Finite Group Databasebeta

← C83×#61846 #35289670

Subgroups — C83×#61846 #35289670

Order 21248.

1664 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3502C211yesmaximal, characteristic, complemented
2Q8×#41×C83C211yesmaximal, characteristic
3C83×#3387C213yesmaximal, complemented
4C83×#3385C213yesmaximal, complemented
5Q8×Q16×C83C211yesmaximal, characteristic, complemented
6Q8×SD16×C83C211yesmaximal, characteristic, complemented
7C83×#3374C213yesmaximal, complemented
8C83×#3372C213yesmaximal, complemented
9C83×#3363C211yesmaximal, characteristic, complemented
10C83×#3359C211yesmaximal, characteristic, complemented
11C83×#3237C213yesmaximal, complemented
12C83×#3231C213yesmaximal, complemented
13C83×#3116C213yesmaximal
14C83×#3107C213yesmaximal
15C83×#2984C211yesmaximal, characteristic, complemented
16Q82×C83C2211yescharacteristic
17Q82×C83C2211yescharacteristic
18Q82×C83C2213yes—
19#557×C83—21no—
20#557×C83—23no—
21#557×C83C2211yescharacteristic
22#557×C83C2213yes—
23#557×C83C2213yes—
24#554×C83—23no—
25#554×C83C2213yes—
26D4×Q8×C83—21no—
27D4×Q8×C83C2211yescharacteristic
28#544×C83—23no—
29#533×C83—23no—
30#533×C83C2213yes—
31#533×C83C2213yes—
32#533×C83C2213yes—
33#519×C83—21no—
34C4×#41×C83C2213yes—
35C2×C4×Q8×C83—21no—
36C2×C4×Q8×C83C2211yescharacteristic
37#501×C83C2216yes—
38#500×C83C2213yes—
39#498×C83C2213yes—
40#491×C83C2216yes—
41#487×C83C2213yes—
42#487×C83C2213yes—
43#485×C83C2213yes—
44#485×C83C2213yes—
45#485×C83C2216yes—
46#479×C83C2216yes—
47#479×C83C2216yes—
48#477×C83C2213yes—
49#477×C83C2213yes—
50#477×C83C2213yes—
51#476×C83C2213yes—
52#475×C83C2213yes—
53#475×C83C2213yes—
54#475×C83C2213yes—
55#474×C83C2213yes—
56#462×C83C2213yes—
57#462×C83C2213yes—
58#461×C83C2213yes—
59#461×C83C2213yes—
60#452×C83C2211yescharacteristic
61#452×C83C2213yes—
62#446×C83C2213yes—
63#446×C83C2213yes—
64C8×Q8×C83C2211yescharacteristic
65C8×Q8×C83C2211yescharacteristic
66#441×C83C2216yes—
67#440×C83C2216yes—
68C4×Q16×C83C2213yes—
69C4×Q16×C83C2213yes—
70C4×SD16×C83C2213yes—
71C4×SD16×C83C2213yes—
72#432×C83C2213yes—
73#431×C83C2213yes—
74C2×#41×C83C2311yescharacteristic
75C22×Q8×C83—21no—
76C22×Q8×C83D411yescharacteristic
77C2×Q16×C83C2311yescharacteristic
78C2×Q16×C83C2313yes—
79C2×SD16×C83C2311yescharacteristic
80C2×SD16×C83C2313yes—
81C4⋊Q8×C83—23no—
82C4⋊Q8×C83—23no—
83C4⋊Q8×C83—23no—
84C4⋊Q8×C83—26no—
85C4⋊Q8×C83—23no—
86C4⋊Q8×C83C2313yes—
87C4⋊Q8×C83C2313yes—
88C4⋊Q8×C83C2313yes—
89C4⋊Q8×C83C2313yes—
90C4⋊Q8×C83C2313yes—
91C4⋊Q8×C83C2313yes—
92#125×C83—26no—
93#125×C83—23no—
94#125×C83—26no—
95#125×C83—23no—
96#125×C83—23no—
97#125×C83—23no—
98#125×C83C2316yes—
99C22⋊Q8×C83—23no—
100C22⋊Q8×C83—23no—
101C22⋊Q8×C83—23no—
102C22⋊Q8×C83—23no—
103C22⋊Q8×C83—23no—
104C22⋊Q8×C83—23no—
105C22⋊Q8×C83—26no—
106C22⋊Q8×C83C2313yes—
107C22⋊Q8×C83C2311yescharacteristic
108C4×Q8×C83—21no—
