Finite Group Databasebeta

← C83×#61823 #35289647

Subgroups — C83×#61823 #35289647

Order 21248.

1750 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3522C211yesmaximal, characteristic, complemented
2Q8×#41×C83C211yesmaximal, characteristic, complemented
3Q8×Q16×C83C211yesmaximal, characteristic, complemented
4C83×#3381C213yesmaximal, complemented
5C83×#3380C213yesmaximal, complemented
6Q8×SD16×C83C211yesmaximal, characteristic, complemented
7C83×#3365C213yesmaximal, complemented
8C83×#3362C211yesmaximal, characteristic, complemented
9C83×#3361C213yesmaximal, complemented
10C83×#3358C211yesmaximal, characteristic, complemented
11C83×#3233C213yesmaximal, complemented
12C83×#3228C213yesmaximal, complemented
13C83×#3106C213yesmaximal, complemented
14C83×#3102C213yesmaximal, complemented
15C83×#2982C211yesmaximal, characteristic, complemented
16#559×C83C2213yes—
17Q82×C83C2211yescharacteristic
18Q82×C83C2211yescharacteristic
19#557×C83C2213yes—
20#554×C83—23no—
21#554×C83C2213yes—
22#554×C83C2213yes—
23#550×C83C2211yescharacteristic
24D4×Q8×C83—21no—
25D4×Q8×C83C2211yescharacteristic
26#548×C83—21no—
27#547×C83—23no—
28#542×C83—23no—
29#533×C83—23no—
30#533×C83C2213yes—
31#532×C83C2213yes—
32#529×C83C2213yes—
33#520×C83—21no—
34C4×#41×C83C2213yes—
35C2×C4×Q8×C83—21no—
36C2×C4×Q8×C83C2211yescharacteristic
37#500×C83C2213yes—
38#499×C83C2216yes—
39#498×C83C2213yes—
40#488×C83C2216yes—
41#486×C83C2213yes—
42#486×C83C2213yes—
43#485×C83C2213yes—
44#482×C83C2213yes—
45#482×C83C2216yes—
46#479×C83C2213yes—
47#478×C83C2213yes—
48#478×C83C2216yes—
49#478×C83C2216yes—
50#477×C83C2213yes—
51#477×C83C2213yes—
52#476×C83C2213yes—
53#476×C83C2213yes—
54#475×C83C2213yes—
55#474×C83C2213yes—
56#464×C83C2213yes—
57#463×C83C2213yes—
58#460×C83C2213yes—
59#459×C83C2213yes—
60#452×C83C2211yescharacteristic
61#449×C83C2213yes—
62C8×Q8×C83C2211yescharacteristic
63C8×Q8×C83C2213yes—
64C8×Q8×C83C2211yescharacteristic
65C8×Q8×C83C2213yes—
66C4×Q16×C83C2213yes—
67C4×SD16×C83C2213yes—
68C4×SD16×C83C2213yes—
69C4×SD16×C83C2216yes—
70C4×D8×C83C2213yes—
71C4×D8×C83C2216yes—
72#431×C83C2213yes—
73#431×C83C2213yes—
74C2×#41×C83—21no—
75C2×#41×C83—23no—
76C2×#41×C83C2311yescharacteristic
77C22×Q8×C83—21no—
78C22×Q8×C83D411yescharacteristic
79C2×Q16×C83C2311yescharacteristic
80C2×SD16×C83C2311yescharacteristic
81C2×SD16×C83C2313yes—
82C2×D8×C83C2313yes—
83C4⋊Q8×C83—23no—
84C4⋊Q8×C83—23no—
85C4⋊Q8×C83—23no—
86C4⋊Q8×C83C2313yes—
