Finite Group Databasebeta

← C83×#8438 #35236262

Subgroups — C83×#8438 #35236262

Order 21248.

792 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3475C211yesmaximal, characteristic
2C83×#1806C211yesmaximal, characteristic, complemented
3C4×#102×C83C211yesmaximal, characteristic, complemented
4C83×#1503C211yesmaximal, characteristic, complemented
5C83×#1493C211yesmaximal, characteristic, complemented
6C83×#1486C211yesmaximal, characteristic, complemented
7C83×#1476C211yesmaximal, characteristic, complemented
8#556×C83C411yescharacteristic, complemented
9#555×C83C411yescharacteristic, complemented
10#554×C83C411yescharacteristic, complemented
11#553×C83—21no—
12#552×C83—21no—
13#552×C83—21no—
14#551×C83—22no—
15#547×C83C411yescharacteristic, complemented
16#546×C83—21no—
17#533×C83C2211yescharacteristic
18#529×C83—22no—
19#529×C83—21no—
20#525×C83—21no—
21C4×#41×C83—21no—
22C4×#41×C83C2211yescharacteristic
23C2×C4×D4×C83C2211yescharacteristic
24C2×#117×C83—21no—
25#431×C83C2211yescharacteristic
26#423×C83C2211yescharacteristic
27C2×#104×C83—21no—
28C2×#102×C83—21no—
29C2×C4×C8×C83C2211yescharacteristic
30C4×C4⋊C4×C83C2211yescharacteristic
31#340×C83—21no—
32#339×C83—21no—
33#328×C83—22no—
34Q8⋊C8×C83—22no—
35Q8⋊C8×C83—21no—
36D4⋊C8×C83—21no—
37D4⋊C8×C83—21no—
38D4⋊C8×C83—21no—
39C2×#41×C83—21no—
40C2×#41×C83D411yescharacteristic
41C22×D4×C83D411yescharacteristic
42C23×C4×C83—21no—
43C2×M16×C83—21no—
44C22×C8×C83—21no—
45C4⋊Q8×C83C2×C411yescharacteristic
46C4⋊Q8×C83C2×C411yescharacteristic
47#126×C83—22no—
48#126×C83—22no—
49#126×C83—22no—
50#125×C83—22no—
51#125×C83—21no—
52#125×C83—21no—
53#124×C83—22no—
54#124×C83—21no—
55#123×C83—22no—
56#123×C83—22no—
57#123×C83—22no—
58#123×C83—22no—
59C22⋊Q8×C83—21no—
60C22⋊Q8×C83—21no—
61C22⋊Q8×C83—22no—
62C22⋊Q8×C83—21no—
63C22⋊Q8×C83—21no—
64C22⋊Q8×C83D412yes—
65#121×C83—22no—
66#121×C83—21no—
67#121×C83—21no—
68#121×C83—21no—
69#120×C83—22no—
70C4×Q8×C83—21no—
71C4×Q8×C83—21no—
72C4×Q8×C83C2×C411yescharacteristic
73C4×Q8×C83D411yescharacteristic
74C4×Q8×C83D412yes—
75C4×D4×C83—22no—
76C4×D4×C83—22no—
77C4×D4×C83—22no—
78C4×D4×C83—21no—
79C4×D4×C83—21no—
80C4×D4×C83—21no—
81C4×D4×C83—21no—
82C4×D4×C83—21no—
83C4×D4×C83C2×C411yescharacteristic
84C4×D4×C83C2×C411yescharacteristic
85C4×D4×C83C2×C411yescharacteristic
86C4×D4×C83D411yescharacteristic
87C4×D4×C83D411yescharacteristic
88C4×D4×C83D411yescharacteristic
89#117×C83—22no—
90#117×C83—22no—
91#117×C83—22no—
92#117×C83—21no—
93#117×C83—21no—
94#117×C83—21no—
95#117×C83—21no—
96#117×C83—21no—
97#117×C83D411yescharacteristic
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83D411yescharacteristic
102C2×C22⋊C4×C83—22no—
103C2×C22⋊C4×C83—21no—
104C2×C42×C83—22no—
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#102×C83—24no—
113C22⋊C8×C83—21no—
114C4×C8×C83—21no—
115C4×C8×C83—21no—
116C4×C8×C83—21no—
117#95×C83—21no—
118C24×C83—21no—
119#41×C83—21no—
120#41×C83—21no—
121#41×C83—21no—
122#41×C83—21no—
123#41×C83—42no—
124#41×C83—42no—
125C2×Q8×C83—21no—
