Finite Group Databasebeta

← C83×#18437 #35246261

Subgroups — C83×#18437 #35246261

Order 21248.

1028 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3558C211yesmaximal, characteristic
2C83×#3239C211yesmaximal, characteristic, complemented
3C83×#3227C211yesmaximal, characteristic, complemented
4C83×#2658C211yesmaximal, characteristic, complemented
5C83×#1618C214yesmaximal, complemented
6C83×#1613C214yesmaximal, complemented
7C83×#1505C211yesmaximal, characteristic, complemented
8C83×#1500C211yesmaximal, characteristic, complemented
9C83×#1469C211yesmaximal, characteristic, complemented
10#562×C83—21no—
11H(3,𝔽4)×C83—21no—
12#559×C83—21no—
13#548×C83—21no—
14#546×C83—21no—
15#546×C83—22no—
16D42×C83—21no—
17#542×C83—21no—
18#538×C83—22no—
19#538×C83C2212yes—
20#535×C83—21no—
21#534×C83C2212yes—
22#532×C83C2211yescharacteristic
23C2×C4⋊Q8×C83—21no—
24C2×C4⋊Q8×C83C2211yescharacteristic, complemented
25#529×C83C2211yescharacteristic
26C2×#124×C83—21no—
27C2×#124×C83C2211yescharacteristic
28#501×C83—21no—
29#498×C83—21no—
30#489×C83—21no—
31#486×C83—21no—
32#484×C83C2212yescomplemented
33#481×C83C2212yes—
34#481×C83C2214yescomplemented
35#476×C83—22no—
36#475×C83—21no—
37#474×C83—21no—
38#432×C83C2211yescharacteristic, complemented
39#431×C83C2211yescharacteristic, complemented
40#399×C83—22no—
41#399×C83C2214yescomplemented
42#376×C83C2211yescharacteristic, complemented
43#328×C83C2212yescomplemented
44#327×C83C2212yes—
45#327×C83C2214yescomplemented
46#323×C83C2214yescomplemented
47C2×#41×C83—21no—
48C2×#41×C83—21no—
49C22×Q8×C83—21no—
50C22×Q8×C83D411yescharacteristic
51C22×D4×C83—21no—
52C22×D4×C83—22no—
53C22×D4×C83D411yescharacteristic
54C4⋊Q8×C83—21no—
55C4⋊Q8×C83—21no—
56C4⋊Q8×C83—21no—
57C4⋊Q8×C83—21no—
58C4⋊Q8×C83—44no—
59C4⋊Q8×C83D411yescharacteristic
60#127×C83—21no—
61#127×C83D411yescharacteristic
62#126×C83—22no—
63#126×C83—21no—
64#125×C83D411yescharacteristic
65#124×C83—21no—
66#124×C83—22no—
67#124×C83—22no—
68#124×C83—22no—
69#124×C83—21no—
70#124×C83—21no—
71#124×C83—21no—
72#124×C83—21no—
73#124×C83—21no—
74#124×C83—44no—
75#124×C83D411yescharacteristic
76#123×C83—22no—
77#123×C83—21no—
78#123×C83—22no—
79#123×C83—22no—
80#123×C83—22no—
81C22⋊Q8×C83—22no—
82C22⋊Q8×C83—21no—
83C22⋊Q8×C83—22no—
84C22⋊Q8×C83C2311yescharacteristic
85#121×C83—22no—
86#121×C83—22no—
87#121×C83—22no—
88#121×C83—21no—
89#121×C83—22no—
90#121×C83—21no—
91#121×C83—22no—
92#121×C83C2311yescharacteristic
93#121×C83C2312yes—
94#120×C83—22no—
95#120×C83—22no—
96#120×C83—22no—
97#120×C83—22no—
98#120×C83—22no—
99C4×Q8×C83—21no—
100C4×Q8×C83D411yescharacteristic
101C4×D4×C83—22no—
102C4×D4×C83—21no—
103C4×D4×C83—21no—
104C4×D4×C83—22no—
105C4×D4×C83—21no—
106C4×D4×C83D411yescharacteristic
107C4×D4×C83D412yes—
108#117×C83—21no—
109C2×C4⋊C4×C83—24no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83C2312yes—
112C2×C22⋊C4×C83—22no—
113C2×C22⋊C4×C83—24no—
114C2×C22⋊C4×C83D412yes—
115C2×C22⋊C4×C83C2312yes—
116C2×C42×C83—21no—
117C2×C42×C83C2311yescharacteristic
118#107×C83—22no—
119#106×C83—22no—
120#106×C83—24no—
121C4⋊C8×C83—21no—
122C4⋊C8×C83—21no—
123#103×C83—22no—
124#102×C83—22no—
125#102×C83—24no—
126C22⋊C8×C83C2312yes—
127C22⋊C8×C83C2312yes—