87C4⋊Q8×C83C2313yes—
88C4⋊Q8×C83C2313yes—
89#127×C83C2313yes—
90#126×C83—23no—
91#125×C83—26no—
92#125×C83C2316yes—
93#125×C83C2313yes—
94#124×C83—23no—
95#124×C83—23no—
96#124×C83C2313yes—
97#123×C83—23no—
98#123×C83—26no—
99C22⋊Q8×C83—23no—
100C22⋊Q8×C83—23no—
101C22⋊Q8×C83—23no—
102C22⋊Q8×C83—23no—
103C22⋊Q8×C83—23no—
104C22⋊Q8×C83—26no—
105C22⋊Q8×C83—26no—
106C22⋊Q8×C83C2311yescharacteristic
107#121×C83—23no—
108#121×C83—23no—
109#121×C83C2313yes—
110C4×Q8×C83—21no—
111C4×Q8×C83—21no—
112C4×Q8×C83—23no—
113C4×Q8×C83—23no—
114C4×Q8×C83—23no—
115C4×Q8×C83D411yescharacteristic
116C4×Q8×C83D413yes—
117C4×Q8×C83C2313yes—
118C4×Q8×C83C2313yes—
119C4×Q8×C83C2313yes—
120C4×Q8×C83C2311yescharacteristic
121C4×Q8×C83C2313yes—
122C4×D4×C83—23no—
123C4×D4×C83—23no—
124C4×D4×C83—23no—
125C4×D4×C83—23no—
126C4×D4×C83—23no—
127C4×D4×C83C2313yes—
128C4×D4×C83C2313yes—
129C4×D4×C83C2313yes—
130C4×D4×C83C2316yes—
131#117×C83—23no—
132#117×C83—23no—
133#117×C83C2313yes—
134C2×C4⋊C4×C83—23no—
135C2×C4⋊C4×C83—23no—
136C2×C4⋊C4×C83D413yes—
137C2×C42×C83—23no—
138C2×C42×C83C2313yes—
139#107×C83C2313yes—
140#107×C83C2316yes—
141#107×C83C2313yes—
142#106×C83C2316yes—
143#106×C83C2313yes—
144#106×C83C2313yes—
145C4⋊C8×C83C2313yes—
146C4⋊C8×C83C2313yes—
147C4⋊C8×C83C2313yes—
148C4⋊C8×C83C2313yes—
149#103×C83C2313yes—
150#103×C83C2313yes—
151#103×C83C2313yes—
152#103×C83C2311yescharacteristic
153#102×C83C2313yes—
154#102×C83C2313yes—
155#102×C83C2316yes—
156#102×C83C2313yes—
157#102×C83C2316yes—
158#102×C83C2311yescharacteristic
159C22⋊C8×C83C2311yescharacteristic
160C22⋊C8×C83C2313yes—
161C4×C8×C83C2313yes—
162C4×C8×C83C2313yes—
163C4×C8×C83C2316yes—
164#41×C83—22no—
165#41×C83—22no—
166#41×C83—44no—
167#41×C83—86no—
168C2×Q8×C83—24no—
169C2×Q8×C83—23no—
170C2×Q8×C83—23no—
171C2×Q8×C83—23no—
172C2×Q8×C83—44no—
173C2×Q8×C83C2×D413yes—
174C2×Q8×C83C2×D411yescharacteristic
175C2×Q8×C83C2411yescharacteristic
176C2×Q8×C83C2411yescharacteristic
177C2×D4×C83—21no—
178C2×D4×C83—23no—
179C2×D4×C83—23no—
180C2×D4×C83—23no—
181C2×D4×C83C2413yes—
182C2×D4×C83C2411yescharacteristic
183C22×C4×C83—21no—
184C22×C4×C83—23no—
185C22×C4×C83—23no—
186C22×C4×C83—23no—
187C22×C4×C83—26no—
188C22×C4×C83C2×D413yes—
189C22×C4×C83C2×D413yes—
190C22×C4×C83C2411yescharacteristic
191Q16×C83—22no—
192SD16×C83—26no—
193SD16×C83—22no—
194D8×C83—26no—
195C2×C8×C83C2411yescharacteristic
196C2×C8×C83C2411yescharacteristic
197C2×C8×C83C2413yes—
198C2×C8×C83C2413yes—
