Finite Group Databasebeta

← C83×#61876 #35289700

Subgroups — C83×#61876 #35289700

Order 21248.

1712 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3548C211yesmaximal, characteristic, complemented
2C83×#3504C211yesmaximal, characteristic, complemented
3C83×#3389C214yesmaximal, complemented
4C83×#3388C214yesmaximal, complemented
5C83×#3368C214yesmaximal, complemented
6C83×#3366C214yesmaximal, complemented
7C83×#3241C214yesmaximal, complemented
8C83×#3234C212yesmaximal, complemented
9C83×#3119C214yesmaximal, complemented
10C83×#3115C212yesmaximal, complemented
11C83×#2987C211yesmaximal, characteristic, complemented
12#562×C83—21no—
13#562×C83—21no—
14#557×C83C2214yes—
15#554×C83—24no—
16#554×C83C2214yes—
17#554×C83C2214yes—
18#550×C83C2214yes—
19#547×C83—22no—
20#544×C83—22no—
21#544×C83C2212yes—
22#540×C83C2214yes—
23#537×C83—24no—
24#533×C83—21no—
25#533×C83C2211yescharacteristic
26#532×C83C2212yes—
27C2×#125×C83—21no—
28C2×#125×C83C2211yescharacteristic
29#520×C83C2212yes—
30C4×#41×C83C2211yescharacteristic
31#501×C83C2214yes—
32#501×C83C2214yes—
33#499×C83C2214yes—
34#490×C83C2214yes—
35#490×C83C2214yes—
36#486×C83C2214yes—
37#482×C83C2214yes—
38#481×C83C2214yes—
39#480×C83C2214yes—
40#479×C83C2214yes—
41#478×C83C2214yes—
42#478×C83C2214yes—
43#478×C83C2214yes—
44#478×C83C2214yes—
45#478×C83C2214yes—
46#476×C83C2214yes—
47#475×C83C2212yes—
48#474×C83C2214yes—
49#474×C83C2212yes—
50#460×C83C2212yes—
51#460×C83C2214yes—
52#459×C83C2212yes—
53#459×C83C2214yes—
54#449×C83C2214yes—
55#446×C83C2214yes—
56#446×C83C2214yes—
57#442×C83C2214yes—
58#442×C83C2214yes—
59#440×C83C2214yes—
60#440×C83C2214yes—
61C4×SD16×C83C2214yes—
62C4×D8×C83C2214yes—
63#433×C83C2214yes—
64#432×C83C2212yes—
65C2×#41×C83—24no—
66C2×#41×C83—22no—
67C2×#41×C83C2311yescharacteristic
68C2×SD16×C83C2314yes—
69C2×D8×C83C2314yes—
70C4⋊Q8×C83—24no—
71C4⋊Q8×C83—22no—
72C4⋊Q8×C83D412yes—
73C4⋊Q8×C83C2312yes—
74C4⋊Q8×C83C2314yes—
75#127×C83C2314yes—
76#127×C83C2312yes—
77#125×C83—22no—
78#125×C83—24no—
79#125×C83—24no—
80#125×C83—21no—
81#125×C83—21no—
82#125×C83D411yescharacteristic
83#125×C83D411yescharacteristic
84#125×C83C2314yes—
85#125×C83C2314yes—
86#125×C83C2314yes—
87#124×C83—22no—
88#123×C83—24no—
89#123×C83—24no—
90C22⋊Q8×C83—24no—
91C22⋊Q8×C83—24no—
92C22⋊Q8×C83—24no—
93C22⋊Q8×C83—24no—
94C22⋊Q8×C83—24no—
95C22⋊Q8×C83—24no—
96C22⋊Q8×C83—24no—
97C22⋊Q8×C83D414yes—
98#121×C83—22no—
99#121×C83—24no—
100#121×C83—24no—
101#121×C83—24no—
102#121×C83—24no—
103#121×C83—24no—
104#121×C83C2314yes—
105C4×Q8×C83—24no—
106C4×Q8×C83C2312yes—
107C4×Q8×C83C2314yes—
108C4×Q8×C83C2314yes—
109C4×D4×C83—22no—
110C4×D4×C83—24no—
111C4×D4×C83—22no—
112C4×D4×C83C2314yes—
113C4×D4×C83C2312yes—
114C4×D4×C83C2314yes—
115C4×D4×C83C2314yes—
116C4×D4×C83C2314yes—
117#117×C83—22no—
118#117×C83—21no—
119#117×C83C2311yescharacteristic
120#117×C83C2312yes—
121C2×C4⋊C4×C83—22no—
122C2×C4⋊C4×C83—22no—
123C2×C4⋊C4×C83—24no—
124C2×C4⋊C4×C83—24no—
125C2×C4⋊C4×C83C2312yes—
126C2×C42×C83—21no—
127C2×C42×C83C2311yescharacteristic
