Finite Group Databasebeta

← C83×#18407 #35246231

Subgroups — C83×#18407 #35246231

Order 21248.

904 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3493C211yesmaximal, characteristic
2C83×#3232C212yesmaximal, complemented
3C83×#2611C211yesmaximal, characteristic, complemented
4C83×#1618C212yesmaximal, complemented
5C83×#1617C211yesmaximal, characteristic, complemented
6C83×#1616C211yesmaximal, characteristic, complemented
7C83×#1612C212yesmaximal, complemented
8C83×#1611C211yesmaximal, characteristic, complemented
9C83×#1610C211yesmaximal, characteristic, complemented
10C83×#1500C212yesmaximal, complemented
11C83×#1458C211yesmaximal, characteristic, complemented
12#556×C83—22no—
13#555×C83—22no—
14#553×C83—22no—
15#552×C83—22no—
16#544×C83C2211yescharacteristic
17#538×C83—21no—
18#538×C83C2212yes—
19#537×C83C2211yescharacteristic
20#529×C83—21no—
21#529×C83C2212yes—
22C2×#125×C83C2211yescharacteristic
23#525×C83—21no—
24#520×C83—21no—
25C2×#117×C83C2211yescharacteristic
26#499×C83—22no—
27#488×C83—22no—
28#484×C83C2212yes—
29#483×C83C2212yes—
30#481×C83C2212yes—
31#480×C83C2212yes—
32#479×C83—22no—
33#478×C83—22no—
34#431×C83C2212yes—
35#400×C83C2211yescharacteristic
36#399×C83C2211yescharacteristic
37#398×C83C2211yescharacteristic
38#397×C83C2211yescharacteristic
39#389×C83—21no—
40#388×C83—21no—
41#387×C83—21no—
42#385×C83—21no—
43#376×C83C2211yescharacteristic
44#328×C83C2212yes—
45#328×C83C2212yes—
46#327×C83C2212yes—
47#327×C83C2212yes—
48#324×C83C2211yescharacteristic
49#324×C83C2211yescharacteristic
50#323×C83C2212yes—
51C2×#41×C83—21no—
52C23×C4×C83D411yescharacteristic
53C4⋊Q8×C83—22no—
54#126×C83—22no—
55#126×C83—22no—
56#126×C83—22no—
57#126×C83—22no—
58#125×C83—21no—
59#125×C83—22no—
60#125×C83—22no—
61#125×C83—22no—
62#125×C83D412yes—
63#124×C83—21no—
64#124×C83—22no—
65#124×C83—22no—
66#124×C83D412yes—
67#123×C83—22no—
68#123×C83—22no—
69#123×C83—22no—
70C22⋊Q8×C83—21no—
71C22⋊Q8×C83—21no—
72C22⋊Q8×C83—21no—
73C22⋊Q8×C83—22no—
74C22⋊Q8×C83C2312yes—
75#121×C83—21no—
76#121×C83—21no—
77#121×C83—22no—
78#121×C83—22no—
79#121×C83—22no—
80#121×C83—21no—
81#121×C83C2312yes—
82#120×C83—22no—
83C4×Q8×C83—22no—
84C4×Q8×C83—21no—
85C4×Q8×C83D412yes—
86C4×D4×C83—22no—
87C4×D4×C83—22no—
88C4×D4×C83—22no—
89C4×D4×C83—21no—
90C4×D4×C83—22no—
91C4×D4×C83—22no—
92C4×D4×C83D412yes—
93#117×C83—22no—
94#117×C83—22no—
95#117×C83—21no—
96#117×C83—22no—
97#117×C83—22no—
98#117×C83—22no—
99C2×C4⋊C4×C83—22no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83D411yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C22⋊C4×C83—21no—
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83D411yescharacteristic
111C2×C22⋊C4×C83C2311yescharacteristic
112C2×C42×C83—21no—
113C2×C42×C83D411yescharacteristic
114C2×C42×C83C2311yescharacteristic
115#107×C83—22no—
116#107×C83—22no—
117#106×C83—22no—
118#106×C83—22no—
