Finite Group Databasebeta

← C83×#61901 #35289725

Subgroups — C83×#61901 #35289725

Order 21248.

2036 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3523C211yesmaximal, characteristic, complemented
2C2×#550×C83C211yesmaximal, characteristic, complemented
3C83×#3397C216yesmaximal, complemented
4C83×#3392C212yesmaximal, complemented
5C83×#3368C216yesmaximal, complemented
6C83×#3358C212yesmaximal, complemented
7C83×#3217C213yesmaximal, complemented
8C83×#3208C213yesmaximal, complemented
9C83×#3110C213yesmaximal, complemented
10C83×#3105C213yesmaximal, complemented
11C83×#2984C211yesmaximal, characteristic, complemented
12C22×#41×C83—21no—
13#557×C83C2211yescharacteristic, complemented
14#557×C83C2213yescomplemented
15#556×C83C2213yescomplemented
16#552×C83—23no—
17#550×C83—21no—
18#550×C83—23no—
19#550×C83C2212yes—
20#550×C83C2216yes—
21D4×Q8×C83—21no—
22D4×Q8×C83C2211yescharacteristic, complemented
23#541×C83—23no—
24#533×C83C2213yes—
25C2×#127×C83C2213yes—
26#529×C83C2213yes—
27C2×#121×C83—23no—
28#519×C83—21no—
29C2×C4×Q8×C83C2211yescharacteristic
30C2×C4×D4×C83C2213yes—
31#496×C83C2216yescomplemented
32#492×C83C2216yescomplemented
33#490×C83C2216yescomplemented
34#486×C83C2216yescomplemented
35#484×C83C2213yes—
36#481×C83C2211yescharacteristic
37#478×C83C2216yescomplemented
38#474×C83C2216yescomplemented
39#463×C83C2216yescomplemented
40#461×C83C2216yescomplemented
41#460×C83C2216yescomplemented
42#460×C83C22112yescomplemented
43#459×C83C2216yescomplemented
44#450×C83C2213yes—
45#450×C83C2216yes—
46#447×C83C2213yes—
47#446×C83C2216yescomplemented
48C8×Q8×C83C2212yescomplemented
49#440×C83C22112yescomplemented
50C4×SD16×C83C2216yescomplemented
51C4×D8×C83C2216yescomplemented
52#432×C83C2213yes—
53#431×C83C2213yes—
54C2×#41×C83—26no—
55C2×#41×C83—22no—
56C2×#41×C83—48no—
57C22×Q8×C83—21no—
58C22×Q8×C83D411yescharacteristic
59C22×D4×C83—23no—
60C22×D4×C83—23no—
61C22×D4×C83C2313yes—
62C23×C4×C83—23no—
63C2×SD16×C83C2316yes—
64C2×SD16×C83C23112yes—
65C2×D8×C83C2316yes—
66C4⋊Q8×C83—23no—
67C4⋊Q8×C83C2313yes—
68C4⋊Q8×C83C2313yes—
69#127×C83—23no—
70#127×C83—23no—
71#127×C83C2316yes—
72#127×C83C2316yes—
73#126×C83—26no—
74#126×C83—23no—
75#125×C83—26no—
76#125×C83—23no—
77#125×C83—23no—
78#125×C83C2313yes—
79#124×C83C2313yes—
80#123×C83—23no—
81C22⋊Q8×C83—23no—
82C22⋊Q8×C83—23no—
83C22⋊Q8×C83—23no—
84C22⋊Q8×C83C2313yes—
85#121×C83—26no—
86#121×C83—26no—
87#121×C83—212no—
88#121×C83C2311yescharacteristic
89C4×Q8×C83—21no—
90C4×Q8×C83—23no—
91C4×Q8×C83—23no—
92C4×Q8×C83D411yescharacteristic
93C4×Q8×C83D413yes—
94C4×Q8×C83C2313yes—
95C4×Q8×C83C2316yes—
96C4×Q8×C83C2311yescharacteristic