126C2×Q8×C83C22⋊C411yescharacteristic
127C2×Q8×C83C22⋊C412yes—
128C2×D4×C83—22no—
129C2×D4×C83—22no—
130C2×D4×C83—21no—
131C2×D4×C83—22no—
132C2×D4×C83—21no—
133C2×D4×C83—41no—
134C2×D4×C83C22⋊C411yescharacteristic
135C2×D4×C83C22⋊C411yescharacteristic
136C2×D4×C83C22⋊C411yescharacteristic
137C22×C4×C83—22no—
138C22×C4×C83—21no—
139C22×C4×C83—22no—
140C22×C4×C83—21no—
141C22×C4×C83—21no—
142C22×C4×C83—21no—
143C22×C4×C83—21no—
144C22×C4×C83—21no—
145C22×C4×C83—21no—
146C22×C4×C83—21no—
147C22×C4×C83—41no—
148C22×C4×C83—41no—
149C22×C4×C83—41no—
150C22×C4×C83—41no—
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—22no—
172C4⋊C4×C83—21no—
173C4⋊C4×C83—22no—
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—22no—
189C22⋊C4×C83—21no—
190C22⋊C4×C83—21no—
191C22⋊C4×C83—21no—
192C22⋊C4×C83—21no—
193C22⋊C4×C83—21no—
194C22⋊C4×C83—21no—
195C42×C83—22no—
196C42×C83—22no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—22no—
201C42×C83—41no—
202C42×C83—41no—
203C42×C83C22×C411yescharacteristic
204C42×C83C2×D411yescharacteristic
205C42×C83C2×D411yescharacteristic
206C42×C83C2×D411yescharacteristic
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—42no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83—81no—
214C23×C83#12011yescharacteristic
215Q8×C83—21no—
216Q8×C83—21no—
217Q8×C83—42no—
218Q8×C83—41no—
219Q8×C83—41no—
220D4×C83—21no—
221D4×C83—21no—
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—21no—
225D4×C83—42no—
226D4×C83—44no—
227D4×C83—41no—
228D4×C83—41no—
229D4×C83—82no—
230D4×C83#10412yescomplemented
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—22no—
237C2×C4×C83—21no—
238C2×C4×C83—22no—
239C2×C4×C83—21no—
240C2×C4×C83—41no—
241C2×C4×C83—44no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—42no—
257C2×C4×C83—81no—
258C2×C4×C83—82no—
259C2×C4×C83—81no—
260C2×C4×C83—81no—
261C2×C4×C83C2×C22⋊C411yescharacteristic
262C2×C4×C83C2×C22⋊C411yescharacteristic
263C2×C4×C83C2×C22⋊C411yescharacteristic
264C2×C4×C83#12011yescharacteristic
265C2×C4×C83#12011yescharacteristic
266C2×C4×C83#12011yescharacteristic
267C2×C4×C83#13511yescharacteristic
268C2×C4×C83#13511yescharacteristic
269C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
270C2×C4×C83#13711yescharacteristic
271C8×C83—42no—
272C8×C83—81no—
273C8×C83—81no—
274C22×C83—21no—
275C22×C83—21no—
276C22×C83—22no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—41no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—82no—
286C22×C83—82no—
287C22×C83#37911yescharacteristic
288C22×C83#44911yescharacteristic
289C22×C83#45211yescharacteristic
290C4×C83—21no—
291C4×C83—21no—
292C4×C83—21no—
293C4×C83—41no—
294C4×C83—42no—
295C4×C83—41no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83—41no—
299C4×C83—42no—
300C4×C83—82no—
301C4×C83#41611yescharacteristic
302C4×C83#41711yescharacteristic
303C4×C83C2×#10411yescharacteristic
304C4×C83#42111yescharacteristic
305C4×C83#44911yescharacteristic
306C4×C83#45211yescharacteristic
307#8438C8311yesmaximal, characteristic, direct product factor, Sylow
308C2×C83—21no—
