Finite Group Databasebeta

← C83×#62248 #35290072

Subgroups — C83×#62248 #35290072

Order 21248.

1902 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3506C212yesmaximal, complemented
2C83×#3340C214yesmaximal, complemented
3C83×#3305C214yesmaximal, complemented
4C83×#3302C214yesmaximal, complemented
5C83×#3284C214yesmaximal, complemented
6C83×#3249C211yesmaximal, characteristic, complemented
7C83×#3245C211yesmaximal, characteristic, complemented
8C83×#3191C214yesmaximal, complemented
9C83×#3167C212yesmaximal, complemented
10C83×#3050C214yesmaximal, complemented
11C2×C8×D4×C83C211yesmaximal, characteristic, complemented
12C2×#135×C83C2214yescomplemented
13C23×C8×C83C2212yes—
14#559×C83—24no—
15#556×C83C2214yes—
16#555×C83C2214yes—
17#553×C83—24no—
18#550×C83C2214yes—
19#547×C83C2214yes—
20#538×C83—24no—
21#535×C83—22no—
22#533×C83C2212yes—
23#532×C83C2212yes—
24#525×C83C2218yes—
25#520×C83—22no—
26C2×C4×D4×C83C2211yescharacteristic
27#498×C83C2212yescomplemented
28#492×C83C2212yescomplemented
29#487×C83C2212yescomplemented
30#486×C83C2212yescomplemented
31#485×C83C2218yescomplemented
32#482×C83C2218yescomplemented
33#477×C83C2214yescomplemented
34#474×C83C2214yescomplemented
35#467×C83C2214yes—
36#467×C83C2214yescomplemented
37#466×C83C2214yes—
38#466×C83C2214yescomplemented
39#465×C83C2214yes—
40#465×C83C2214yes—
41#465×C83C2218yescomplemented
42#462×C83C2214yescomplemented
43#459×C83C2214yescomplemented
44#452×C83C2218yescomplemented
45#449×C83C2218yescomplemented
46C4×SD16×C83C2214yescomplemented
47C4×SD16×C83C2214yescomplemented
48C8×D4×C83C2214yes—
49C8×D4×C83C2214yes—
50#427×C83C2214yescomplemented
51C2×C4⋊C8×C83C2211yescharacteristic
52#416×C83C2218yescomplemented
53C2×C22⋊C8×C83C2212yes—
54C2×C4×C8×C83C2211yescharacteristic
55C2×#41×C83—22no—
56C2×#41×C83C2318yes—
57C22×D4×C83D411yescharacteristic
58C23×C4×C83C2312yes—
59#135×C83—48no—
60C2×Q16×C83C2314yes—
61C2×SD16×C83C2314yes—
62C2×SD16×C83C2314yes—
63C2×D8×C83C2314yes—
64C22×C8×C83—44no—
65C22×C8×C83C2311yescharacteristic
66C22×C8×C83C2314yes—
67C22×C8×C83C2311yescharacteristic
68C22×C8×C83C2312yes—
69C22×C8×C83C2314yes—
70C4⋊Q8×C83C2312yes—
71C4⋊Q8×C83C2312yes—
72C4⋊Q8×C83C2312yes—
73#127×C83—24no—
74#127×C83C2312yes—
75#126×C83—28no—
76#125×C83—22no—
77#124×C83—24no—
78#124×C83—28no—
79#124×C83—22no—
80#123×C83—28no—
81#123×C83—28no—
82C22⋊Q8×C83—22no—
83C22⋊Q8×C83C2314yes—
84C22⋊Q8×C83C2318yes—
85#121×C83—24no—
86#121×C83—24no—
87#121×C83—22no—
88#121×C83—28no—
89#121×C83C2314yes—
90#121×C83C2318yes—
91#120×C83—28no—
92C4×Q8×C83—24no—
93C4×Q8×C83—22no—
94C4×Q8×C83C2314yes—
95C4×D4×C83—28no—
96C4×D4×C83—28no—
97C4×D4×C83—24no—
98C4×D4×C83—22no—
99C4×D4×C83D414yes—
100C4×D4×C83C2314yes—
101C4×D4×C83C2314yes—
102#117×C83—22no—
103#117×C83—24no—
104#117×C83C2318yes—
105C2×C4⋊C4×C83—24no—
106C2×C4⋊C4×C83D411yescharacteristic
107C2×C22⋊C4×C83D412yes—
108C2×C42×C83C2311yescharacteristic
109#107×C83C2314yes—
110#107×C83C2314yes—
111#106×C83C2314yes—
112#106×C83C2314yes—
113C4⋊C8×C83C2314yes—
114#103×C83C2314yes—
115#103×C83C2314yes—
116#103×C83C2318yes—
117#102×C83C2314yes—
118#102×C83C2314yes—
119#102×C83C2318yes—
120C22⋊C8×C83C2318yes—
121C4×C8×C83C2312yes—
122C4×C8×C83C2312yes—
123C24×C83C2×D412yes—
124#41×C83—48no—
