Finite Group Databasebeta

← C83×#8415 #35236239

Subgroups — C83×#8415 #35236239

Order 21248.

704 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3515C211yesmaximal, characteristic
2C83×#1759C211yesmaximal, characteristic, complemented
3C83×#1529C211yesmaximal, characteristic, complemented
4C83×#1506C211yesmaximal, characteristic, complemented
5C83×#1500C211yesmaximal, characteristic, complemented
6C83×#1488C211yesmaximal, characteristic, complemented
7C83×#1485C211yesmaximal, characteristic, complemented
8#557×C83C411yescharacteristic, complemented
9#554×C83C411yescharacteristic, complemented
10#554×C83C411yescharacteristic, complemented
11#553×C83—21no—
12#552×C83—21no—
13#552×C83—22no—
14#552×C83—21no—
15#547×C83—21no—
16#547×C83C411yescharacteristic, complemented
17#541×C83—21no—
18#539×C83—22no—
19#533×C83C2211yescharacteristic
20#529×C83—21no—
21#529×C83C2211yescharacteristic
22C2×#125×C83—21no—
23#520×C83—21no—
24C4×#41×C83C2211yescharacteristic
25#432×C83C2211yescharacteristic
26#431×C83C2211yescharacteristic
27C2×#104×C83—21no—
28C4×M16×C83C2211yescharacteristic
29#376×C83C2211yescharacteristic
30#339×C83—21no—
31#330×C83—21no—
32#329×C83—21no—
33#328×C83—21no—
34#328×C83—22no—
35#327×C83—21no—
36Q8⋊C8×C83—22no—
37Q8⋊C8×C83—21no—
38D4⋊C8×C83—21no—
39C2×#41×C83—21no—
40C2×#41×C83D411yescharacteristic
41C2×M16×C83—21no—
42C4⋊Q8×C83—21no—
43C4⋊Q8×C83C2×C411yescharacteristic
44C4⋊Q8×C83C2×C411yescharacteristic
45#126×C83—22no—
46#126×C83—22no—
47#126×C83—22no—
48#126×C83—21no—
49#125×C83—22no—
50#125×C83—22no—
51#125×C83—22no—
52#125×C83—21no—
53#125×C83—21no—
54#125×C83—22no—
55#125×C83—21no—
56#125×C83—21no—
57#125×C83D411yescharacteristic
58#124×C83D411yescharacteristic
59#123×C83—22no—
60#123×C83—22no—
61#123×C83—22no—
62#123×C83—21no—
63#123×C83—22no—
64C22⋊Q8×C83—21no—
65C22⋊Q8×C83—22no—
66C22⋊Q8×C83—22no—
67C22⋊Q8×C83—21no—
68C22⋊Q8×C83—21no—
69C22⋊Q8×C83—21no—
70C22⋊Q8×C83D411yescharacteristic
71C22⋊Q8×C83D412yes—
72#121×C83—22no—
73#121×C83—21no—
74#121×C83D411yescharacteristic
75C4×Q8×C83—21no—
76C4×Q8×C83—21no—
77C4×Q8×C83—21no—
78C4×Q8×C83C2×C411yescharacteristic
79C4×Q8×C83C2×C411yescharacteristic
80C4×Q8×C83D411yescharacteristic
81C4×Q8×C83D412yes—
82C4×D4×C83—22no—
83C4×D4×C83—22no—
84C4×D4×C83—21no—
85C4×D4×C83—21no—
86C4×D4×C83—21no—
87C4×D4×C83—21no—
88C4×D4×C83—21no—
89C4×D4×C83C2×C411yescharacteristic
90C4×D4×C83C2×C411yescharacteristic
91C4×D4×C83D411yescharacteristic
92#117×C83—22no—
93#117×C83—22no—
94#117×C83—21no—
95#117×C83—21no—
96#117×C83—21no—
97#117×C83D411yescharacteristic