109C4×Q8×C83—23no—
110C4×Q8×C83—21no—
111C4×Q8×C83—23no—
112C4×Q8×C83—21no—
113C4×Q8×C83—23no—
114C4×Q8×C83—23no—
115C4×Q8×C83—23no—
116C4×Q8×C83—23no—
117C4×Q8×C83—23no—
118C4×Q8×C83D411yescharacteristic
119C4×Q8×C83D413yes—
120C4×Q8×C83C2313yes—
121C4×Q8×C83C2313yes—
122C4×Q8×C83C2313yes—
123C4×Q8×C83C2313yes—
124C4×Q8×C83C2316yes—
125C4×Q8×C83C2311yescharacteristic
126C4×Q8×C83C2313yes—
127C4×D4×C83—23no—
128C4×D4×C83—23no—
129C4×D4×C83C2313yes—
130C4×D4×C83C2313yes—
131#117×C83—23no—
132#117×C83—23no—
133#117×C83—23no—
134#117×C83C2313yes—
135C2×C4⋊C4×C83—23no—
136C2×C4⋊C4×C83D413yes—
137C2×C42×C83—23no—
138C2×C42×C83C2313yes—
139#107×C83C2313yes—
140#107×C83C2313yes—
141#107×C83C2316yes—
142#106×C83C2313yes—
143#106×C83C2316yes—
144#106×C83C2313yes—
145C4⋊C8×C83C2313yes—
146C4⋊C8×C83C2313yes—
147C4⋊C8×C83C2313yes—
148C4⋊C8×C83C2313yes—
149#103×C83C2311yescharacteristic
150#103×C83C2313yes—
151#103×C83C2313yes—
152#103×C83C2316yes—
153#103×C83C2313yes—
154#103×C83C2316yes—
155#103×C83C2313yes—
156#103×C83C2313yes—
157#102×C83C2311yescharacteristic
158#102×C83C2313yes—
159C22⋊C8×C83C2311yescharacteristic
160C22⋊C8×C83C2313yes—
161#97×C83C2316yes—
162C4×C8×C83C2313yes—
163C4×C8×C83C2313yes—
164#41×C83—22no—
165#41×C83—22no—
166C2×Q8×C83—24no—
167C2×Q8×C83—23no—
168C2×Q8×C83—23no—
169C2×Q8×C83—23no—
170C2×Q8×C83—23no—
171C2×Q8×C83—44no—
172C2×Q8×C83C2×D413yes—
173C2×Q8×C83C2×D411yescharacteristic
174C2×Q8×C83C2411yescharacteristic
175C2×Q8×C83C2413yes—
176C2×Q8×C83C2411yescharacteristic
177C2×D4×C83—21no—
178C2×D4×C83C2411yescharacteristic
179C22×C4×C83—21no—
180C22×C4×C83—23no—
181C22×C4×C83—23no—
182C22×C4×C83C2×D413yes—
183C22×C4×C83C2×D413yes—
184C22×C4×C83C2411yescharacteristic
185Q16×C83—22no—
186Q16×C83—26no—
187SD16×C83—22no—
188SD16×C83—26no—
189C2×C8×C83C2411yescharacteristic
190C2×C8×C83C2411yescharacteristic
191C2×C8×C83C2413yes—
192C2×C8×C83C2413yes—
193C4⋊C4×C83—26no—
194C4⋊C4×C83—23no—
195C4⋊C4×C83—23no—
196C4⋊C4×C83—23no—
197C4⋊C4×C83—23no—
198C4⋊C4×C83—23no—
199C4⋊C4×C83—23no—
200C4⋊C4×C83—23no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—23no—
203C4⋊C4×C83—23no—
204C4⋊C4×C83—26no—
205C4⋊C4×C83—23no—
206C4⋊C4×C83—23no—
207C4⋊C4×C83—26no—
208C4⋊C4×C83—23no—
209C4⋊C4×C83—23no—
210C4⋊C4×C83C2×D413yes—
211C4⋊C4×C83C2×D416yes—
212C4⋊C4×C83C2×D413yes—
213C4⋊C4×C83C2411yescharacteristic
214C4⋊C4×C83C2413yes—
215C4⋊C4×C83C2413yes—
216C4⋊C4×C83C2413yes—
217C22⋊C4×C83—21no—
218C22⋊C4×C83—23no—
219C22⋊C4×C83—23no—
220C22⋊C4×C83—23no—
221C22⋊C4×C83—23no—
222C42×C83—23no—
223C42×C83—23no—
224C42×C83—23no—
225C42×C83—26no—
226C42×C83—23no—
227C42×C83—23no—
228C42×C83C2×D413yes—
229C42×C83C2×D413yes—
230C42×C83C2413yes—
231C42×C83C2413yes—
232C23×C83—21no—
233C23×C83C22×D411yescharacteristic
234Q8×C83—26no—
235Q8×C83—21no—
236Q8×C83—21no—
237Q8×C83—23no—
238Q8×C83—21no—
