Finite Group Databasebeta

← C83×#8437 #35236261

Subgroups — C83×#8437 #35236261

Order 21248.

782 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3480C211yesmaximal, characteristic
2C83×#1806C211yesmaximal, characteristic, complemented
3C4×#103×C83C211yesmaximal, characteristic, complemented
4C83×#1502C211yesmaximal, characteristic, complemented
5C83×#1492C211yesmaximal, characteristic, complemented
6C83×#1487C211yesmaximal, characteristic, complemented
7C83×#1476C211yesmaximal, characteristic, complemented
8#559×C83—21no—
9#559×C83C411yescharacteristic, complemented
10#557×C83C411yescharacteristic, complemented
11#556×C83—21no—
12#556×C83C411yescharacteristic, complemented
13#553×C83—22no—
14#553×C83—21no—
15#550×C83C411yescharacteristic, complemented
16#548×C83—21no—
17#532×C83—21no—
18#532×C83C2211yescharacteristic
19#529×C83—21no—
20#529×C83—22no—
21C4×#41×C83—21no—
22C4×#41×C83—21no—
23C4×#41×C83C2211yescharacteristic
24C2×C4×Q8×C83C2211yescharacteristic
25#431×C83C2211yescharacteristic
26#423×C83C2211yescharacteristic
27C2×#104×C83—21no—
28C2×#103×C83—21no—
29C2×C4×C8×C83C2211yescharacteristic
30C4×C4⋊C4×C83C2211yescharacteristic
31#340×C83—21no—
32#339×C83—21no—
33#327×C83—22no—
34Q8⋊C8×C83—21no—
35Q8⋊C8×C83—21no—
36Q8⋊C8×C83—21no—
37D4⋊C8×C83—22no—
38D4⋊C8×C83—21no—
39C2×#41×C83—21no—
40C2×#41×C83—21no—
41C2×#41×C83D411yescharacteristic
42C22×Q8×C83D411yescharacteristic
43C2×M16×C83—21no—
44C22×C8×C83—21no—
45C4⋊Q8×C83—21no—
46C4⋊Q8×C83C2×C411yescharacteristic
47#127×C83—21no—
48#127×C83C2×C411yescharacteristic
49#126×C83—22no—
50#126×C83—22no—
51#126×C83—22no—
52#125×C83—22no—
53#125×C83—21no—
54#124×C83—22no—
55#124×C83—21no—
56#124×C83—21no—
57#124×C83—21no—
58#124×C83—21no—
59#123×C83—22no—
60#123×C83—22no—
61#123×C83—22no—
62C22⋊Q8×C83—22no—
63C22⋊Q8×C83—21no—
64#121×C83—22no—
65#121×C83—22no—
66#121×C83—22no—
67#121×C83—21no—
68#121×C83—21no—
69#121×C83D412yes—
70C4×Q8×C83—21no—
71C4×Q8×C83—21no—
72C4×Q8×C83—21no—
73C4×Q8×C83—21no—
74C4×Q8×C83—21no—
75C4×Q8×C83C2×C411yescharacteristic
76C4×Q8×C83C2×C411yescharacteristic
77C4×Q8×C83C2×C411yescharacteristic
78C4×Q8×C83D411yescharacteristic
79C4×Q8×C83D411yescharacteristic
80C4×Q8×C83D411yescharacteristic
81C4×D4×C83—22no—
82C4×D4×C83—22no—
83C4×D4×C83—22no—
84C4×D4×C83—22no—
85C4×D4×C83—22no—
86C4×D4×C83—21no—
87C4×D4×C83—21no—
88C4×D4×C83—21no—
89C4×D4×C83—21no—
90C4×D4×C83—21no—
91C4×D4×C83—21no—
92C4×D4×C83C2×C411yescharacteristic
93C4×D4×C83D411yescharacteristic
94C4×D4×C83D412yes—
95#117×C83—22no—
96#117×C83—22no—
97#117×C83—21no—
98#117×C83—21no—
99#117×C83D411yescharacteristic
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83D411yescharacteristic
103C2×C42×C83—22no—
104C2×C42×C83—21no—
105C2×C42×C83—21no—
106C2×C42×C83—21no—
107C2×C42×C83C2311yescharacteristic
108C4⋊C8×C83—21no—
109C4⋊C8×C83—21no—
110C4⋊C8×C83—21no—
111#104×C83—42no—
112#103×C83—24no—
113C22⋊C8×C83—21no—
114C4×C8×C83—21no—
115C4×C8×C83—21no—
116C4×C8×C83—21no—
117#95×C83—21no—
118#41×C83—21no—
119#41×C83—21no—
120#41×C83—21no—
121#41×C83—21no—
122#41×C83—42no—
123#41×C83—42no—
124#41×C83—41no—
125#41×C83—41no—
126#41×C83—41no—
127#41×C83—41no—
128C2×Q8×C83—21no—
129C2×Q8×C83—21no—
130C2×Q8×C83—22no—