199C4⋊C4×C83—23no—
200C4⋊C4×C83—23no—
201C4⋊C4×C83—26no—
202C4⋊C4×C83—23no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—23no—
205C4⋊C4×C83—23no—
206C4⋊C4×C83—23no—
207C4⋊C4×C83—23no—
208C4⋊C4×C83—26no—
209C4⋊C4×C83—23no—
210C4⋊C4×C83C2×D413yes—
211C4⋊C4×C83C2×D413yes—
212C4⋊C4×C83C2×D416yes—
213C4⋊C4×C83C2413yes—
214C4⋊C4×C83C2413yes—
215C4⋊C4×C83C2411yescharacteristic
216C4⋊C4×C83C2413yes—
217C22⋊C4×C83—23no—
218C22⋊C4×C83—21no—
219C22⋊C4×C83—23no—
220C22⋊C4×C83—23no—
221C22⋊C4×C83—23no—
222C22⋊C4×C83—23no—
223C22⋊C4×C83—23no—
224C22⋊C4×C83—23no—
225C22⋊C4×C83—26no—
226C42×C83—23no—
227C42×C83—23no—
228C42×C83—23no—
229C42×C83—23no—
230C42×C83C2×D413yes—
231C42×C83C2×D413yes—
232C42×C83C2413yes—
233C42×C83C2413yes—
234C23×C83—21no—
235C23×C83—23no—
236C23×C83C22×D411yescharacteristic
237Q8×C83—26no—
238Q8×C83—21no—
239Q8×C83—21no—
240Q8×C83—21no—
241Q8×C83—46no—
242Q8×C83—83no—
243Q8×C83—83no—
244Q8×C83#13514yes—
245Q8×C832−1+412yes—
246D4×C83—24no—
247D4×C83—21no—
248D4×C83—23no—
249D4×C83—23no—
250D4×C83—46no—
251D4×C83—46no—
252D4×C83—83no—
253D4×C832−1+412yes—
254C2×C4×C83—24no—
255C2×C4×C83—21no—
256C2×C4×C83—26no—
257C2×C4×C83—21no—
258C2×C4×C83—21no—
259C2×C4×C83—21no—
260C2×C4×C83—23no—
261C2×C4×C83—23no—
262C2×C4×C83—23no—
263C2×C4×C83—21no—
264C2×C4×C83—43no—
265C2×C4×C83—46no—
266C2×C4×C83—46no—
267C2×C4×C83—83no—
268C2×C4×C83—86no—
269C2×C4×C83C22×D411yescharacteristic
270C2×C4×C83C22×D413yes—
271C2×C4×C83C22×D413yes—
272C2×C4×C83C22×D413yes—
273C2×C4×C83C22×D413yes—
274C2×C4×C83C2511yescharacteristic
275C8×C83—23no—
276C8×C83—23no—
277C8×C83—21no—
278C8×C83—21no—
279C22×C83—21no—
280C22×C83—21no—
281C22×C83—22no—
282C22×C83—21no—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—43no—
286C22×C83—43no—
287C22×C83—43no—
288C22×C83C23×D411yescharacteristic
289C4×C83—23no—
290C4×C83—23no—
291C4×C83—21no—
292C4×C83—22no—
293C4×C83—23no—
294C4×C83—43no—
295C4×C83—43no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83—43no—
299C4×C83C2×#13516yes—
300C4×C83C2×2−1+411yescharacteristic
301C4×C83C2×2−1+411yescharacteristic
302#61823C8311yesmaximal, characteristic, direct product factor, Sylow
303C2×C83—41no—
304C2×C83—42no—
305C2×C83—83no—
306C2×C83#348611yescharacteristic
307C2×C83C22×#13511yescharacteristic
308C2×C83#358511yescharacteristic
309#3522C2×C8311yescharacteristic, complemented
310Q8×#41C2×C8311yescharacteristic, complemented
311Q8×Q16C2×C8311yescharacteristic, complemented