97C4×Q8×C83C2313yes—
98C4×D4×C83—23no—
99C4×D4×C83—23no—
100C4×D4×C83—23no—
101C4×D4×C83C2316yes—
102C4×D4×C83C2316yes—
103C4×D4×C83C2313yes—
104#117×C83—23no—
105#117×C83—23no—
106#117×C83—23no—
107C2×C4⋊C4×C83D413yes—
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C22⋊C4×C83—23no—
110C2×C42×C83C2313yes—
111#107×C83C2316yes—
112#106×C83C2312yes—
113C4⋊C8×C83C2316yes—
114C4⋊C8×C83C2316yes—
115#103×C83C2316yes—
116#102×C83C2312yes—
117#102×C83C23112yes—
118#102×C83C2316yes—
119#102×C83C2316yes—
120C22⋊C8×C83C2311yescharacteristic
121C22⋊C8×C83C2313yes—
122#97×C83C2316yes—
123C4×C8×C83C2316yes—
124C24×C83—23no—
125#41×C83—424no—
126#41×C83—48no—
127C2×Q8×C83—24no—
128C2×Q8×C83—23no—
129C2×Q8×C83—44no—
130C2×Q8×C83C2×D413yes—
131C2×Q8×C83C2×D411yescharacteristic
132C2×Q8×C83C2413yes—
133C2×D4×C83—21no—
134C2×D4×C83—23no—
135C2×D4×C83—26no—
136C2×D4×C83—26no—
137C2×D4×C83—26no—
138C2×D4×C83—26no—
139C2×D4×C83—46no—
140C2×D4×C83—412no—
141C2×D4×C83—86no—
142C2×D4×C83C2416yes—
143C2×D4×C83C2411yescharacteristic
144C22×C4×C83—21no—
145C22×C4×C83—23no—
146C22×C4×C83—26no—
147C22×C4×C83—26no—
148C22×C4×C83—412no—
149C22×C4×C83C2×D413yes—
150C22×C4×C83C2×D413yes—
151C22×C4×C83C2411yescharacteristic
152SD16×C83—412no—
153SD16×C83—46no—
154D8×C83—46no—
155C2×C8×C83C2412yes—
156C2×C8×C83C2416yes—
157C4⋊C4×C83—21no—
158C4⋊C4×C83—23no—
159C4⋊C4×C83—23no—
160C4⋊C4×C83—23no—
161C4⋊C4×C83—23no—
162C4⋊C4×C83—23no—
163C4⋊C4×C83—23no—
164C4⋊C4×C83—26no—
165C4⋊C4×C83—23no—
166C4⋊C4×C83C2×D413yes—
167C4⋊C4×C83C2×D413yes—
168C4⋊C4×C83C2×D416yes—
169C4⋊C4×C83C2411yescharacteristic
170C4⋊C4×C83C2413yes—
171C4⋊C4×C83C2412yes—
172C22⋊C4×C83—23no—
173C22⋊C4×C83—26no—
174C22⋊C4×C83—26no—
175C22⋊C4×C83—21no—
176C22⋊C4×C83—23no—
177C22⋊C4×C83—23no—
178C22⋊C4×C83—23no—
179C42×C83—26no—
180C42×C83—23no—
181C42×C83C2×D413yes—
182C42×C83C2×D413yes—
183C42×C83C2413yes—
184C42×C83C2413yes—
185C23×C83—21no—
186C23×C83—26no—
187C23×C83—46no—
188C23×C83—46no—
189C23×C83—46no—
190C23×C83C22×D411yescharacteristic
191Q8×C83—26no—
192Q8×C83—26no—
193Q8×C83—46no—
194Q8×C83AGL(1,ℤ/8)14yes—
195D4×C83—22no—
196D4×C83—26no—
197D4×C83—26no—
198D4×C83—46no—
199D4×C83—412no—
200D4×C83—42no—
201D4×C83—412no—
202D4×C83—86no—
203D4×C83—86no—
204C2×C4×C83—26no—
205C2×C4×C83—26no—
206C2×C4×C83—22no—
207C2×C4×C83—23no—
208C2×C4×C83—21no—
209C2×C4×C83—23no—
210C2×C4×C83—22no—
211C2×C4×C83—412no—
212C2×C4×C83—42no—
213C2×C4×C83—412no—
214C2×C4×C83—46no—
215C2×C4×C83C22×D413yes—
216C2×C4×C83C22×D411yescharacteristic
217C2×C4×C83C22×D413yes—
218C2×C4×C83C22×D413yes—
219C2×C4×C83C22×D413yes—
220C2×C4×C83C2511yescharacteristic