128#97×C83—21no—
129C4×C8×C83—21no—
130#95×C83—21no—
131#95×C83C2312yes—
132C24×C83—22no—
133C24×C83C2×D412yes—
134#41×C83—82no—
135#41×C83—82no—
136C2×Q8×C83—21no—
137C2×Q8×C83—21no—
138C2×Q8×C83—21no—
139C2×Q8×C83—42no—
140C2×Q8×C83—42no—
141C2×Q8×C83—42no—
142C2×Q8×C83—41no—
143C2×Q8×C83—41no—
144C2×Q8×C83C2×D411yescharacteristic
145C2×D4×C83—22no—
146C2×D4×C83—21no—
147C2×D4×C83—22no—
148C2×D4×C83—22no—
149C2×D4×C83—22no—
150C2×D4×C83—22no—
151C2×D4×C83—22no—
152C2×D4×C83—21no—
153C2×D4×C83—21no—
154C2×D4×C83—21no—
155C2×D4×C83—21no—
156C2×D4×C83—42no—
157C2×D4×C83—82no—
158C2×D4×C83—84no—
159C2×D4×C83C2×D411yescharacteristic
160C2×D4×C83C2×D412yes—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—21no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D412yes—
170C22×C4×C83C2411yescharacteristic
171C2×C8×C83—22no—
172C2×C8×C83—22no—
173C4⋊C4×C83—22no—
174C4⋊C4×C83—22no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—22no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83—48no—
183C4⋊C4×C83—41no—
184C4⋊C4×C83—41no—
185C4⋊C4×C83C2×D412yes—
186C4⋊C4×C83C2×D411yescharacteristic
187C4⋊C4×C83C2×D411yescharacteristic
188C22⋊C4×C83—22no—
189C22⋊C4×C83—22no—
190C22⋊C4×C83—22no—
191C22⋊C4×C83—22no—
192C22⋊C4×C83—22no—
193C22⋊C4×C83—21no—
194C22⋊C4×C83—21no—
195C22⋊C4×C83—22no—
196C22⋊C4×C83—22no—
197C22⋊C4×C83—21no—
198C22⋊C4×C83—21no—
199C22⋊C4×C83—22no—
200C22⋊C4×C83—22no—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—21no—
203C22⋊C4×C83—22no—
204C22⋊C4×C83—48no—
205C42×C83—24no—
206C42×C83—22no—
207C42×C83—21no—
208C42×C83C2×D411yescharacteristic
209C42×C83C2×D411yescharacteristic
210C42×C83C2×D411yescharacteristic
211C42×C83C2×D411yescharacteristic
212C23×C83—22no—
213C23×C83—21no—
214C23×C83—22no—
215C23×C83—24no—
216C23×C83—42no—
217C23×C83—42no—
218C23×C83—82no—
219C23×C83—82no—
220C23×C83—82no—
221C23×C83C22×D411yescharacteristic
222Q8×C83—41no—
223Q8×C83—42no—
224Q8×C83—81no—
225Q8×C83—81no—
226Q8×C83—82no—
227Q8×C83—82no—
228D4×C83—42no—
229D4×C83—41no—
230D4×C83—42no—
231D4×C83—44no—
232D4×C83—42no—
233D4×C83—42no—
234D4×C83—81no—
235D4×C83—81no—
236D4×C83—82no—
237D4×C83—82no—
238D4×C83—82no—
239D4×C83—82no—
240D4×C83—82no—
241C2×C4×C83—22no—
242C2×C4×C83—22no—
243C2×C4×C83—24no—
244C2×C4×C83—21no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—41no—
250C2×C4×C83—42no—
251C2×C4×C83—42no—
252C2×C4×C83—42no—
253C2×C4×C83—42no—
254C2×C4×C83—42no—
255C2×C4×C83—42no—
256C2×C4×C83—42no—
257C2×C4×C83—82no—
258C2×C4×C83—82no—
259C2×C4×C83—82no—
260C2×C4×C83#12011yescharacteristic
261C2×C4×C83#12011yescharacteristic
262C2×C4×C83#12011yescharacteristic
263C2×C4×C83#12011yescharacteristic
264C2×C4×C83C22×D411yescharacteristic
265C2×C4×C83C22×D411yescharacteristic
266C8×C83—42no—
267C8×C83—42no—
268C22×C83—21no—
269C22×C83—21no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—41no—
278C22×C83—42no—
279C22×C83—84no—
280C22×C83—164no—
281C22×C83C2×#12011yescharacteristic
282C22×C83#57712yes—
283C4×C83—21no—
284C4×C83—21no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—21no—
288C4×C83—44no—
289C4×C83—41no—
290C4×C83—42no—
291C4×C83—41no—
292C4×C83—42no—
293C4×C83—41no—
294C4×C83—82no—