312#3381C2×C8313yescomplemented
313#3380C2×C8313yescomplemented
314Q8×SD16C2×C8311yescharacteristic, complemented
315#3365C2×C8313yescomplemented
316#3362C2×C8311yescharacteristic, complemented
317#3361C2×C8313yescomplemented
318#3358C2×C8311yescharacteristic, complemented
319#3233C2×C8313yescomplemented
320#3228C2×C8313yescomplemented
321#3106C2×C8313yescomplemented
322#3102C2×C8313yescomplemented
323#2982C2×C8311yescharacteristic, complemented
324C83#6182311yescharacteristic, direct product factor, Sylow
325#559C22×C8313yes—
326Q82C22×C8311yescharacteristic
327Q82C22×C8311yescharacteristic
328#557C22×C8313yes—
329#554—23no—
330#554C22×C8313yes—
331#554C22×C8313yes—
332#550C22×C8311yescharacteristic
333D4×Q8—21no—
334D4×Q8C22×C8311yescharacteristic
335#548—21no—
336#547—23no—
337#542—23no—
338#533—23no—
339#533C22×C8313yes—
340#532C22×C8313yes—
341#529C22×C8313yes—
342#520—21no—
343C4×#41C22×C8313yes—
344C2×C4×Q8—21no—
345C2×C4×Q8C22×C8311yescharacteristic
346#500C22×C8313yes—
347#499C22×C8316yes—
348#498C22×C8313yes—
349#488C22×C8316yes—
350#486C22×C8313yes—
351#486C22×C8313yes—
352#485C22×C8313yes—
353#482C22×C8313yes—
354#482C22×C8316yes—
355#479C22×C8313yes—
356#478C22×C8313yes—
357#478C22×C8316yes—
358#478C22×C8316yes—
359#477C22×C8313yes—
360#477C22×C8313yes—
361#476C22×C8313yes—
362#476C22×C8313yes—
363#475C22×C8313yes—
364#474C22×C8313yes—
365#464C22×C8313yes—
366#463C22×C8313yes—
367#460C22×C8313yes—
368#459C22×C8313yes—
369#452C22×C8311yescharacteristic
370#449C22×C8313yes—
371C8×Q8C22×C8311yescharacteristic
372C8×Q8C22×C8313yes—
373C8×Q8C22×C8311yescharacteristic
374C8×Q8C22×C8313yes—
375C4×Q16C22×C8313yes—
376C4×SD16C22×C8313yes—
377C4×SD16C22×C8313yes—
378C4×SD16C22×C8316yes—
379C4×D8C22×C8313yes—
380C4×D8C22×C8316yes—
381#431C22×C8313yes—
382#431C22×C8313yes—
383C2×#41—21no—
384C2×#41—23no—
385C2×#41C23×C8311yescharacteristic
386C22×Q8—21no—
387C22×Q8D4×C8311yescharacteristic
388C2×Q16C23×C8311yescharacteristic
389C2×SD16C23×C8313yes—
390C2×SD16C23×C8311yescharacteristic
391C2×D8C23×C8313yes—
392C4⋊Q8—23no—
393C4⋊Q8—23no—
394C4⋊Q8—23no—
395C4⋊Q8C23×C8313yes—
396C4⋊Q8C23×C8313yes—
397C4⋊Q8C23×C8313yes—
398#127C23×C8313yes—
399#126—23no—
400#125—26no—
401#125C23×C8313yes—
402#125C23×C8316yes—
403#124—23no—
404#124—23no—
405#124C23×C8313yes—
406#123—23no—
407#123—26no—
408C22⋊Q8—23no—
409C22⋊Q8—23no—
410C22⋊Q8—23no—
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—26no—
414C22⋊Q8—26no—