221C8×C83—26no—
222C8×C83—22no—
223C22×C83—21no—
224C22×C83—21no—
225C22×C83—46no—
226C22×C83—41no—
227C22×C83—41no—
228C22×C83—46no—
229C22×C83—41no—
230C22×C83—46no—
231C22×C83—812no—
232C22×C83#57712yes—
233C22×C83C23×D411yescharacteristic
234C4×C83—21no—
235C4×C83—23no—
236C4×C83—23no—
237C4×C83—23no—
238C4×C83—43no—
239C4×C83—42no—
240C4×C83—43no—
241C4×C83—41no—
242C4×C83C2×AGL(1,ℤ/8)16yes—
243C4×C83#58512yes—
244#61901C8311yesmaximal, characteristic, direct product factor, Sylow
245C2×C83—22no—
246C2×C83—81no—
247C2×C83—86no—
248C2×C83#348711yescharacteristic
249C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
250C2×C83C2×#57711yescharacteristic
251#3523C2×C8311yescharacteristic, complemented
252C2×#550C2×C8311yescharacteristic, complemented
253#3397C2×C8316yescomplemented
254#3392C2×C8312yescomplemented
255#3368C2×C8316yescomplemented
256#3358C2×C8312yescomplemented
257#3217C2×C8313yescomplemented
258#3208C2×C8313yescomplemented
259#3110C2×C8313yescomplemented
260#3105C2×C8313yescomplemented
261#2984C2×C8311yescharacteristic, complemented
262C83#6190111yescharacteristic, direct product factor, Sylow
263C22×#41—21no—
264#557C22×C8311yescharacteristic, complemented
265#557C22×C8313yescomplemented
266#556C22×C8313yescomplemented
267#552—23no—
268#550—21no—
269#550—23no—
270#550C22×C8312yes—
271#550C22×C8316yes—
272D4×Q8—21no—
273D4×Q8C22×C8311yescharacteristic, complemented
274#541—23no—
275#533C22×C8313yes—
276C2×#127C22×C8313yes—
277#529C22×C8313yes—
278C2×#121—23no—
279#519—21no—
280C2×C4×Q8C22×C8311yescharacteristic
281C2×C4×D4C22×C8313yes—
282#496C22×C8316yescomplemented
283#492C22×C8316yescomplemented
284#490C22×C8316yescomplemented
285#486C22×C8316yescomplemented
286#484C22×C8313yes—
287#481C22×C8311yescharacteristic
288#478C22×C8316yescomplemented
289#474C22×C8316yescomplemented
290#463C22×C8316yescomplemented
291#461C22×C8316yescomplemented
292#460C22×C8316yescomplemented
293#460C22×C83112yescomplemented
294#459C22×C8316yescomplemented
295#450C22×C8313yes—
296#450C22×C8316yes—
297#447C22×C8313yes—
298#446C22×C8316yescomplemented
299C8×Q8C22×C8312yescomplemented
300#440C22×C83112yescomplemented
301C4×SD16C22×C8316yescomplemented
302C4×D8C22×C8316yescomplemented
303#432C22×C8313yes—
304#431C22×C8313yes—
305C2×#41—26no—
306C2×#41—22no—
307C2×#41—48no—
308C22×Q8—21no—
309C22×Q8D4×C8311yescharacteristic
310C22×D4—23no—
311C22×D4—23no—
312C22×D4C23×C8313yes—
313C23×C4—23no—
314C2×SD16C23×C8316yes—
315C2×SD16C23×C83112yes—
316C2×D8C23×C8316yes—
317C4⋊Q8—23no—
318C4⋊Q8C23×C8313yes—
319C4⋊Q8C23×C8313yes—
320#127—23no—
321#127—23no—
322#127C23×C8316yes—
