—
74C22⋊Q8×C83—21no—
75C22⋊Q8×C83—22no—
76C22⋊Q8×C83—22no—
77C22⋊Q8×C83—21no—
78C22⋊Q8×C83—22no—
79C22⋊Q8×C83—22no—
80C22⋊Q8×C83—22no—
81C22⋊Q8×C83D412yes—
82C22⋊Q8×C83C2312yes—
83#121×C83—21no—
84#121×C83—21no—
85#121×C83—21no—
86#121×C83—22no—
87#121×C83—21no—
88#121×C83D412yes—
89#121×C83C2312yes—
90#120×C83—22no—
91C4×Q8×C83—22no—
92C4×D4×C83—22no—
93C4×D4×C83—22no—
94C4×D4×C83—22no—
95C4×D4×C83—22no—
96C4×D4×C83—22no—
97C4×D4×C83—22no—
98C4×D4×C83—22no—
99#117×C83—22no—
100#117×C83—21no—
101#117×C83—21no—
102#117×C83D412yes—
103C2×C4⋊C4×C83—22no—
104C2×C4⋊C4×C83—22no—
105C2×C4⋊C4×C83—22no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83D411yescharacteristic
111C2×C4⋊C4×C83D411yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117#107×C83—22no—
118#106×C83—22no—
119#103×C83—22no—
120#103×C83—22no—
121#103×C83—22no—
122#102×C83—22no—
123C22⋊C8×C83—22no—
124C22⋊C8×C83C2312yes—
125C22⋊C8×C83C2312yes—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83—21no—
130#95×C83—21no—
131#95×C83—21no—
132#95×C83C2311yescharacteristic
133#95×C83C2311yescharacteristic
134C24×C83C2×D411yescharacteristic
135#41×C83—42no—
136#41×C83—42no—
137#41×C83—82no—
138#41×C83—82no—
139C2×Q8×C83—21no—
140C2×Q8×C83—21no—
141C2×Q8×C83—22no—
142C2×Q8×C83—81no—
143C2×Q8×C83C2×D412yes—
144C2×D4×C83—21no—
145C2×D4×C83—22no—
146C2×D4×C83—22no—
147C2×D4×C83—22no—
148C2×D4×C83—21no—
149C2×D4×C83C2×D412yes—
150C22×C4×C83—22no—
151C22×C4×C83—21no—
152C22×C4×C83—21no—
153C22×C4×C83—21no—
154C22×C4×C83—22no—
155C22×C4×C83—21no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—41no—
163C22×C4×C83—81no—
164C22×C4×C83—81no—
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D412yes—
171C22×C4×C83C2×D412yes—
172C22×C4×C83C2411yescharacteristic
173Q16×C83—82no—
174SD16×C83—82no—
175M16×C83—82no—
176C2×C8×C83—22no—
177C2×C8×C83—22no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—22no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—22no—
184C4⋊C4×C83—22no—
185C4⋊C4×C83—22no—
186C4⋊C4×C83—22no—
187C4⋊C4×C83—22no—
188C4⋊C4×C83—22no—
189C4⋊C4×C83—22no—
190C4⋊C4×C83—22no—
191C4⋊C4×C83—22no—
192C4⋊C4×C83—22no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83—81no—
195C4⋊C4×C83—81no—
196C4⋊C4×C83—81no—
197C4⋊C4×C83C2×D412yes—
198C4⋊C4×C83C2×D412yes—
199C22⋊C4×C83—22no—
200C22⋊C4×C83—22no—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—22no—
203C22⋊C4×C83—22no—
204C22⋊C4×C83—22no—
205C22⋊C4×C83—22no—
206C22⋊C4×C83—22no—
207C22⋊C4×C83—22no—
208C22⋊C4×C83—42no—
209C22⋊C4×C83—41no—
210C22⋊C4×C83—41no—
211C42×C83—22no—
212C42×C83—22no—
213C42×C83—22no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83—22no—
217C23×C83—41no—
218C23×C83—41no—
219C23×C83#12011yescharacteristic
220C23×C83#12011yescharacteristic
221C23×C83C22×D411yescharacteristic
222Q8×C83—42no—
223Q8×C83—42no—
224Q8×C83—42no—
225Q8×C83—42no—
226Q8×C83—81no—
227D4×C83—42no—
228D4×C83—42no—
229D4×C83—44no—
230D4×C83—44no—
231D4×C83—42no—
232D4×C83—42no—
233D4×C83—81no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—21no—
239C2×C4×C83—22no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—41no—
245C2×C4×C83—42no—
246C2×C4×C83—42no—
247C2×C4×C83—41no—
248C2×C4×C83—42no—