119C4⋊C8×C83—22no—
120#103×C83—22no—
121#103×C83—22no—
122#102×C83—22no—
123#102×C83—22no—
124C22⋊C8×C83C2312yes—
125C22⋊C8×C83C2312yes—
126C4×C8×C83—22no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83—21no—
130#95×C83—21no—
131#95×C83—21no—
132#95×C83C2311yescharacteristic
133#95×C83C2311yescharacteristic
134C24×C83C2×D411yescharacteristic
135#41×C83—42no—
136#41×C83—42no—
137C2×Q8×C83—21no—
138C2×Q8×C83—21no—
139C2×Q8×C83C2×D412yes—
140C2×D4×C83—21no—
141C2×D4×C83—22no—
142C2×D4×C83—22no—
143C2×D4×C83—22no—
144C2×D4×C83—21no—
145C2×D4×C83C2×D412yes—
146C22×C4×C83—22no—
147C22×C4×C83—21no—
148C22×C4×C83—22no—
149C22×C4×C83—21no—
150C22×C4×C83—21no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—21no—
154C22×C4×C83—21no—
155C22×C4×C83—21no—
156C22×C4×C83—41no—
157C22×C4×C83C2×D411yescharacteristic
158C22×C4×C83C2×D411yescharacteristic
159C22×C4×C83C2×D411yescharacteristic
160C22×C4×C83C2×D411yescharacteristic
161C22×C4×C83C2×D411yescharacteristic
162C22×C4×C83C2411yescharacteristic
163C2×C8×C83—22no—
164C2×C8×C83—22no—
165C4⋊C4×C83—21no—
166C4⋊C4×C83—22no—
167C4⋊C4×C83—21no—
168C4⋊C4×C83—22no—
169C4⋊C4×C83—21no—
170C4⋊C4×C83—21no—
171C4⋊C4×C83—22no—
172C4⋊C4×C83—22no—
173C4⋊C4×C83—22no—
174C4⋊C4×C83—22no—
175C4⋊C4×C83—22no—
176C4⋊C4×C83—22no—
177C4⋊C4×C83—22no—
178C4⋊C4×C83—22no—
179C4⋊C4×C83—42no—
180C4⋊C4×C83—41no—
181C4⋊C4×C83—41no—
182C4⋊C4×C83—42no—
183C4⋊C4×C83—42no—
184C4⋊C4×C83C2×D412yes—
185C4⋊C4×C83C2×D412yes—
186C22⋊C4×C83—22no—
187C22⋊C4×C83—22no—
188C22⋊C4×C83—22no—
189C22⋊C4×C83—22no—
190C22⋊C4×C83—22no—
191C22⋊C4×C83—22no—
192C22⋊C4×C83—22no—
193C22⋊C4×C83—22no—
194C22⋊C4×C83—22no—
195C22⋊C4×C83—22no—
196C22⋊C4×C83—22no—
197C22⋊C4×C83—42no—
198C22⋊C4×C83—42no—
199C42×C83—22no—
200C42×C83—22no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—22no—
204C42×C83—42no—
205C42×C83C2×D412yes—
206C42×C83C2×D412yes—
207C23×C83—21no—
208C23×C83—22no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83C22×D411yescharacteristic
214Q8×C83—42no—
215Q8×C83—42no—
216Q8×C83—81no—
217D4×C83—42no—
218D4×C83—42no—
219D4×C83—42no—
220D4×C83—42no—
221D4×C83—82no—
222D4×C83—81no—
223D4×C83—82no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—22no—
231C2×C4×C83—21no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—42no—
237C2×C4×C83—42no—
238C2×C4×C83—44no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
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—82no—
253C2×C4×C83—82no—
254C2×C4×C83#12011yescharacteristic
255C2×C4×C83#12011yescharacteristic
256C2×C4×C83#12012yes—
257C2×C4×C83#13512yes—
258C2×C4×C83C22×D412yes—
259C8×C83—42no—
260C8×C83—42no—
261C22×C83—21no—
262C22×C83—21no—
263C22×C83—42no—
264C22×C83—41no—
265C22×C83—42no—
266C22×C83—42no—
267C22×C83—41no—
268C22×C83—42no—
269C22×C83—41no—
270C22×C83—81no—
271C22×C83C2×#12011yescharacteristic
272C22×C83C2×#13511yescharacteristic