131C2×Q8×C83—41no—
132C2×Q8×C83C22⋊C411yescharacteristic
133C2×Q8×C83C22⋊C411yescharacteristic
134C2×Q8×C83C22⋊C411yescharacteristic
135C2×D4×C83—22no—
136C2×D4×C83—21no—
137C2×D4×C83—22no—
138C2×D4×C83—22no—
139C2×D4×C83—21no—
140C2×D4×C83—21no—
141C2×D4×C83C22⋊C411yescharacteristic
142C2×D4×C83C22⋊C412yes—
143C22×C4×C83—22no—
144C22×C4×C83—21no—
145C22×C4×C83—22no—
146C22×C4×C83—21no—
147C22×C4×C83—21no—
148C22×C4×C83—21no—
149C22×C4×C83—22no—
150C22×C4×C83—21no—
151C22×C4×C83—41no—
152C22×C4×C83C2×D411yescharacteristic
153C22×C4×C83C2×D411yescharacteristic
154C22×C4×C83C2×D411yescharacteristic
155M16×C83—41no—
156M16×C83—41no—
157C2×C8×C83—22no—
158C2×C8×C83—41no—
159C2×C8×C83—41no—
160C2×C8×C83—41no—
161C2×C8×C83—41no—
162C4⋊C4×C83—22no—
163C4⋊C4×C83—22no—
164C4⋊C4×C83—22no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83—21no—
167C4⋊C4×C83—21no—
168C4⋊C4×C83—21no—
169C4⋊C4×C83—21no—
170C4⋊C4×C83—21no—
171C4⋊C4×C83—21no—
172C4⋊C4×C83—21no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—22no—
175C4⋊C4×C83—41no—
176C4⋊C4×C83C22⋊C411yescharacteristic
177C4⋊C4×C83C22⋊C411yescharacteristic
178C4⋊C4×C83C22⋊C411yescharacteristic
179C4⋊C4×C83C22⋊C411yescharacteristic
180C4⋊C4×C83C22⋊C412yes—
181C22⋊C4×C83—22no—
182C22⋊C4×C83—22no—
183C22⋊C4×C83—22no—
184C22⋊C4×C83—22no—
185C22⋊C4×C83—22no—
186C22⋊C4×C83—22no—
187C22⋊C4×C83—21no—
188C22⋊C4×C83—21no—
189C22⋊C4×C83—22no—
190C22⋊C4×C83—21no—
191C22⋊C4×C83—22no—
192C42×C83—22no—
193C42×C83—22no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—41no—
202C42×C83—41no—
203C42×C83C22×C411yescharacteristic
204C42×C83C2×D411yescharacteristic
205C42×C83C2×D411yescharacteristic
206C42×C83C2×D411yescharacteristic
207C23×C83—22no—
208C23×C83—21no—
209C23×C83#12011yescharacteristic
210Q8×C83—21no—
211Q8×C83—21no—
212Q8×C83—21no—
213Q8×C83—21no—
214Q8×C83—21no—
215Q8×C83—42no—
216Q8×C83—41no—
217Q8×C83—41no—
218Q8×C83#10412yescomplemented
219D4×C83—21no—
220D4×C83—21no—
221D4×C83—42no—
222D4×C83—44no—
223D4×C83—41no—
224D4×C83—42no—
225D4×C83—41no—
226D4×C83—42no—
227D4×C83—41no—
228D4×C83—41no—
229D4×C83—41no—
230D4×C83—41no—
231D4×C83—82no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—22no—
241C2×C4×C83—21no—
242C2×C4×C83—41no—
243C2×C4×C83—42no—
244C2×C4×C83—44no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—42no—
249C2×C4×C83—41no—
250C2×C4×C83—42no—
251C2×C4×C83—42no—
252C2×C4×C83—81no—
253C2×C4×C83—81no—
254C2×C4×C83—82no—
255C2×C4×C83—81no—
256C2×C4×C83—81no—
257C2×C4×C83C2×C22⋊C411yescharacteristic
258C2×C4×C83C2×C22⋊C411yescharacteristic
259C2×C4×C83C2×C22⋊C411yescharacteristic
260C2×C4×C83#12011yescharacteristic
261C2×C4×C83#12011yescharacteristic
262C2×C4×C83#12011yescharacteristic
263C2×C4×C83#13511yescharacteristic
264C2×C4×C83#13511yescharacteristic
265C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
266C2×C4×C83#13711yescharacteristic
267C8×C83—42no—
268C8×C83—81no—
269C8×C83—81no—
270C22×C83—21no—
271C22×C83—21no—
272C22×C83—42no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—41no—
278C22×C83#37911yescharacteristic
279C22×C83#44911yescharacteristic
280C22×C83#45211yescharacteristic
281C4×C83—21no—
282C4×C83—21no—
283C4×C83—21no—
284C4×C83—22no—