239Q8×C83—23no—
240Q8×C83—46no—
241Q8×C83—83no—
242Q8×C83—83no—
243Q8×C83—83no—
244Q8×C83#13714yes—
245Q8×C832−1+412yes—
246D4×C83—24no—
247D4×C83—21no—
248D4×C832−1+412yes—
249C2×C4×C83—24no—
250C2×C4×C83—21no—
251C2×C4×C83—26no—
252C2×C4×C83—21no—
253C2×C4×C83—21no—
254C2×C4×C83—23no—
255C2×C4×C83—21no—
256C2×C4×C83—23no—
257C2×C4×C83—21no—
258C2×C4×C83—23no—
259C2×C4×C83—23no—
260C2×C4×C83—43no—
261C2×C4×C83—46no—
262C2×C4×C83—83no—
263C2×C4×C83C22×D411yescharacteristic
264C2×C4×C83C22×D413yes—
265C2×C4×C83C22×D413yes—
266C2×C4×C83C22×D413yes—
267C2×C4×C83C22×D413yes—
268C2×C4×C83C2511yescharacteristic
269C8×C83—21no—
270C8×C83—23no—
271C8×C83—21no—
272C8×C83—23no—
273C22×C83—21no—
274C22×C83—21no—
275C22×C83—22no—
276C22×C83—21no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83C23×D411yescharacteristic
280C4×C83—23no—
281C4×C83—23no—
282C4×C83—21no—
283C4×C83—22no—
284C4×C83—23no—
285C4×C83—43no—
286C4×C83—41no—
287C4×C83—43no—
288C4×C83—43no—
289C4×C83—41no—
290C4×C83—43no—
291C4×C83C2×#13716yes—
292C4×C83C2×2−1+411yescharacteristic
293C4×C83C2×2−1+411yescharacteristic
294#61846C8311yesmaximal, characteristic, direct product factor, Sylow
295C2×C83—41no—
296C2×C83—42no—
297C2×C83#348611yescharacteristic
298C2×C83C22×#13711yescharacteristic
299C2×C83#358411yescharacteristic
300#3502C2×C8311yescharacteristic, complemented
301Q8×#41C2×C8311yescharacteristic
302#3387C2×C8313yescomplemented
303#3385C2×C8313yescomplemented
304Q8×Q16C2×C8311yescharacteristic, complemented
305Q8×SD16C2×C8311yescharacteristic, complemented
306#3374C2×C8313yescomplemented
307#3372C2×C8313yescomplemented
308#3363C2×C8311yescharacteristic, complemented
309#3359C2×C8311yescharacteristic, complemented
310#3237C2×C8313yescomplemented
311#3231C2×C8313yescomplemented
312#3116C2×C8313yes—
313#3107C2×C8313yes—
314#2984C2×C8311yescharacteristic, complemented
315C83#6184611yescharacteristic, direct product factor, Sylow
316Q82C22×C8311yescharacteristic
317Q82C22×C8311yescharacteristic
318Q82C22×C8313yes—
319#557—21no—
320#557—23no—
321#557C22×C8311yescharacteristic
322#557C22×C8313yes—
323#557C22×C8313yes—
324#554—23no—
325#554C22×C8313yes—
326D4×Q8—21no—
327D4×Q8C22×C8311yescharacteristic
328#544—23no—
329#533—23no—
330#533C22×C8313yes—
331#533C22×C8313yes—
332#533C22×C8313yes—
333#519—21no—
334C4×#41C22×C8313yes—
335C2×C4×Q8—21no—
336C2×C4×Q8C22×C8311yescharacteristic
337#501C22×C8316yes—
338#500C22×C8313yes—
339#498C22×C8313yes—
340#491C22×C8316yes—
341#487C22×C8313yes—
342#487C22×C8313yes—
343#485C22×C8313yes—
344#485C22×C8313yes—
345#485C22×C8316yes—
346#479C22×C8316yes—
347#479C22×C8316yes—
348#477C22×C8313yes—
349#477C22×C8313yes—
350#477C22×C8313yes—
351#476C22×C8313yes—
352#475C22×C8313yes—
353#475C22×C8313yes—
354#475C22×C8313yes—
355#474C22×C8313yes—
356#462C22×C8313yes—
357#462C22×C8313yes—
358#461C22×C8313yes—