295C4×C83#45312yes—
296#18437C8311yesmaximal, characteristic, direct product factor, Sylow
297C2×C83—22no—
298C2×C83—82no—
299C2×C83—82no—
300C2×C83—81no—
301C2×C83#300911yescharacteristic
302C2×C83C2×#45311yescharacteristic
303C2×C83#302811yescharacteristic
304#3558C2×C8311yescharacteristic
305#3239C2×C8311yescharacteristic, complemented
306#3227C2×C8311yescharacteristic, complemented
307#2658C2×C8311yescharacteristic, complemented
308#1618C2×C8314yescomplemented
309#1613C2×C8314yescomplemented
310#1505C2×C8311yescharacteristic, complemented
311#1500C2×C8311yescharacteristic, complemented
312#1469C2×C8311yescharacteristic, complemented
313C83#1843711yescharacteristic, direct product factor, Sylow
314#562—21no—
315H(3,𝔽4)—21no—
316#559—21no—
317#548—21no—
318#546—21no—
319#546—22no—
320D42—21no—
321#542—21no—
322#538—22no—
323#538C22×C8312yes—
324#535—21no—
325#534C22×C8312yes—
326#532C22×C8311yescharacteristic
327C2×C4⋊Q8—21no—
328C2×C4⋊Q8C22×C8311yescharacteristic, complemented
329#529C22×C8311yescharacteristic
330C2×#124—21no—
331C2×#124C22×C8311yescharacteristic
332#501—21no—
333#498—21no—
334#489—21no—
335#486—21no—
336#484C22×C8312yescomplemented
337#481C22×C8312yes—
338#481C22×C8314yescomplemented
339#476—22no—
340#475—21no—
341#474—21no—
342#432C22×C8311yescharacteristic, complemented
343#431C22×C8311yescharacteristic, complemented
344#399—22no—
345#399C22×C8314yescomplemented
346#376C22×C8311yescharacteristic, complemented
347#328C22×C8312yescomplemented
348#327C22×C8312yes—
349#327C22×C8314yescomplemented
350#323C22×C8314yescomplemented
351C2×#41—21no—
352C2×#41—21no—
353C22×Q8—21no—
354C22×Q8D4×C8311yescharacteristic
355C22×D4—21no—
356C22×D4—22no—
357C22×D4D4×C8311yescharacteristic
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—44no—
363C4⋊Q8D4×C8311yescharacteristic
364#127—21no—
365#127D4×C8311yescharacteristic
366#126—22no—
367#126—21no—
368#125D4×C8311yescharacteristic
369#124—21no—
370#124—22no—
371#124—22no—
372#124—22no—
373#124—21no—
374#124—21no—
375#124—21no—
376#124—21no—
377#124—21no—
378#124—44no—
379#124D4×C8311yescharacteristic
380#123—22no—
381#123—21no—
382#123—22no—
383#123—22no—
384#123—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8C23×C8311yescharacteristic
389#121—22no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—22no—
394#121—21no—
395#121—22no—
396#121C23×C8311yescharacteristic
397#121C23×C8312yes—
398#120—22no—
399#120—22no—
400#120—22no—
401#120—22no—
402#120—22no—
403C4×Q8—21no—
404C4×Q8D4×C8311yescharacteristic
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—22no—
409C4×D4—21no—
410C4×D4D4×C8311yescharacteristic
411C4×D4D4×C8312yes—
412#117—21no—
413C2×C4⋊C4—24no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4C23×C8312yes—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4D4×C8312yes—
419C2×C22⋊C4C23×C8312yes—
420C2×C42—21no—
421C2×C42C23×C8311yescharacteristic
422#107—22no—
423#106—22no—
424#106—24no—
425C4⋊C8—21no—
426C4⋊C8—21no—
427#103—22no—
428#102—22no—
429#102—24no—
430C22⋊C8C23×C8312yes—
431C22⋊C8C23×C8312yes—
432#97—21no—
433C4×C8—21no—
434#95—21no—
435#95C23×C8312yes—
436C24—22no—
437C24C2×D4×C8312yes—
438#41—82no—