128#107×C83C2314yes—
129#107×C83C2314yes—
130#107×C83C2314yes—
131#106×C83C2314yes—
132#106×C83C2314yes—
133#106×C83C2314yes—
134C4⋊C8×C83C2312yes—
135C4⋊C8×C83C2314yes—
136C4⋊C8×C83C2312yes—
137C4⋊C8×C83C2314yes—
138#103×C83C2314yes—
139#102×C83C2314yes—
140#102×C83C2314yes—
141#102×C83C2314yes—
142#102×C83C2314yes—
143#102×C83C2314yes—
144#102×C83C2314yes—
145#102×C83C2314yes—
146C22⋊C8×C83C2314yes—
147#97×C83C2314yes—
148#97×C83C2314yes—
149C4×C8×C83C2314yes—
150#41×C83—22no—
151#41×C83—22no—
152#41×C83—84no—
153#41×C83—88no—
154C2×Q8×C83—24no—
155C2×Q8×C83—22no—
156C2×Q8×C83C2×D414yes—
157C2×Q8×C83C2411yescharacteristic
158C2×D4×C83—21no—
159C2×D4×C83—24no—
160C2×D4×C83—22no—
161C2×D4×C83—24no—
162C2×D4×C83—24no—
163C2×D4×C83—22no—
164C2×D4×C83C2414yes—
165C2×D4×C83C2411yescharacteristic
166C22×C4×C83—21no—
167C22×C4×C83—22no—
168C22×C4×C83—24no—
169C22×C4×C83—22no—
170C22×C4×C83—24no—
171C22×C4×C83—24no—
172C22×C4×C83—24no—
173C22×C4×C83C2×D412yes—
174C22×C4×C83C2411yescharacteristic
175SD16×C83—28no—
176D8×C83—28no—
177C2×C8×C83C2414yes—
178C2×C8×C83C2414yes—
179C4⋊C4×C83—22no—
180C4⋊C4×C83—24no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83—24no—
183C4⋊C4×C83—24no—
184C4⋊C4×C83—24no—
185C4⋊C4×C83—24no—
186C4⋊C4×C83—24no—
187C4⋊C4×C83—24no—
188C4⋊C4×C83—22no—
189C4⋊C4×C83—24no—
190C4⋊C4×C83C2×D412yes—
191C4⋊C4×C83C2×D414yes—
192C4⋊C4×C83C2×D414yes—
193C4⋊C4×C83C2×D412yes—
194C4⋊C4×C83C2×D414yes—
195C4⋊C4×C83C2414yes—
196C4⋊C4×C83C2412yes—
197C4⋊C4×C83C2414yes—
198C22⋊C4×C83—21no—
199C22⋊C4×C83—22no—
200C22⋊C4×C83—24no—
201C22⋊C4×C83—24no—
202C22⋊C4×C83—24no—
203C22⋊C4×C83—24no—
204C22⋊C4×C83—24no—
205C22⋊C4×C83C2×D414yes—
206C42×C83—21no—
207C42×C83—22no—
208C42×C83—24no—
209C42×C83C2×D411yescharacteristic
210C42×C83C2×D411yescharacteristic
211C42×C83C2414yes—
212C42×C83C2412yes—
213C23×C83—21no—
214C23×C83—24no—
215C23×C83C22×D411yescharacteristic
216Q8×C83—21no—
217Q8×C83—44no—
218Q8×C83—84no—
219Q8×C83—82no—
220Q8×C832−1+412yes—
221D4×C83—24no—
222D4×C83—21no—
223D4×C83—24no—
224D4×C83—24no—
225D4×C83—44no—
226D4×C83—82no—
227D4×C83—84no—
228D4×C83—84no—
229D4×C83—84no—
230D4×C832−1+412yes—
231C2×C4×C83—24no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—22no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—42no—
239C2×C4×C83—48no—
240C2×C4×C83—82no—
241C2×C4×C83—84no—
242C2×C4×C83—84no—
243C2×C4×C83—84no—
244C2×C4×C83—82no—
245C2×C4×C83C22×D411yescharacteristic
246C2×C4×C83C22×D412yes—
247C2×C4×C83C22×D414yes—
248C2×C4×C83C22×D412yes—
249C2×C4×C83C22×D414yes—
250C2×C4×C83C2511yescharacteristic
251C8×C83—24no—
252C8×C83—24no—
253C22×C83—22no—
254C22×C83—21no—
255C22×C83—22no—
256C22×C83—42no—
257C22×C83—44no—
258C22×C83—44no—
259C22×C83—44no—
260C22×C83C23×D411yescharacteristic
261C4×C83—22no—
262C4×C83—22no—
263C4×C83—24no—
264C4×C83—24no—
265C4×C83—21no—
266C4×C83—22no—
267C4×C83—44no—
268C4×C83—44no—
269C4×C83—42no—
270C4×C83C2×2−1+411yescharacteristic
271C4×C83C2×2−1+411yescharacteristic