415C22⋊Q8C23×C8311yescharacteristic
416#121—23no—
417#121—23no—
418#121C23×C8313yes—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—23no—
422C4×Q8—23no—
423C4×Q8—23no—
424C4×Q8D4×C8311yescharacteristic
425C4×Q8D4×C8313yes—
426C4×Q8C23×C8313yes—
427C4×Q8C23×C8313yes—
428C4×Q8C23×C8313yes—
429C4×Q8C23×C8311yescharacteristic
430C4×Q8C23×C8313yes—
431C4×D4—23no—
432C4×D4—23no—
433C4×D4—23no—
434C4×D4—23no—
435C4×D4—23no—
436C4×D4C23×C8313yes—
437C4×D4C23×C8313yes—
438C4×D4C23×C8313yes—
439C4×D4C23×C8316yes—
440#117—23no—
441#117—23no—
442#117C23×C8313yes—
443C2×C4⋊C4—23no—
444C2×C4⋊C4—23no—
445C2×C4⋊C4D4×C8313yes—
446C2×C42—23no—
447C2×C42C23×C8313yes—
448#107C23×C8313yes—
449#107C23×C8316yes—
450#107C23×C8313yes—
451#106C23×C8316yes—
452#106C23×C8313yes—
453#106C23×C8313yes—
454C4⋊C8C23×C8313yes—
455C4⋊C8C23×C8313yes—
456C4⋊C8C23×C8313yes—
457C4⋊C8C23×C8313yes—
458#103C23×C8311yescharacteristic
459#103C23×C8313yes—
460#103C23×C8313yes—
461#103C23×C8313yes—
462#102C23×C8311yescharacteristic
463#102C23×C8313yes—
464#102C23×C8316yes—
465#102C23×C8313yes—
466#102C23×C8316yes—
467#102C23×C8313yes—
468C22⋊C8C23×C8313yes—
469C22⋊C8C23×C8311yescharacteristic
470C4×C8C23×C8316yes—
471C4×C8C23×C8313yes—
472C4×C8C23×C8313yes—
473#41—22no—
474#41—22no—
475#41—44no—
476#41—86no—
477C2×Q8—24no—
478C2×Q8—23no—
479C2×Q8—23no—
480C2×Q8—23no—
481C2×Q8—44no—
482C2×Q8C2×D4×C8313yes—
483C2×Q8C2×D4×C8311yescharacteristic
484C2×Q8C24×C8311yescharacteristic
485C2×Q8C24×C8311yescharacteristic
486C2×D4—21no—
487C2×D4—23no—
488C2×D4—23no—
489C2×D4—23no—
490C2×D4C24×C8313yes—
491C2×D4C24×C8311yescharacteristic
492C22×C4—21no—
493C22×C4—23no—
494C22×C4—23no—
495C22×C4—23no—
496C22×C4—26no—
497C22×C4C2×D4×C8313yes—
498C22×C4C2×D4×C8313yes—
499C22×C4C24×C8311yescharacteristic
500Q16—22no—
501SD16—26no—
502SD16—22no—
503D8—26no—
504C2×C8C24×C8311yescharacteristic
505C2×C8C24×C8313yes—
506C2×C8C24×C8311yescharacteristic
507C2×C8C24×C8313yes—
508C4⋊C4—23no—
509C4⋊C4—23no—
510C4⋊C4—26no—
511C4⋊C4—23no—
512C4⋊C4—21no—
513C4⋊C4—23no—
514C4⋊C4—23no—
515C4⋊C4—23no—
516C4⋊C4—23no—
517C4⋊C4—26no—
518C4⋊C4—23no—
519C4⋊C4C2×D4×C8313yes—
520C4⋊C4C2×D4×C8313yes—
521C4⋊C4C2×D4×C8316yes—
522C4⋊C4C24×C8313yes—
523C4⋊C4C24×C8313yes—
524C4⋊C4C24×C8311yescharacteristic
525C4⋊C4C24×C8313yes—
526C22⋊C4—23no—
527C22⋊C4—21no—
528C22⋊C4—23no—
529C22⋊C4—23no—
530C22⋊C4—23no—
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—23no—
534C22⋊C4—26no—