285C4×C83—41no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—42no—
290C4×C83—41no—
291C4×C83—41no—
292C4×C83—82no—
293C4×C83#41611yescharacteristic
294C4×C83#41711yescharacteristic
295C4×C83C2×#10411yescharacteristic
296C4×C83#42111yescharacteristic
297C4×C83#44911yescharacteristic
298C4×C83#45211yescharacteristic
299#8437C8311yesmaximal, characteristic, direct product factor, Sylow
300C2×C83—21no—
301C2×C83—21no—
302C2×C83—82no—
303C2×C83—81no—
304C2×C83#178811yescharacteristic
305C2×C83#178911yescharacteristic
306C2×C83#179411yescharacteristic
307#3480C2×C8311yescharacteristic
308#1806C2×C8311yescharacteristic, complemented
309C4×#103C2×C8311yescharacteristic, complemented
310#1502C2×C8311yescharacteristic, complemented
311#1492C2×C8311yescharacteristic, complemented
312#1487C2×C8311yescharacteristic, complemented
313#1476C2×C8311yescharacteristic, complemented
314C83#843711yescharacteristic, direct product factor, Sylow
315#559—21no—
316#559C4×C8311yescharacteristic, complemented
317#557C4×C8311yescharacteristic, complemented
318#556—21no—
319#556C4×C8311yescharacteristic, complemented
320#553—22no—
321#553—21no—
322#550C4×C8311yescharacteristic, complemented
323#548—21no—
324#532—21no—
325#532C22×C8311yescharacteristic
326#529—21no—
327#529—22no—
328C4×#41—21no—
329C4×#41—21no—
330C4×#41C22×C8311yescharacteristic
331C2×C4×Q8C22×C8311yescharacteristic
332#431C22×C8311yescharacteristic
333#423C22×C8311yescharacteristic
334C2×#104—21no—
335C2×#103—21no—
336C2×C4×C8C22×C8311yescharacteristic
337C4×C4⋊C4C22×C8311yescharacteristic
338#340—21no—
339#339—21no—
340#327—22no—
341Q8⋊C8—21no—
342Q8⋊C8—21no—
343Q8⋊C8—21no—
344D4⋊C8—21no—
345D4⋊C8—22no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41D4×C8311yescharacteristic
349C22×Q8D4×C8311yescharacteristic
350C2×M16—21no—
351C22×C8—21no—
352C4⋊Q8—21no—
353C4⋊Q8C2×C4×C8311yescharacteristic
354#127—21no—
355#127C2×C4×C8311yescharacteristic
356#126—22no—
357#126—22no—
358#126—22no—
359#125—22no—
360#125—21no—
361#124—22no—
362#124—21no—
363#124—21no—
364#124—21no—
365#124—21no—
366#123—22no—
367#123—22no—
368#123—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—21no—
375#121—21no—
376#121D4×C8312yes—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C2×C4×C8311yescharacteristic
383C4×Q8C2×C4×C8311yescharacteristic
384C4×Q8C2×C4×C8311yescharacteristic
385C4×Q8D4×C8311yescharacteristic
386C4×Q8D4×C8311yescharacteristic
387C4×Q8D4×C8311yescharacteristic
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C2×C4×C8311yescharacteristic
400C4×D4D4×C8311yescharacteristic
401C4×D4D4×C8312yes—
402#117—22no—
403#117—22no—
404#117—21no—
405#117—21no—
406#117D4×C8311yescharacteristic
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4D4×C8311yescharacteristic
410C2×C42—22no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42C23×C8311yescharacteristic
415C4⋊C8—21no—
416C4⋊C8—21no—
417C4⋊C8—21no—
418#104—42no—
419#103—24no—
420C22⋊C8—21no—
421C4×C8—21no—
422C4×C8—21no—
423C4×C8—21no—
424#95—21no—
425#41—21no—
426#41—21no—
427#41—21no—
428#41—21no—
429#41—42no—
430#41—42no—
431#41—41no—
432#41—41no—
433#41—41no—
434#41—41no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8C22⋊C4×C8311yescharacteristic
440C2×Q8C22⋊C4×C8311yescharacteristic