272#61876C8311yesmaximal, characteristic, direct product factor, Sylow
273C2×C83—41no—
274C2×C83—42no—
275C2×C83—84no—
276C2×C83#348611yescharacteristic
277C2×C83#358512yes—
278#3548C2×C8311yescharacteristic, complemented
279#3504C2×C8311yescharacteristic, complemented
280#3389C2×C8314yescomplemented
281#3388C2×C8314yescomplemented
282#3368C2×C8314yescomplemented
283#3366C2×C8314yescomplemented
284#3241C2×C8314yescomplemented
285#3234C2×C8312yescomplemented
286#3119C2×C8314yescomplemented
287#3115C2×C8312yescomplemented
288#2987C2×C8311yescharacteristic, complemented
289C83#6187611yescharacteristic, direct product factor, Sylow
290#562—21no—
291#562—21no—
292#557C22×C8314yes—
293#554—24no—
294#554C22×C8314yes—
295#554C22×C8314yes—
296#550C22×C8314yes—
297#547—22no—
298#544—22no—
299#544C22×C8312yes—
300#540C22×C8314yes—
301#537—24no—
302#533—21no—
303#533C22×C8311yescharacteristic
304#532C22×C8312yes—
305C2×#125—21no—
306C2×#125C22×C8311yescharacteristic
307#520C22×C8312yes—
308C4×#41C22×C8311yescharacteristic
309#501C22×C8314yes—
310#501C22×C8314yes—
311#499C22×C8314yes—
312#490C22×C8314yes—
313#490C22×C8314yes—
314#486C22×C8314yes—
315#482C22×C8314yes—
316#481C22×C8314yes—
317#480C22×C8314yes—
318#479C22×C8314yes—
319#478C22×C8314yes—
320#478C22×C8314yes—
321#478C22×C8314yes—
322#478C22×C8314yes—
323#478C22×C8314yes—
324#476C22×C8314yes—
325#475C22×C8312yes—
326#474C22×C8314yes—
327#474C22×C8312yes—
328#460C22×C8312yes—
329#460C22×C8314yes—
330#459C22×C8312yes—
331#459C22×C8314yes—
332#449C22×C8314yes—
333#446C22×C8314yes—
334#446C22×C8314yes—
335#442C22×C8314yes—
336#442C22×C8314yes—
337#440C22×C8314yes—
338#440C22×C8314yes—
339C4×SD16C22×C8314yes—
340C4×D8C22×C8314yes—
341#433C22×C8314yes—
342#432C22×C8312yes—
343C2×#41—24no—
344C2×#41—22no—
345C2×#41C23×C8311yescharacteristic
346C2×SD16C23×C8314yes—
347C2×D8C23×C8314yes—
348C4⋊Q8—24no—
349C4⋊Q8—22no—
350C4⋊Q8D4×C8312yes—
351C4⋊Q8C23×C8312yes—
352C4⋊Q8C23×C8314yes—
353#127C23×C8312yes—
354#127C23×C8314yes—
355#125—22no—
356#125—24no—
357#125—24no—
358#125—21no—
359#125—21no—
360#125D4×C8311yescharacteristic
361#125D4×C8311yescharacteristic
362#125C23×C8314yes—
363#125C23×C8314yes—
364#125C23×C8314yes—
365#124—22no—
366#123—24no—
367#123—24no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370C22⋊Q8—24no—
371C22⋊Q8—24no—
372C22⋊Q8—24no—
373C22⋊Q8—24no—
374C22⋊Q8—24no—
375C22⋊Q8D4×C8314yes—
376#121—22no—
377#121—24no—
378#121—24no—
379#121—24no—
380#121—24no—
381#121—24no—
382#121C23×C8314yes—
383C4×Q8—24no—
384C4×Q8C23×C8312yes—
385C4×Q8C23×C8314yes—
386C4×Q8C23×C8314yes—
387C4×D4—22no—
388C4×D4—24no—
389C4×D4—22no—
390C4×D4C23×C8312yes—
391C4×D4C23×C8314yes—
392C4×D4C23×C8314yes—
393C4×D4C23×C8314yes—
394C4×D4C23×C8314yes—
395#117—22no—
396#117—21no—
397#117C23×C8311yescharacteristic
398#117C23×C8312yes—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4C23×C8312yes—
404C2×C42—21no—
405C2×C42C23×C8311yescharacteristic
406#107C23×C8314yes—
407#107C23×C8314yes—
408#107C23×C8314yes—
409#106C23×C8314yes—
410#106C23×C8314yes—
411#106C23×C8314yes—