323#127C23×C8316yes—
324#126—26no—
325#126—23no—
326#125—26no—
327#125—23no—
328#125—23no—
329#125C23×C8313yes—
330#124C23×C8313yes—
331#123—23no—
332C22⋊Q8—23no—
333C22⋊Q8—23no—
334C22⋊Q8—23no—
335C22⋊Q8C23×C8313yes—
336#121—26no—
337#121—26no—
338#121—212no—
339#121C23×C8311yescharacteristic
340C4×Q8—21no—
341C4×Q8—23no—
342C4×Q8—23no—
343C4×Q8D4×C8311yescharacteristic
344C4×Q8D4×C8313yes—
345C4×Q8C23×C8313yes—
346C4×Q8C23×C8316yes—
347C4×Q8C23×C8311yescharacteristic
348C4×Q8C23×C8313yes—
349C4×D4—23no—
350C4×D4—23no—
351C4×D4—23no—
352C4×D4C23×C8316yes—
353C4×D4C23×C8316yes—
354C4×D4C23×C8313yes—
355#117—23no—
356#117—23no—
357#117—23no—
358C2×C4⋊C4D4×C8313yes—
359C2×C4⋊C4C23×C8311yescharacteristic
360C2×C22⋊C4—23no—
361C2×C42C23×C8313yes—
362#107C23×C8316yes—
363#106C23×C8312yes—
364C4⋊C8C23×C8316yes—
365C4⋊C8C23×C8316yes—
366#103C23×C8316yes—
367#102C23×C8312yes—
368#102C23×C8316yes—
369#102C23×C8316yes—
370#102C23×C83112yes—
371C22⋊C8C23×C8311yescharacteristic
372C22⋊C8C23×C8313yes—
373#97C23×C8316yes—
374C4×C8C23×C8316yes—
375C24—23no—
376#41—424no—
377#41—48no—
378C2×Q8—24no—
379C2×Q8—23no—
380C2×Q8—44no—
381C2×Q8C2×D4×C8313yes—
382C2×Q8C2×D4×C8311yescharacteristic
383C2×Q8C24×C8313yes—
384C2×D4—21no—
385C2×D4—23no—
386C2×D4—26no—
387C2×D4—26no—
388C2×D4—26no—
389C2×D4—26no—
390C2×D4—46no—
391C2×D4—412no—
392C2×D4—86no—
393C2×D4C24×C8316yes—
394C2×D4C24×C8311yescharacteristic
395C22×C4—21no—
396C22×C4—23no—
397C22×C4—26no—
398C22×C4—26no—
399C22×C4—412no—
400C22×C4C2×D4×C8313yes—
401C22×C4C2×D4×C8313yes—
402C22×C4C24×C8311yescharacteristic
403SD16—412no—
404SD16—46no—
405D8—46no—
406C2×C8C24×C8312yes—
407C2×C8C24×C8316yes—
408C4⋊C4—21no—
409C4⋊C4—23no—
410C4⋊C4—23no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4—23no—
414C4⋊C4—23no—
415C4⋊C4—26no—
416C4⋊C4—23no—
417C4⋊C4C2×D4×C8313yes—
418C4⋊C4C2×D4×C8313yes—
419C4⋊C4C2×D4×C8316yes—
420C4⋊C4C24×C8311yescharacteristic
421C4⋊C4C24×C8313yes—
422C4⋊C4C24×C8312yes—
423C22⋊C4—23no—
424C22⋊C4—26no—
425C22⋊C4—26no—
426C22⋊C4—21no—
427C22⋊C4—23no—
428C22⋊C4—23no—
429C22⋊C4—23no—
430C42—26no—
431C42—23no—
432C42C2×D4×C8313yes—
433C42C2×D4×C8313yes—
434C42C24×C8313yes—
435C42C24×C8313yes—
436C23—21no—
437C23—26no—
438C23—46no—
439C23—46no—
440C23—46no—
441C23C22×D4×C8311yescharacteristic
442Q8—26no—
443Q8—26no—
444Q8—46no—
445Q8AGL(1,ℤ/8)×C8314yes—
446D4—22no—
447D4—26no—
448D4—26no—
449D4—412no—
450D4—46no—
451D4—42no—
452D4—412no—
453D4—86no—
454D4—86no—
455C2×C4—26no—
456C2×C4—26no—
457C2×C4—22no—
458C2×C4—23no—
459C2×C4—21no—
460C2×C4—23no—
461C2×C4—22no—
462C2×C4—412no—
463C2×C4—42no—
464C2×C4—412no—
465C2×C4—46no—