439#41—82no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8C2×D4×C8311yescharacteristic
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—42no—
461C2×D4—82no—
462C2×D4—84no—
463C2×D4C2×D4×C8311yescharacteristic
464C2×D4C2×D4×C8312yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4C2×D4×C8311yescharacteristic
472C22×C4C2×D4×C8311yescharacteristic
473C22×C4C2×D4×C8312yes—
474C22×C4C24×C8311yescharacteristic
475C2×C8—22no—
476C2×C8—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—48no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C8312yes—
490C4⋊C4C2×D4×C8311yescharacteristic
491C4⋊C4C2×D4×C8311yescharacteristic
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—48no—
509C42—24no—
510C42—22no—
511C42—21no—
512C42C2×D4×C8311yescharacteristic
513C42C2×D4×C8311yescharacteristic
514C42C2×D4×C8311yescharacteristic
515C42C2×D4×C8311yescharacteristic
516C23—22no—
517C23—21no—
518C23—22no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23C22×D4×C8311yescharacteristic
526Q8—41no—
527Q8—42no—
528Q8—81no—
529Q8—81no—
530Q8—82no—
531Q8—82no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—44no—
536D4—42no—
537D4—42no—
538D4—81no—
539D4—81no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#120×C8311yescharacteristic
565C2×C4#120×C8311yescharacteristic
566C2×C4#120×C8311yescharacteristic
567C2×C4#120×C8311yescharacteristic
568C2×C4C22×D4×C8311yescharacteristic
569C2×C4C22×D4×C8311yescharacteristic
570C8—42no—
571C8—42no—
572C22—21no—
573C22—21no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—42no—
583C22—84no—
584C22—164no—
585C22C2×#120×C8311yescharacteristic
586C22#577×C8312yes—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—44no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—82no—
599C4#453×C8312yes—
600C2—22no—
601C2—82no—
602C2—81no—
603C2—82no—
604C2C83×#300911yescharacteristic
605C2C2×#453×C8311yescharacteristic
606C2C83×#302811yescharacteristic
subgroupG/H
center, socleC22×C83C2×#120
commutator, Frattini, cosocleC22×C4C24×C83
prime radicalC83×#3558C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#3239C2semidirect
3. C83×#3227C2semidirect
4. C83×#2658C2semidirect
5. C83×#1618C2semidirect
6. C83×#1613C2semidirect
7. C83×#1505C2semidirect
8. C83×#1500C2semidirect
9. C83×#1469C2semidirect
24. C2×C4⋊Q8×C83C22semidirect
32. #484×C83C22semidirect
34. #481×C83C22semidirect
38. #432×C83C22semidirect
39. #431×C83C22semidirect
41. #399×C83C22semidirect
42. #376×C83C22semidirect
43. #328×C83C22semidirect
45. #327×C83C22semidirect
46. #323×C83C22semidirect
296. #18437C83direct (N × H)
305. #3239C2×C83semidirect
306. #3227C2×C83semidirect
307. #2658C2×C83semidirect
308. #1618C2×C83semidirect
309. #1613C2×C83semidirect
310. #1505C2×C83semidirect
311. #1500C2×C83semidirect
312. #1469C2×C83semidirect
313. C83#18437direct (N × H)
328. C2×C4⋊Q8C22×C83semidirect
336. #484C22×C83semidirect
338. #481C22×C83semidirect
342. #432C22×C83semidirect
343. #431C22×C83semidirect
345. #399C22×C83semidirect
346. #376C22×C83semidirect
347. #328C22×C83semidirect
349. #327C22×C83semidirect
350. #323C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35582
C83×#32392
C83×#32272
C83×#26582
C83×#16182
C83×#16132
C83×#15052
C83×#15002
C83×#14692
#1843783