249C2×C4×C83—42no—
250C2×C4×C83—44no—
251C2×C4×C83—42no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83—41no—
260C2×C4×C83—42no—
261C2×C4×C83—41no—
262C2×C4×C83—41no—
263C2×C4×C83—41no—
264C2×C4×C83—41no—
265C2×C4×C83—82no—
266C2×C4×C83—81no—
267C2×C4×C83—82no—
268C2×C4×C83#12012yes—
269C2×C4×C83#13712yes—
270C2×C4×C83C22×D412yes—
271C8×C83—42no—
272C8×C83—42no—
273C22×C83—21no—
274C22×C83—21no—
275C22×C83—21no—
276C22×C83—21no—
277C22×C83—21no—
278C22×C83—21no—
279C22×C83—21no—
280C22×C83—21no—
281C22×C83—42no—
282C22×C83—42no—
283C22×C83—42no—
284C22×C83—42no—
285C22×C83—42no—
286C22×C83C2×#12011yescharacteristic
287C22×C83C2×#13711yescharacteristic
288C22×C83#57511yescharacteristic
289C4×C83—22no—
290C4×C83—22no—
291C4×C83—22no—
292C4×C83—41no—
293C4×C83—42no—
294C4×C83—42no—
295C4×C83—41no—
296C4×C83—41no—
297C4×C83—42no—
298C4×C83—81no—
299C4×C83—81no—
300C4×C83—81no—
301C4×C83—81no—
302#18349C8311yesmaximal, characteristic, direct product factor, Sylow
303C2×C83—21no—
304C2×C83—21no—
305C2×C83—41no—
306C2×C83—41no—
307C2×C83—82no—
308C2×C83#300711yescharacteristic
309C2×C83#302011yescharacteristic
310C2×C83#302611yescharacteristic
311#3492C2×C8311yescharacteristic
312#3196C2×C8312yescomplemented
313#2627C2×C8311yescharacteristic, complemented
314#1617C2×C8311yescharacteristic, complemented
315#1616C2×C8311yescharacteristic, complemented
316#1615C2×C8312yescomplemented
317#1605C2×C8311yescharacteristic, complemented
318#1604C2×C8311yescharacteristic, complemented
319#1603C2×C8312yescomplemented
320#1513C2×C8312yescomplemented
321#1463C2×C8311yescharacteristic, complemented
322C83#1834911yescharacteristic, direct product factor, Sylow
323#554—22no—
324#553—22no—
325#552—22no—
326#547—22no—
327#544C22×C8311yescharacteristic
328#539—21no—
329#539C22×C8312yes—
330#537C22×C8311yescharacteristic
331#525—21no—
332#525—21no—
333#525C22×C8312yes—
334C2×C22⋊Q8C22×C8311yescharacteristic
335#520—21no—
336C22×C4⋊C4C22×C8311yescharacteristic
337#485—22no—
338#484C22×C8312yes—
339#481C22×C8312yes—
340#470—22no—
341#468—22no—
342#452—22no—
343#451C22×C8312yes—
344#448C22×C8312yes—
345#407C22×C8312yes—
346#400C22×C8311yescharacteristic
347#400C22×C8311yescharacteristic
348#398—21no—
349#398—21no—
350#395C22×C8311yescharacteristic
351#395C22×C8311yescharacteristic
352#385—21no—
353#381—21no—
354#380C22×C8311yescharacteristic
355#328C22×C8312yes—
356#328C22×C8312yes—
357#327C22×C8312yes—
358#327C22×C8312yes—
359#324C22×C8312yes—
360#324C22×C8311yescharacteristic
361#324C22×C8311yescharacteristic
362C2×#41—21no—
363C2×#41—21no—
364C2×#41D4×C8312yes—
365C22×Q8C23×C8311yescharacteristic
366C23×C4D4×C8311yescharacteristic
367C23×C4D4×C8311yescharacteristic
368C23×C4C23×C8311yescharacteristic
369C2×Q16—22no—
370C2×SD16—22no—
371C2×M16—22no—
372C4⋊Q8—22no—
373#126—22no—
374#126—22no—
375#125—22no—
376#125—22no—
377#125—22no—
378#123—22no—
379#123—22no—
380#123—22no—
381#123—22no—
382#123—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—21no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8D4×C8312yes—
393C22⋊Q8C23×C8312yes—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—22no—
398#121—21no—
399#121D4×C8312yes—
400#121C23×C8312yes—