359#461C22×C8313yes—
360#452C22×C8311yescharacteristic
361#452C22×C8313yes—
362#446C22×C8313yes—
363#446C22×C8313yes—
364C8×Q8C22×C8311yescharacteristic
365C8×Q8C22×C8311yescharacteristic
366#441C22×C8316yes—
367#440C22×C8316yes—
368C4×Q16C22×C8313yes—
369C4×Q16C22×C8313yes—
370C4×SD16C22×C8313yes—
371C4×SD16C22×C8313yes—
372#432C22×C8313yes—
373#431C22×C8313yes—
374C2×#41C23×C8311yescharacteristic
375C22×Q8—21no—
376C22×Q8D4×C8311yescharacteristic
377C2×Q16C23×C8311yescharacteristic
378C2×Q16C23×C8313yes—
379C2×SD16C23×C8311yescharacteristic
380C2×SD16C23×C8313yes—
381C4⋊Q8—23no—
382C4⋊Q8—23no—
383C4⋊Q8—23no—
384C4⋊Q8—26no—
385C4⋊Q8—23no—
386C4⋊Q8C23×C8313yes—
387C4⋊Q8C23×C8313yes—
388C4⋊Q8C23×C8313yes—
389C4⋊Q8C23×C8313yes—
390C4⋊Q8C23×C8313yes—
391C4⋊Q8C23×C8313yes—
392#125—26no—
393#125—23no—
394#125—26no—
395#125—23no—
396#125—23no—
397#125—23no—
398#125C23×C8316yes—
399C22⋊Q8—23no—
400C22⋊Q8—23no—
401C22⋊Q8—23no—
402C22⋊Q8—23no—
403C22⋊Q8—23no—
404C22⋊Q8—23no—
405C22⋊Q8—26no—
406C22⋊Q8C23×C8311yescharacteristic
407C22⋊Q8C23×C8313yes—
408C4×Q8—21no—
409C4×Q8—23no—
410C4×Q8—21no—
411C4×Q8—23no—
412C4×Q8—21no—
413C4×Q8—23no—
414C4×Q8—23no—
415C4×Q8—23no—
416C4×Q8—23no—
417C4×Q8—23no—
418C4×Q8D4×C8311yescharacteristic
419C4×Q8D4×C8313yes—
420C4×Q8C23×C8313yes—
421C4×Q8C23×C8313yes—
422C4×Q8C23×C8313yes—
423C4×Q8C23×C8313yes—
424C4×Q8C23×C8316yes—
425C4×Q8C23×C8311yescharacteristic
426C4×Q8C23×C8313yes—
427C4×D4—23no—
428C4×D4—23no—
429C4×D4C23×C8313yes—
430C4×D4C23×C8313yes—
431#117—23no—
432#117—23no—
433#117—23no—
434#117C23×C8313yes—
435C2×C4⋊C4—23no—
436C2×C4⋊C4D4×C8313yes—
437C2×C42—23no—
438C2×C42C23×C8313yes—
439#107C23×C8313yes—
440#107C23×C8313yes—
441#107C23×C8316yes—
442#106C23×C8313yes—
443#106C23×C8313yes—
444#106C23×C8316yes—
445C4⋊C8C23×C8313yes—
446C4⋊C8C23×C8313yes—
447C4⋊C8C23×C8313yes—
448C4⋊C8C23×C8313yes—
449#103C23×C8313yes—
450#103C23×C8313yes—
451#103C23×C8311yescharacteristic
452#103C23×C8313yes—
453#103C23×C8313yes—
454#103C23×C8313yes—
455#103C23×C8316yes—
456#103C23×C8316yes—
457#102C23×C8311yescharacteristic
458#102C23×C8313yes—
459C22⋊C8C23×C8311yescharacteristic
460C22⋊C8C23×C8313yes—
461#97C23×C8316yes—
462C4×C8C23×C8313yes—
463C4×C8C23×C8313yes—
464#41—22no—
465#41—22no—
466C2×Q8—24no—
467C2×Q8—23no—
468C2×Q8—23no—
469C2×Q8—23no—
470C2×Q8—23no—
471C2×Q8—44no—
472C2×Q8C2×D4×C8313yes—
473C2×Q8C2×D4×C8311yescharacteristic
474C2×Q8C24×C8311yescharacteristic
475C2×Q8C24×C8313yes—
476C2×Q8C24×C8311yescharacteristic
477C2×D4—21no—
478C2×D4C24×C8311yescharacteristic
479C22×C4—21no—
480C22×C4—23no—
481C22×C4—23no—
482C22×C4C2×D4×C8313yes—
483C22×C4C2×D4×C8313yes—
484C22×C4C24×C8311yescharacteristic
485Q16—22no—
486Q16—26no—
487SD16—22no—
488SD16—26no—
489C2×C8C24×C8311yescharacteristic
490C2×C8C24×C8311yescharacteristic
491C2×C8C24×C8313yes—
492C2×C8C24×C8313yes—