125#41×C83—48no—
126#41×C83—48no—
127C2×Q8×C83—22no—
128C2×Q8×C83—22no—
129C2×Q8×C83C2414yes—
130C2×D4×C83—24no—
131C2×D4×C83—28no—
132C2×D4×C83—28no—
133C2×D4×C83—24no—
134C2×D4×C83—22no—
135C2×D4×C83—22no—
136C2×D4×C83C2×D414yes—
137C2×D4×C83C2414yes—
138C22×C4×C83—24no—
139C22×C4×C83—28no—
140C22×C4×C83—42no—
141C22×C4×C83—42no—
142C22×C4×C83C2×D411yescharacteristic
143C22×C4×C83C2×D412yes—
144C22×C4×C83C2×D414yes—
145C22×C4×C83C2×D411yescharacteristic
146C22×C4×C83C2414yes—
147C22×C4×C83C2411yescharacteristic
148Q16×C83—44no—
149SD16×C83—28no—
150SD16×C83—44no—
151D8×C83—44no—
152C2×C8×C83—22no—
153C2×C8×C83—28no—
154C2×C8×C83—44no—
155C2×C8×C83—42no—
156C2×C8×C83—41no—
157C2×C8×C83C2412yes—
158C2×C8×C83C2411yescharacteristic
159C2×C8×C83C2414yes—
160C2×C8×C83C2411yescharacteristic
161C4⋊C4×C83—22no—
162C4⋊C4×C83—28no—
163C4⋊C4×C83—22no—
164C4⋊C4×C83—24no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83—22no—
167C4⋊C4×C83—24no—
168C4⋊C4×C83C2×D414yes—
169C4⋊C4×C83C2414yes—
170C4⋊C4×C83C2414yes—
171C22⋊C4×C83—24no—
172C22⋊C4×C83—28no—
173C22⋊C4×C83—28no—
174C22⋊C4×C83—28no—
175C22⋊C4×C83—24no—
176C22⋊C4×C83—24no—
177C22⋊C4×C83C2×D418yes—
178C42×C83—28no—
179C42×C83—24no—
180C42×C83C2×D412yes—
181C42×C83C2412yes—
182C23×C83—24no—
183C23×C83—24no—
184C23×C83—42no—
185C23×C83—42no—
186C23×C83C22×D414yes—
187C23×C83C22×D411yescharacteristic
188Q8×C83—24no—
189Q8×C83—24no—
190Q8×C83—44no—
191Q8×C83—82no—
192D4×C83—28no—
193D4×C83—24no—
194D4×C83—24no—
195D4×C83—48no—
196D4×C83—44no—
197D4×C83—82no—
198D4×C83—88no—
199D4×C83—88no—
200C2×C4×C83—21no—
201C2×C4×C83—24no—
202C2×C4×C83—21no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—44no—
208C2×C4×C83—42no—
209C2×C4×C83—41no—
210C2×C4×C83—48no—
211C2×C4×C83—88no—
212C2×C4×C83#13514yescomplemented
213C2×C4×C83C22×D414yes—
214C2×C4×C83C22×D411yescharacteristic
215C2×C4×C83C22×D412yes—
216C2×C4×C83C22×D412yes—
217C2×C4×C83C2511yescharacteristic
218C8×C83—21no—
219C8×C83—24no—
220C8×C83—22no—
221C8×C83H(5,𝔽2)12yes—
222C22×C83—21no—
223C22×C83—21no—
224C22×C83—24no—
225C22×C83—24no—
226C22×C83—24no—
227C22×C83—48no—
228C22×C83—44no—
229C22×C83—44no—
230C22×C83—44no—
231C22×C83C2×#13512yes—
232C22×C83C23×D411yescharacteristic
233C4×C83—22no—
234C4×C83—21no—
235C4×C83—24no—
236C4×C83—44no—
237C4×C83—44no—
238C4×C83—44no—
239C4×C83C2×#13514yes—
240C4×C83C2×H(5,𝔽2)11yescharacteristic
241C4×C83#58511yescharacteristic
242#62248C8311yesmaximal, characteristic, direct product factor, Sylow
243C2×C83—22no—
244C2×C83—44no—
245C2×C83—84no—
246C2×C83#349611yescharacteristic
247C2×C83C22×#13511yescharacteristic
248C2×C83#358411yescharacteristic
249#3506C2×C8312yescomplemented
250#3340C2×C8314yescomplemented
251#3305C2×C8314yescomplemented
252#3302C2×C8314yescomplemented
253#3284C2×C8314yescomplemented
254#3249C2×C8311yescharacteristic, complemented
255#3245C2×C8311yescharacteristic, complemented
256#3191C2×C8314yescomplemented
257#3167C2×C8312yescomplemented
258#3050C2×C8314yescomplemented
259C2×C8×D4C2×C8311yescharacteristic, complemented
260C83#6224811yescharacteristic, direct product factor, Sylow
261C2×#135C22×C8314yescomplemented
262C23×C8C22×C8312yes—
263#559—24no—
264#556C22×C8314yes—
265#555C22×C8314yes—
266#553—24no—
267#550C22×C8314yes—