401#120—22no—
402C4×Q8—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410#117—22no—
411#117—21no—
412#117—21no—
413#117D4×C8312yes—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4D4×C8311yescharacteristic
422C2×C4⋊C4D4×C8311yescharacteristic
423C2×C4⋊C4C23×C8311yescharacteristic
424C2×C4⋊C4C23×C8311yescharacteristic
425C2×C4⋊C4C23×C8311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428#107—22no—
429#106—22no—
430#103—22no—
431#103—22no—
432#103—22no—
433#102—22no—
434C22⋊C8—22no—
435C22⋊C8C23×C8312yes—
436C22⋊C8C23×C8312yes—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95C23×C8311yescharacteristic
444#95C23×C8311yescharacteristic
445C24C2×D4×C8311yescharacteristic
446#41—42no—
447#41—42no—
448#41—82no—
449#41—82no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—81no—
454C2×Q8C2×D4×C8312yes—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4C2×D4×C8312yes—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—41no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4C2×D4×C8311yescharacteristic
477C22×C4C2×D4×C8311yescharacteristic
478C22×C4C2×D4×C8311yescharacteristic
479C22×C4C2×D4×C8311yescharacteristic
480C22×C4C2×D4×C8311yescharacteristic
481C22×C4C2×D4×C8312yes—
482C22×C4C2×D4×C8312yes—
483C22×C4C24×C8311yescharacteristic
484Q16—82no—
485SD16—82no—
486M16—82no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4C2×D4×C8312yes—
509C4⋊C4C2×D4×C8312yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C42—22no—
523C42—22no—
524C42—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23#120×C8311yescharacteristic
531C23#120×C8311yescharacteristic
532C23C22×D4×C8311yescharacteristic
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—81no—
538D4—42no—
539D4—42no—
540D4—44no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—81no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4#120×C8312yes—
580C2×C4#137×C8312yes—
581C2×C4C22×D4×C8312yes—
582C8—42no—
583C8—42no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22C2×#120×C8311yescharacteristic
598C22C2×#137×C8311yescharacteristic
599C22#575×C8311yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C2—21no—
614C2—21no—
615C2—41no—
616C2—41no—
617C2—82no—
618C2C83×#300711yescharacteristic
619C2C83×#302011yescharacteristic
620C2C83×#302611yescharacteristic
subgroupG/H
center, socleC22×C83C2×#120
commutator, Frattini, cosocleC22×C4C24×C83
prime radicalC83×#3492C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#3196C2semidirect
3. C83×#2627C2semidirect
4. C83×#1617C2semidirect
5. C83×#1616C2semidirect
6. C83×#1615C2semidirect
7. C83×#1605C2semidirect
8. C83×#1604C2semidirect
9. C83×#1603C2semidirect
10. C83×#1513C2semidirect
11. C83×#1463C2semidirect
302. #18349C83direct (N × H)
312. #3196C2×C83semidirect
313. #2627C2×C83semidirect
314. #1617C2×C83semidirect
315. #1616C2×C83semidirect
316. #1615C2×C83semidirect
317. #1605C2×C83semidirect
318. #1604C2×C83semidirect
319. #1603C2×C83semidirect
320. #1513C2×C83semidirect
321. #1463C2×C83semidirect
322. C83#18349direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#34922
C83×#31962
C83×#26272
C83×#16172
C83×#16162
C83×#16152
C83×#16052
C83×#16042
C83×#16032
C83×#15132
C83×#14632
#1834983