493C4⋊C4—26no—
494C4⋊C4—23no—
495C4⋊C4—23no—
496C4⋊C4—23no—
497C4⋊C4—23no—
498C4⋊C4—23no—
499C4⋊C4—23no—
500C4⋊C4—23no—
501C4⋊C4—21no—
502C4⋊C4—23no—
503C4⋊C4—23no—
504C4⋊C4—26no—
505C4⋊C4—23no—
506C4⋊C4—23no—
507C4⋊C4—26no—
508C4⋊C4—23no—
509C4⋊C4—23no—
510C4⋊C4C2×D4×C8313yes—
511C4⋊C4C2×D4×C8316yes—
512C4⋊C4C2×D4×C8313yes—
513C4⋊C4C24×C8311yescharacteristic
514C4⋊C4C24×C8313yes—
515C4⋊C4C24×C8313yes—
516C4⋊C4C24×C8313yes—
517C22⋊C4—21no—
518C22⋊C4—23no—
519C22⋊C4—23no—
520C22⋊C4—23no—
521C22⋊C4—23no—
522C42—23no—
523C42—23no—
524C42—23no—
525C42—26no—
526C42—23no—
527C42—23no—
528C42C2×D4×C8313yes—
529C42C2×D4×C8313yes—
530C42C24×C8313yes—
531C42C24×C8313yes—
532C23—21no—
533C23C22×D4×C8311yescharacteristic
534Q8—26no—
535Q8—21no—
536Q8—21no—
537Q8—23no—
538Q8—21no—
539Q8—23no—
540Q8—46no—
541Q8—83no—
542Q8—83no—
543Q8—83no—
544Q8#137×C8314yes—
545Q82−1+4×C8312yes—
546D4—24no—
547D4—21no—
548D42−1+4×C8312yes—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—26no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—23no—
555C2×C4—21no—
556C2×C4—23no—
557C2×C4—21no—
558C2×C4—23no—
559C2×C4—23no—
560C2×C4—43no—
561C2×C4—46no—
562C2×C4—83no—
563C2×C4C22×D4×C8311yescharacteristic
564C2×C4C22×D4×C8313yes—
565C2×C4C22×D4×C8313yes—
566C2×C4C22×D4×C8313yes—
567C2×C4C22×D4×C8313yes—
568C2×C4C25×C8311yescharacteristic
569C8—21no—
570C8—23no—
571C8—21no—
572C8—23no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22C23×D4×C8311yescharacteristic
580C4—23no—
581C4—23no—
582C4—23no—
583C4—21no—
584C4—22no—
585C4—43no—
586C4—43no—
587C4—41no—
588C4—43no—
589C4—43no—
590C4—41no—
591C4C2×#137×C8316yes—
592C4C2×2−1+4×C8311yescharacteristic
593C4C2×2−1+4×C8311yescharacteristic
594C2—41no—
595C2—42no—
596C2C83×#348611yescharacteristic
597C2C22×#137×C8311yescharacteristic
598C2C83×#358411yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC2×#41×C83C23
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×#3502C2semidirect
3. C83×#3387C2semidirect
4. C83×#3385C2semidirect
5. Q8×Q16×C83C2semidirect
6. Q8×SD16×C83C2semidirect
7. C83×#3374C2semidirect
8. C83×#3372C2semidirect
9. C83×#3363C2semidirect
10. C83×#3359C2semidirect
11. C83×#3237C2semidirect
12. C83×#3231C2semidirect
15. C83×#2984C2semidirect
294. #61846C83direct (N × H)
300. #3502C2×C83semidirect
302. #3387C2×C83semidirect
303. #3385C2×C83semidirect
304. Q8×Q16C2×C83semidirect
305. Q8×SD16C2×C83semidirect
306. #3374C2×C83semidirect
307. #3372C2×C83semidirect
308. #3363C2×C83semidirect
309. #3359C2×C83semidirect
310. #3237C2×C83semidirect
311. #3231C2×C83semidirect
314. #2984C2×C83semidirect
315. C83#61846direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35022
Q8×#41×C832
C83×#33872
C83×#33852
Q8×Q16×C832
Q8×SD16×C832
C83×#33742
C83×#33722
C83×#33632
C83×#33592
C83×#32372
C83×#32312
C83×#31162
C83×#31072
C83×#29842
#6184683