273C22×C83#57511yescharacteristic
274C4×C83—22no—
275C4×C83—21no—
276C4×C83—21no—
277C4×C83—22no—
278C4×C83—22no—
279C4×C83—42no—
280C4×C83—42no—
281C4×C83—41no—
282C4×C83—42no—
283C4×C83—41no—
284C4×C83—42no—
285C4×C83—81no—
286C4×C83—81no—
287C4×C83—81no—
288#18407C8311yesmaximal, characteristic, direct product factor, Sylow
289C2×C83—21no—
290C2×C83—21no—
291C2×C83—82no—
292C2×C83—81no—
293C2×C83#300211yescharacteristic
294C2×C83#301811yescharacteristic
295C2×C83#302811yescharacteristic
296#3493C2×C8311yescharacteristic
297#3232C2×C8312yescomplemented
298#2611C2×C8311yescharacteristic, complemented
299#1618C2×C8312yescomplemented
300#1617C2×C8311yescharacteristic, complemented
301#1616C2×C8311yescharacteristic, complemented
302#1612C2×C8312yescomplemented
303#1611C2×C8311yescharacteristic, complemented
304#1610C2×C8311yescharacteristic, complemented
305#1500C2×C8312yescomplemented
306#1458C2×C8311yescharacteristic, complemented
307C83#1840711yescharacteristic, direct product factor, Sylow
308#556—22no—
309#555—22no—
310#553—22no—
311#552—22no—
312#544C22×C8311yescharacteristic
313#538—21no—
314#538C22×C8312yes—
315#537C22×C8311yescharacteristic
316#529—21no—
317#529C22×C8312yes—
318C2×#125C22×C8311yescharacteristic
319#525—21no—
320#520—21no—
321C2×#117C22×C8311yescharacteristic
322#499—22no—
323#488—22no—
324#484C22×C8312yes—
325#483C22×C8312yes—
326#481C22×C8312yes—
327#480C22×C8312yes—
328#479—22no—
329#478—22no—
330#431C22×C8312yes—
331#400C22×C8311yescharacteristic
332#399C22×C8311yescharacteristic
333#398C22×C8311yescharacteristic
334#397C22×C8311yescharacteristic
335#389—21no—
336#388—21no—
337#387—21no—
338#385—21no—
339#376C22×C8311yescharacteristic
340#328C22×C8312yes—
341#328C22×C8312yes—
342#327C22×C8312yes—
343#327C22×C8312yes—
344#324C22×C8311yescharacteristic
345#324C22×C8311yescharacteristic
346#323C22×C8312yes—
347C2×#41—21no—
348C23×C4D4×C8311yescharacteristic
349C4⋊Q8—22no—
350#126—22no—
351#126—22no—
352#126—22no—
353#126—22no—
354#125—21no—
355#125—22no—
356#125—22no—
357#125—22no—
358#125D4×C8312yes—
359#124—21no—
360#124—22no—
361#124—22no—
362#124D4×C8312yes—
363#123—22no—
364#123—22no—
365#123—22no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8C23×C8312yes—
371#121—21no—
372#121—21no—
373#121—22no—
374#121—22no—
375#121—22no—
376#121—21no—
377#121C23×C8312yes—
378#120—22no—
379C4×Q8—22no—
380C4×Q8—21no—
381C4×Q8D4×C8312yes—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—21no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4D4×C8312yes—
389#117—22no—
390#117—22no—
391#117—21no—
392#117—22no—
393#117—22no—
394#117—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4D4×C8311yescharacteristic
401C2×C4⋊C4C23×C8311yescharacteristic
402C2×C4⋊C4C23×C8311yescharacteristic
403C2×C4⋊C4C23×C8311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4D4×C8311yescharacteristic