309C2×C83—21no—
310C2×C83—42no—
311C2×C83—81no—
312C2×C83—81no—
313C2×C83#178811yescharacteristic
314C2×C83#178911yescharacteristic
315C2×C83#179411yescharacteristic
316#3475C2×C8311yescharacteristic
317#1806C2×C8311yescharacteristic, complemented
318C4×#102C2×C8311yescharacteristic, complemented
319#1503C2×C8311yescharacteristic, complemented
320#1493C2×C8311yescharacteristic, complemented
321#1486C2×C8311yescharacteristic, complemented
322#1476C2×C8311yescharacteristic, complemented
323C83#843811yescharacteristic, direct product factor, Sylow
324#556C4×C8311yescharacteristic, complemented
325#555C4×C8311yescharacteristic, complemented
326#554C4×C8311yescharacteristic, complemented
327#553—21no—
328#552—21no—
329#552—21no—
330#551—22no—
331#547C4×C8311yescharacteristic, complemented
332#546—21no—
333#533C22×C8311yescharacteristic
334#529—22no—
335#529—21no—
336#525—21no—
337C4×#41—21no—
338C4×#41C22×C8311yescharacteristic
339C2×C4×D4C22×C8311yescharacteristic
340C2×#117—21no—
341#431C22×C8311yescharacteristic
342#423C22×C8311yescharacteristic
343C2×#104—21no—
344C2×#102—21no—
345C2×C4×C8C22×C8311yescharacteristic
346C4×C4⋊C4C22×C8311yescharacteristic
347#340—21no—
348#339—21no—
349#328—22no—
350Q8⋊C8—21no—
351Q8⋊C8—22no—
352D4⋊C8—21no—
353D4⋊C8—21no—
354D4⋊C8—21no—
355C2×#41—21no—
356C2×#41D4×C8311yescharacteristic
357C22×D4D4×C8311yescharacteristic
358C23×C4—21no—
359C2×M16—21no—
360C22×C8—21no—
361C4⋊Q8C2×C4×C8311yescharacteristic
362C4⋊Q8C2×C4×C8311yescharacteristic
363#126—22no—
364#126—22no—
365#126—22no—
366#125—22no—
367#125—21no—
368#125—21no—
369#124—22no—
370#124—21no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8D4×C8312yes—
381#121—22no—
382#121—21no—
383#121—21no—
384#121—21no—
385#120—22no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8C2×C4×C8311yescharacteristic
389C4×Q8D4×C8311yescharacteristic
390C4×Q8D4×C8312yes—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C2×C4×C8311yescharacteristic
400C4×D4C2×C4×C8311yescharacteristic
401C4×D4C2×C4×C8311yescharacteristic
402C4×D4D4×C8311yescharacteristic
403C4×D4D4×C8311yescharacteristic
404C4×D4D4×C8311yescharacteristic
405#117—22no—
406#117—22no—
407#117—22no—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—21no—
413#117D4×C8311yescharacteristic
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4D4×C8311yescharacteristic
418C2×C22⋊C4—22no—
419C2×C22⋊C4—21no—
420C2×C42—22no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42C23×C8311yescharacteristic
424C4⋊C8—21no—
425C4⋊C8—21no—
426C4⋊C8—21no—
427#104—42no—
428#102—24no—
429C22⋊C8—21no—
430C4×C8—21no—
431C4×C8—21no—
432C4×C8—21no—
433#95—21no—
434C24—21no—
435#41—21no—
436#41—21no—
437#41—21no—
438#41—21no—
439#41—42no—
440#41—42no—
441C2×Q8—21no—
442C2×Q8C22⋊C4×C8311yescharacteristic
443C2×Q8C22⋊C4×C8312yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—41no—
450C2×D4C22⋊C4×C8311yescharacteristic
451C2×D4C22⋊C4×C8311yescharacteristic
452C2×D4C22⋊C4×C8311yescharacteristic
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4C2×D4×C8311yescharacteristic