412C4⋊C8C23×C8312yes—
413C4⋊C8C23×C8312yes—
414C4⋊C8C23×C8314yes—
415C4⋊C8C23×C8314yes—
416#103C23×C8314yes—
417#102C23×C8314yes—
418#102C23×C8314yes—
419#102C23×C8314yes—
420#102C23×C8314yes—
421#102C23×C8314yes—
422#102C23×C8314yes—
423#102C23×C8314yes—
424C22⋊C8C23×C8314yes—
425#97C23×C8314yes—
426#97C23×C8314yes—
427C4×C8C23×C8314yes—
428#41—22no—
429#41—22no—
430#41—84no—
431#41—88no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8C2×D4×C8314yes—
435C2×Q8C24×C8311yescharacteristic
436C2×D4—21no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4C24×C8314yes—
443C2×D4C24×C8311yescharacteristic
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4C2×D4×C8312yes—
452C22×C4C24×C8311yescharacteristic
453SD16—28no—
454D8—28no—
455C2×C8C24×C8314yes—
456C2×C8C24×C8314yes—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4C2×D4×C8312yes—
469C4⋊C4C2×D4×C8314yes—
470C4⋊C4C2×D4×C8314yes—
471C4⋊C4C2×D4×C8312yes—
472C4⋊C4C2×D4×C8314yes—
473C4⋊C4C24×C8314yes—
474C4⋊C4C24×C8312yes—
475C4⋊C4C24×C8314yes—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4C2×D4×C8314yes—
484C42—21no—
485C42—22no—
486C42—24no—
487C42C2×D4×C8311yescharacteristic
488C42C2×D4×C8311yescharacteristic
489C42C24×C8314yes—
490C42C24×C8312yes—
491C23—21no—
492C23—24no—
493C23C22×D4×C8311yescharacteristic
494Q8—21no—
495Q8—44no—
496Q8—82no—
497Q8—84no—
498Q82−1+4×C8312yes—
499D4—24no—
500D4—21no—
501D4—24no—
502D4—24no—
503D4—44no—
504D4—82no—
505D4—84no—
506D4—84no—
507D4—84no—
508D42−1+4×C8312yes—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4C22×D4×C8311yescharacteristic
524C2×C4C22×D4×C8312yes—
525C2×C4C22×D4×C8314yes—
526C2×C4C22×D4×C8312yes—
527C2×C4C22×D4×C8314yes—
528C2×C4C25×C8311yescharacteristic
529C8—24no—
530C8—24no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22C23×D4×C8311yescharacteristic
539C4—22no—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—21no—
544C4—22no—
545C4—44no—
546C4—44no—
547C4—42no—
548C4C2×2−1+4×C8311yescharacteristic
549C4C2×2−1+4×C8311yescharacteristic
550C2—41no—
551C2—42no—
552C2—84no—
553C2C83×#348611yescharacteristic
554C2C83×#358512yes—
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ #533×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3548C2semidirect
2. C83×#3504C2semidirect
3. C83×#3389C2semidirect
4. C83×#3388C2semidirect
5. C83×#3368C2semidirect
6. C83×#3366C2semidirect
7. C83×#3241C2semidirect
8. C83×#3234C2semidirect
9. C83×#3119C2semidirect
10. C83×#3115C2semidirect
11. C83×#2987C2semidirect
272. #61876C83direct (N × H)
278. #3548C2×C83semidirect
279. #3504C2×C83semidirect
280. #3389C2×C83semidirect
281. #3388C2×C83semidirect
282. #3368C2×C83semidirect
283. #3366C2×C83semidirect
284. #3241C2×C83semidirect
285. #3234C2×C83semidirect
286. #3119C2×C83semidirect
287. #3115C2×C83semidirect
288. #2987C2×C83semidirect
289. C83#61876direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35482
C83×#35042
C83×#33892
C83×#33882
C83×#33682
C83×#33662
C83×#32412
C83×#32342
C83×#31192
C83×#31152
C83×#29872
#6187683