441C2×Q8C22⋊C4×C8311yescharacteristic
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C22⋊C4×C8311yescharacteristic
449C2×D4C22⋊C4×C8312yes—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—41no—
459C22×C4C2×D4×C8311yescharacteristic
460C22×C4C2×D4×C8311yescharacteristic
461C22×C4C2×D4×C8311yescharacteristic
462M16—41no—
463M16—41no—
464C2×C8—22no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—41no—
483C4⋊C4C22⋊C4×C8311yescharacteristic
484C4⋊C4C22⋊C4×C8311yescharacteristic
485C4⋊C4C22⋊C4×C8311yescharacteristic
486C4⋊C4C22⋊C4×C8311yescharacteristic
487C4⋊C4C22⋊C4×C8312yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—41no—
509C42—41no—
510C42C22×C4×C8311yescharacteristic
511C42C2×D4×C8311yescharacteristic
512C42C2×D4×C8311yescharacteristic
513C42C2×D4×C8311yescharacteristic
514C23—21no—
515C23—22no—
516C23#120×C8311yescharacteristic
517Q8—21no—
518Q8—21no—
519Q8—21no—
520Q8—21no—
521Q8—21no—
522Q8—42no—
523Q8—41no—
524Q8—41no—
525Q8#104×C8312yescomplemented
526D4—21no—
527D4—21no—
528D4—42no—
529D4—44no—
530D4—41no—
531D4—42no—
532D4—41no—
533D4—42no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—82no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4C2×C22⋊C4×C8311yescharacteristic
565C2×C4C2×C22⋊C4×C8311yescharacteristic
566C2×C4C2×C22⋊C4×C8311yescharacteristic
567C2×C4#120×C8311yescharacteristic
568C2×C4#120×C8311yescharacteristic
569C2×C4#120×C8311yescharacteristic
570C2×C4#135×C8311yescharacteristic
571C2×C4#135×C8311yescharacteristic
572C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
573C2×C4#137×C8311yescharacteristic
574C8—42no—
575C8—81no—
576C8—81no—
577C22—21no—
578C22—21no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—42no—
584C22—41no—
585C22#379×C8311yescharacteristic
586C22#449×C8311yescharacteristic
587C22#452×C8311yescharacteristic
588C4—21no—
589C4—22no—
590C4—21no—
591C4—21no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—82no—
600C4#416×C8311yescharacteristic
601C4#417×C8311yescharacteristic
602C4C2×#104×C8311yescharacteristic
603C4#421×C8311yescharacteristic
604C4#449×C8311yescharacteristic
605C4#452×C8311yescharacteristic
606C2—21no—
607C2—21no—
608C2—81no—
609C2—82no—
610C2C83×#178811yescharacteristic
611C2C83×#178911yescharacteristic
612C2C83×#179411yescharacteristic
subgroupG/H
centerC2×C4×C83#120
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
socleC22×C83#379
prime radicalC83×#3480C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1806C2semidirect
3. C4×#103×C83C2semidirect
4. C83×#1502C2semidirect
5. C83×#1492C2semidirect
6. C83×#1487C2semidirect
7. C83×#1476C2semidirect
9. #559×C83C4semidirect
10. #557×C83C4semidirect
12. #556×C83C4semidirect
15. #550×C83C4semidirect
218. Q8×C83#104semidirect
299. #8437C83direct (N × H)
308. #1806C2×C83semidirect
309. C4×#103C2×C83semidirect
310. #1502C2×C83semidirect
311. #1492C2×C83semidirect
312. #1487C2×C83semidirect
313. #1476C2×C83semidirect
314. C83#8437direct (N × H)
316. #559C4×C83semidirect
317. #557C4×C83semidirect
319. #556C4×C83semidirect
322. #550C4×C83semidirect
525. Q8#104×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#34802
C83×#18062
C4×#103×C832
C83×#15022
C83×#14922
C83×#14872
C83×#14762
#843783