268#547C22×C8314yes—
269#538—24no—
270#535—22no—
271#533C22×C8312yes—
272#532C22×C8312yes—
273#525C22×C8318yes—
274#520—22no—
275C2×C4×D4C22×C8311yescharacteristic
276#498C22×C8312yescomplemented
277#492C22×C8312yescomplemented
278#487C22×C8312yescomplemented
279#486C22×C8312yescomplemented
280#485C22×C8318yescomplemented
281#482C22×C8318yescomplemented
282#477C22×C8314yescomplemented
283#474C22×C8314yescomplemented
284#467C22×C8314yes—
285#467C22×C8314yescomplemented
286#466C22×C8314yes—
287#466C22×C8314yescomplemented
288#465C22×C8314yes—
289#465C22×C8314yes—
290#465C22×C8318yescomplemented
291#462C22×C8314yescomplemented
292#459C22×C8314yescomplemented
293#452C22×C8318yescomplemented
294#449C22×C8318yescomplemented
295C4×SD16C22×C8314yescomplemented
296C4×SD16C22×C8314yescomplemented
297C8×D4C22×C8314yes—
298C8×D4C22×C8314yes—
299#427C22×C8314yescomplemented
300C2×C4⋊C8C22×C8311yescharacteristic
301#416C22×C8318yescomplemented
302C2×C22⋊C8C22×C8312yes—
303C2×C4×C8C22×C8311yescharacteristic
304C2×#41—22no—
305C2×#41C23×C8318yes—
306C22×D4D4×C8311yescharacteristic
307C23×C4C23×C8312yes—
308#135—48no—
309C2×Q16C23×C8314yes—
310C2×SD16C23×C8314yes—
311C2×SD16C23×C8314yes—
312C2×D8C23×C8314yes—
313C22×C8—44no—
314C22×C8C23×C8311yescharacteristic
315C22×C8C23×C8314yes—
316C22×C8C23×C8312yes—
317C22×C8C23×C8314yes—
318C22×C8C23×C8311yescharacteristic
319C4⋊Q8C23×C8312yes—
320C4⋊Q8C23×C8312yes—
321C4⋊Q8C23×C8312yes—
322#127—24no—
323#127C23×C8312yes—
324#126—28no—
325#125—22no—
326#124—24no—
327#124—28no—
328#124—22no—
329#123—28no—
330#123—28no—
331C22⋊Q8—22no—
332C22⋊Q8C23×C8314yes—
333C22⋊Q8C23×C8318yes—
334#121—24no—
335#121—24no—
336#121—22no—
337#121—28no—
338#121C23×C8314yes—
339#121C23×C8318yes—
340#120—28no—
341C4×Q8—24no—
342C4×Q8—22no—
343C4×Q8C23×C8314yes—
344C4×D4—28no—
345C4×D4—28no—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4D4×C8314yes—
349C4×D4C23×C8314yes—
350C4×D4C23×C8314yes—
351#117—22no—
352#117—24no—
353#117C23×C8318yes—
354C2×C4⋊C4—24no—
355C2×C4⋊C4D4×C8311yescharacteristic
356C2×C22⋊C4D4×C8312yes—
357C2×C42C23×C8311yescharacteristic
358#107C23×C8314yes—
359#107C23×C8314yes—
360#106C23×C8314yes—
361#106C23×C8314yes—
362C4⋊C8C23×C8314yes—
363#103C23×C8314yes—
364#103C23×C8318yes—
365#103C23×C8314yes—
366#102C23×C8314yes—
367#102C23×C8318yes—
368#102C23×C8314yes—
369C22⋊C8C23×C8318yes—
370C4×C8C23×C8312yes—
371C4×C8C23×C8312yes—
372C24C2×D4×C8312yes—
373#41—48no—
374#41—48no—
375#41—48no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8C24×C8314yes—
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4C2×D4×C8314yes—
386C2×D4C24×C8314yes—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4C2×D4×C8311yescharacteristic
392C22×C4C2×D4×C8312yes—
393C22×C4C2×D4×C8314yes—
394C22×C4C2×D4×C8311yescharacteristic
395C22×C4C24×C8314yes—
396C22×C4C24×C8311yescharacteristic
397Q16—44no—
398SD16—28no—
399SD16—44no—
400D8—44no—
401C2×C8—22no—
402C2×C8—28no—
403C2×C8—44no—
404C2×C8—42no—
405C2×C8—41no—
406C2×C8C24×C8312yes—
407C2×C8C24×C8311yescharacteristic
408C2×C8C24×C8314yes—
409C2×C8C24×C8311yescharacteristic
410C4⋊C4—22no—
411C4⋊C4—28no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4C2×D4×C8314yes—
418C4⋊C4C24×C8314yes—
419C4⋊C4C24×C8314yes—