466C2×C4C22×D4×C8313yes—
467C2×C4C22×D4×C8311yescharacteristic
468C2×C4C22×D4×C8313yes—
469C2×C4C22×D4×C8313yes—
470C2×C4C22×D4×C8313yes—
471C2×C4C25×C8311yescharacteristic
472C8—26no—
473C8—22no—
474C22—21no—
475C22—21no—
476C22—41no—
477C22—46no—
478C22—41no—
479C22—46no—
480C22—46no—
481C22—41no—
482C22—812no—
483C22#577×C8312yes—
484C22C23×D4×C8311yescharacteristic
485C4—23no—
486C4—23no—
487C4—23no—
488C4—21no—
489C4—42no—
490C4—43no—
491C4—41no—
492C4—43no—
493C4C2×AGL(1,ℤ/8)×C8316yes—
494C4#585×C8312yes—
495C2—22no—
496C2—86no—
497C2—81no—
498C2C83×#348711yescharacteristic
499C2C22×AGL(1,ℤ/8)×C8311yescharacteristic
500C2C2×#577×C8311yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×Q8×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3523C2semidirect
2. C2×#550×C83C2semidirect
3. C83×#3397C2semidirect
4. C83×#3392C2semidirect
5. C83×#3368C2semidirect
6. C83×#3358C2semidirect
7. C83×#3217C2semidirect
8. C83×#3208C2semidirect
9. C83×#3110C2semidirect
10. C83×#3105C2semidirect
11. C83×#2984C2semidirect
13. #557×C83C22semidirect
14. #557×C83C22semidirect
15. #556×C83C22semidirect
22. D4×Q8×C83C22semidirect
31. #496×C83C22semidirect
32. #492×C83C22semidirect
33. #490×C83C22semidirect
34. #486×C83C22semidirect
37. #478×C83C22semidirect
38. #474×C83C22semidirect
39. #463×C83C22semidirect
40. #461×C83C22semidirect
41. #460×C83C22semidirect
42. #460×C83C22semidirect
43. #459×C83C22semidirect
47. #446×C83C22semidirect
48. C8×Q8×C83C22semidirect
49. #440×C83C22semidirect
50. C4×SD16×C83C22semidirect
51. C4×D8×C83C22semidirect
244. #61901C83direct (N × H)
251. #3523C2×C83semidirect
252. C2×#550C2×C83semidirect
253. #3397C2×C83semidirect
254. #3392C2×C83semidirect
255. #3368C2×C83semidirect
256. #3358C2×C83semidirect
257. #3217C2×C83semidirect
258. #3208C2×C83semidirect
259. #3110C2×C83semidirect
260. #3105C2×C83semidirect
261. #2984C2×C83semidirect
262. C83#61901direct (N × H)
264. #557C22×C83semidirect
265. #557C22×C83semidirect
266. #556C22×C83semidirect
273. D4×Q8C22×C83semidirect
282. #496C22×C83semidirect
283. #492C22×C83semidirect
284. #490C22×C83semidirect
285. #486C22×C83semidirect
288. #478C22×C83semidirect
289. #474C22×C83semidirect
290. #463C22×C83semidirect
291. #461C22×C83semidirect
292. #460C22×C83semidirect
293. #460C22×C83semidirect
294. #459C22×C83semidirect
298. #446C22×C83semidirect
299. C8×Q8C22×C83semidirect
300. #440C22×C83semidirect
301. C4×SD16C22×C83semidirect
302. C4×D8C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35232
C2×#550×C832
C83×#33972
C83×#33922
C83×#33682
C83×#33582
C83×#32172
C83×#32082
C83×#31102
C83×#31052
C83×#29842
#6190183