407C2×C22⋊C4C23×C8311yescharacteristic
408C2×C42—21no—
409C2×C42D4×C8311yescharacteristic
410C2×C42C23×C8311yescharacteristic
411#107—22no—
412#107—22no—
413#106—22no—
414#106—22no—
415C4⋊C8—22no—
416#103—22no—
417#103—22no—
418#102—22no—
419#102—22no—
420C22⋊C8C23×C8312yes—
421C22⋊C8C23×C8312yes—
422C4×C8—22no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95C23×C8311yescharacteristic
429#95C23×C8311yescharacteristic
430C24C2×D4×C8311yescharacteristic
431#41—42no—
432#41—42no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8C2×D4×C8312yes—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4C2×D4×C8312yes—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4C2×D4×C8311yescharacteristic
454C22×C4C2×D4×C8311yescharacteristic
455C22×C4C2×D4×C8311yescharacteristic
456C22×C4C2×D4×C8311yescharacteristic
457C22×C4C2×D4×C8311yescharacteristic
458C22×C4C24×C8311yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4C2×D4×C8312yes—
481C4⋊C4C2×D4×C8312yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C42—22no—
500C42—42no—
501C42C2×D4×C8312yes—
502C42C2×D4×C8312yes—
503C23—21no—
504C23—22no—
505C23—21no—
506C23—21no—
507C23—41no—
508C23—41no—
509C23C22×D4×C8311yescharacteristic
510Q8—42no—
511Q8—42no—
512Q8—81no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—81no—
518D4—82no—
519D4—82no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#120×C8311yescharacteristic
551C2×C4#120×C8311yescharacteristic
552C2×C4#120×C8312yes—
553C2×C4#135×C8312yes—
554C2×C4C22×D4×C8312yes—
555C8—42no—
556C8—42no—
557C22—21no—
558C22—21no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—41no—
566C22—81no—
567C22C2×#120×C8311yescharacteristic
568C22C2×#135×C8311yescharacteristic
569C22#575×C8311yescharacteristic
570C4—22no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—81no—
582C4—81no—
583C4—81no—
584C2—21no—
585C2—21no—
586C2—81no—
587C2—82no—
588C2C83×#300211yescharacteristic
589C2C83×#301811yescharacteristic
590C2C83×#302811yescharacteristic
subgroupG/H
center, socleC22×C83C2×#120
commutator, Frattini, cosocleC22×C4C24×C83
prime radicalC83×#3493C2
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×#3232C2semidirect
3. C83×#2611C2semidirect
4. C83×#1618C2semidirect
5. C83×#1617C2semidirect
6. C83×#1616C2semidirect
7. C83×#1612C2semidirect
8. C83×#1611C2semidirect
9. C83×#1610C2semidirect
10. C83×#1500C2semidirect
11. C83×#1458C2semidirect
288. #18407C83direct (N × H)
297. #3232C2×C83semidirect
298. #2611C2×C83semidirect
299. #1618C2×C83semidirect
300. #1617C2×C83semidirect
301. #1616C2×C83semidirect
302. #1612C2×C83semidirect
303. #1611C2×C83semidirect
304. #1610C2×C83semidirect
305. #1500C2×C83semidirect
306. #1458C2×C83semidirect
307. C83#18407direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#34932
C83×#32322
C83×#26112
C83×#16182
C83×#16172
C83×#16162
C83×#16122
C83×#16112
C83×#16102
C83×#15002
C83×#14582
#1840783