420C22⋊C4—24no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4C2×D4×C8318yes—
427C42—28no—
428C42—24no—
429C42C2×D4×C8312yes—
430C42C24×C8312yes—
431C23—24no—
432C23—24no—
433C23—42no—
434C23—42no—
435C23C22×D4×C8314yes—
436C23C22×D4×C8311yescharacteristic
437Q8—24no—
438Q8—24no—
439Q8—44no—
440Q8—82no—
441D4—28no—
442D4—24no—
443D4—24no—
444D4—48no—
445D4—44no—
446D4—82no—
447D4—88no—
448D4—88no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—88no—
461C2×C4#135×C8314yescomplemented
462C2×C4C22×D4×C8314yes—
463C2×C4C22×D4×C8311yescharacteristic
464C2×C4C22×D4×C8312yes—
465C2×C4C22×D4×C8312yes—
466C2×C4C25×C8311yescharacteristic
467C8—21no—
468C8—24no—
469C8—22no—
470C8H(5,𝔽2)×C8312yes—
471C22—21no—
472C22—21no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—48no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22C2×#135×C8312yes—
481C22C23×D4×C8311yescharacteristic
482C4—22no—
483C4—21no—
484C4—24no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4C2×#135×C8314yes—
489C4C2×H(5,𝔽2)×C8311yescharacteristic
490C4#585×C8311yescharacteristic
491C2—22no—
492C2—44no—
493C2—84no—
494C2C83×#349611yescharacteristic
495C2C22×#135×C8311yescharacteristic
496C2C83×#358411yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×D4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3506C2semidirect
2. C83×#3340C2semidirect
3. C83×#3305C2semidirect
4. C83×#3302C2semidirect
5. C83×#3284C2semidirect
6. C83×#3249C2semidirect
7. C83×#3245C2semidirect
8. C83×#3191C2semidirect
9. C83×#3167C2semidirect
10. C83×#3050C2semidirect
11. C2×C8×D4×C83C2semidirect
12. C2×#135×C83C22semidirect
27. #498×C83C22semidirect
28. #492×C83C22semidirect
29. #487×C83C22semidirect
30. #486×C83C22semidirect
31. #485×C83C22semidirect
32. #482×C83C22semidirect
33. #477×C83C22semidirect
34. #474×C83C22semidirect
36. #467×C83C22semidirect
38. #466×C83C22semidirect
41. #465×C83C22semidirect
42. #462×C83C22semidirect
43. #459×C83C22semidirect
44. #452×C83C22semidirect
45. #449×C83C22semidirect
46. C4×SD16×C83C22semidirect
47. C4×SD16×C83C22semidirect
50. #427×C83C22semidirect
52. #416×C83C22semidirect
212. C2×C4×C83#135semidirect
242. #62248C83direct (N × H)
249. #3506C2×C83semidirect
250. #3340C2×C83semidirect
251. #3305C2×C83semidirect
252. #3302C2×C83semidirect
253. #3284C2×C83semidirect
254. #3249C2×C83semidirect
255. #3245C2×C83semidirect
256. #3191C2×C83semidirect
257. #3167C2×C83semidirect
258. #3050C2×C83semidirect
259. C2×C8×D4C2×C83semidirect
260. C83#62248direct (N × H)
261. C2×#135C22×C83semidirect
276. #498C22×C83semidirect
277. #492C22×C83semidirect
278. #487C22×C83semidirect
279. #486C22×C83semidirect
280. #485C22×C83semidirect
281. #482C22×C83semidirect
282. #477C22×C83semidirect
283. #474C22×C83semidirect
285. #467C22×C83semidirect
287. #466C22×C83semidirect
290. #465C22×C83semidirect
291. #462C22×C83semidirect
292. #459C22×C83semidirect
293. #452C22×C83semidirect
294. #449C22×C83semidirect
295. C4×SD16C22×C83semidirect
296. C4×SD16C22×C83semidirect
299. #427C22×C83semidirect
301. #416C22×C83semidirect
461. C2×C4#135×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35062
C83×#33402
C83×#33052
C83×#33022
C83×#32842
C83×#32492
C83×#32452
C83×#31912
C83×#31672
C83×#30502
C2×C8×D4×C832
#6224883