469C22×C4C2×D4×C8311yescharacteristic
470C22×C4C2×D4×C8311yescharacteristic
471M16—41no—
472M16—41no—
473C2×C8—22no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—41no—
492C4⋊C4C22⋊C4×C8311yescharacteristic
493C4⋊C4C22⋊C4×C8311yescharacteristic
494C4⋊C4C22⋊C4×C8311yescharacteristic
495C4⋊C4C22⋊C4×C8311yescharacteristic
496C4⋊C4C22⋊C4×C8312yes—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—22no—
516C42—21no—
517C42—41no—
518C42—41no—
519C42C22×C4×C8311yescharacteristic
520C42C2×D4×C8311yescharacteristic
521C42C2×D4×C8311yescharacteristic
522C42C2×D4×C8311yescharacteristic
523C23—21no—
524C23—21no—
525C23—21no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—81no—
530C23#120×C8311yescharacteristic
531Q8—21no—
532Q8—21no—
533Q8—42no—
534Q8—41no—
535Q8—41no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—42no—
542D4—44no—
543D4—41no—
544D4—41no—
545D4—82no—
546D4#104×C8312yescomplemented
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
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—41no—
572C2×C4—42no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4C2×C22⋊C4×C8311yescharacteristic
578C2×C4C2×C22⋊C4×C8311yescharacteristic
579C2×C4C2×C22⋊C4×C8311yescharacteristic
580C2×C4#120×C8311yescharacteristic
581C2×C4#120×C8311yescharacteristic
582C2×C4#120×C8311yescharacteristic
583C2×C4#135×C8311yescharacteristic
584C2×C4#135×C8311yescharacteristic
585C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
586C2×C4#137×C8311yescharacteristic
587C8—42no—
588C8—81no—
589C8—81no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—82no—
602C22—82no—
603C22#379×C8311yescharacteristic
604C22#449×C8311yescharacteristic
605C22#452×C8311yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—82no—
617C4#416×C8311yescharacteristic
618C4#417×C8311yescharacteristic
619C4C2×#104×C8311yescharacteristic
620C4#421×C8311yescharacteristic
621C4#449×C8311yescharacteristic
622C4#452×C8311yescharacteristic
623C2—21no—
624C2—21no—
625C2—42no—
626C2—81no—
627C2—81no—
628C2C83×#178811yescharacteristic
629C2C83×#178911yescharacteristic
630C2C83×#179411yescharacteristic
subgroupG/H
centerC2×C4×C83#120
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
socleC22×C83#379
prime radicalC83×#3475C2
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×#102×C83C2semidirect
4. C83×#1503C2semidirect
5. C83×#1493C2semidirect
6. C83×#1486C2semidirect
7. C83×#1476C2semidirect
8. #556×C83C4semidirect
9. #555×C83C4semidirect
10. #554×C83C4semidirect
15. #547×C83C4semidirect
230. D4×C83#104semidirect
307. #8438C83direct (N × H)
317. #1806C2×C83semidirect
318. C4×#102C2×C83semidirect
319. #1503C2×C83semidirect
320. #1493C2×C83semidirect
321. #1486C2×C83semidirect
322. #1476C2×C83semidirect
323. C83#8438direct (N × H)
324. #556C4×C83semidirect
325. #555C4×C83semidirect
326. #554C4×C83semidirect
331. #547C4×C83semidirect
546. D4#104×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#34752
C83×#18062
C4×#102×C832
C83×#15032
C83×#14932
C83×#14862
C83×#14762
#843883