535C42—23no—
536C42—23no—
537C42—23no—
538C42—23no—
539C42C2×D4×C8313yes—
540C42C2×D4×C8313yes—
541C42C24×C8313yes—
542C42C24×C8313yes—
543C23—21no—
544C23—23no—
545C23C22×D4×C8311yescharacteristic
546Q8—26no—
547Q8—21no—
548Q8—21no—
549Q8—21no—
550Q8—46no—
551Q8—83no—
552Q8—83no—
553Q8#135×C8314yes—
554Q82−1+4×C8312yes—
555D4—24no—
556D4—21no—
557D4—23no—
558D4—23no—
559D4—46no—
560D4—46no—
561D4—83no—
562D42−1+4×C8312yes—
563C2×C4—24no—
564C2×C4—21no—
565C2×C4—26no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—21no—
573C2×C4—43no—
574C2×C4—46no—
575C2×C4—46no—
576C2×C4—86no—
577C2×C4—83no—
578C2×C4C22×D4×C8311yescharacteristic
579C2×C4C22×D4×C8313yes—
580C2×C4C22×D4×C8313yes—
581C2×C4C22×D4×C8313yes—
582C2×C4C22×D4×C8313yes—
583C2×C4C25×C8311yescharacteristic
584C8—23no—
585C8—23no—
586C8—21no—
587C8—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—43no—
595C22—43no—
596C22—43no—
597C22C23×D4×C8311yescharacteristic
598C4—23no—
599C4—23no—
600C4—23no—
601C4—21no—
602C4—22no—
603C4—41no—
604C4—43no—
605C4—43no—
606C4—43no—
607C4—41no—
608C4C2×#135×C8316yes—
609C4C2×2−1+4×C8311yescharacteristic
610C4C2×2−1+4×C8311yescharacteristic
611C2—41no—
612C2—42no—
613C2—83no—
614C2C83×#348611yescharacteristic
615C2C22×#135×C8311yescharacteristic
616C2C83×#358511yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×Q8×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3522C2semidirect
2. Q8×#41×C83C2semidirect
3. Q8×Q16×C83C2semidirect
4. C83×#3381C2semidirect
5. C83×#3380C2semidirect
6. Q8×SD16×C83C2semidirect
7. C83×#3365C2semidirect
8. C83×#3362C2semidirect
9. C83×#3361C2semidirect
10. C83×#3358C2semidirect
11. C83×#3233C2semidirect
12. C83×#3228C2semidirect
13. C83×#3106C2semidirect
14. C83×#3102C2semidirect
15. C83×#2982C2semidirect
302. #61823C83direct (N × H)
309. #3522C2×C83semidirect
310. Q8×#41C2×C83semidirect
311. Q8×Q16C2×C83semidirect
312. #3381C2×C83semidirect
313. #3380C2×C83semidirect
314. Q8×SD16C2×C83semidirect
315. #3365C2×C83semidirect
316. #3362C2×C83semidirect
317. #3361C2×C83semidirect
318. #3358C2×C83semidirect
319. #3233C2×C83semidirect
320. #3228C2×C83semidirect
321. #3106C2×C83semidirect
322. #3102C2×C83semidirect
323. #2982C2×C83semidirect
324. C83#61823direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35222
Q8×#41×C832
Q8×Q16×C832
C83×#33812
C83×#33802
Q8×SD16×C832
C83×#33652
C83×#33622
C83×#33612
C83×#33582
C83×#32332
C83×#32282
C83×#31062
C83×#31022
C83×#29822
#6182383