98C2×C4⋊C4×C83—22no—
99C2×C4⋊C4×C83—22no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83—21no—
103C2×C42×C83—21no—
104C2×C42×C83—21no—
105C2×C42×C83C2311yescharacteristic
106C4⋊C8×C83—21no—
107C4⋊C8×C83—21no—
108#104×C83—24no—
109C22⋊C8×C83—21no—
110C22⋊C8×C83—21no—
111#97×C83—21no—
112#97×C83—21no—
113C4×C8×C83—21no—
114C4×C8×C83—21no—
115#95×C83—21no—
116#95×C83—21no—
117#41×C83—22no—
118#41×C83—41no—
119#41×C83—42no—
120#41×C83—42no—
121C2×Q8×C83—21no—
122C2×Q8×C83C22⋊C411yescharacteristic
123C2×Q8×C83C22⋊C411yescharacteristic
124C2×Q8×C83C22⋊C412yes—
125C2×D4×C83—22no—
126C2×D4×C83—21no—
127C2×D4×C83—21no—
128C2×D4×C83C22⋊C411yescharacteristic
129C2×D4×C83C22⋊C411yescharacteristic
130C22×C4×C83—22no—
131C22×C4×C83—22no—
132C22×C4×C83—22no—
133C22×C4×C83—21no—
134C22×C4×C83—21no—
135C22×C4×C83—21no—
136C22×C4×C83—21no—
137C22×C4×C83—41no—
138C22×C4×C83C2×D411yescharacteristic
139C22×C4×C83C2×D411yescharacteristic
140C22×C4×C83C2×D411yescharacteristic
141M16×C83—22no—
142M16×C83—41no—
143C2×C8×C83—41no—
144C2×C8×C83—41no—
145C2×C8×C83—41no—
146C4⋊C4×C83—22no—
147C4⋊C4×C83—22no—
148C4⋊C4×C83—22no—
149C4⋊C4×C83—22no—
150C4⋊C4×C83—22no—
151C4⋊C4×C83—22no—
152C4⋊C4×C83—22no—
153C4⋊C4×C83—22no—
154C4⋊C4×C83—21no—
155C4⋊C4×C83—21no—
156C4⋊C4×C83—21no—
157C4⋊C4×C83—21no—
158C4⋊C4×C83—21no—
159C4⋊C4×C83—21no—
160C4⋊C4×C83—21no—
161C4⋊C4×C83—22no—
162C4⋊C4×C83—22no—
163C4⋊C4×C83—21no—
164C4⋊C4×C83—21no—
165C4⋊C4×C83C22⋊C411yescharacteristic
166C4⋊C4×C83C22⋊C411yescharacteristic
167C4⋊C4×C83C22⋊C411yescharacteristic
168C4⋊C4×C83C22⋊C411yescharacteristic
169C4⋊C4×C83C22⋊C412yes—
170C22⋊C4×C83—22no—
171C22⋊C4×C83—22no—
172C22⋊C4×C83—22no—
173C22⋊C4×C83—22no—
174C22⋊C4×C83—21no—
175C22⋊C4×C83—21no—
176C22⋊C4×C83—21no—
177C22⋊C4×C83—21no—
178C22⋊C4×C83—21no—
179C22⋊C4×C83—21no—
180C22⋊C4×C83—21no—
181C22⋊C4×C83—21no—
182C42×C83—22no—
183C42×C83—21no—
184C42×C83—21no—
185C42×C83—21no—
186C42×C83—21no—
187C42×C83—22no—
188C42×C83—22no—
189C42×C83—41no—
190C42×C83C22×C411yescharacteristic
191C42×C83C2×D411yescharacteristic
192C42×C83C2×D411yescharacteristic
193C42×C83C2×D411yescharacteristic
194C23×C83—21no—
195C23×C83—21no—
196C23×C83#12011yescharacteristic
197Q8×C83—21no—
198Q8×C83—21no—
199Q8×C83—42no—
200Q8×C83—41no—
201Q8×C83—41no—
202Q8×C83—41no—
203D4×C83—21no—
204D4×C83—41no—
205D4×C83—42no—
206D4×C83—44no—
207D4×C83—81no—
208D4×C83#10412yes—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—42no—
222C2×C4×C83—42no—
223C2×C4×C83—44no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—42no—
227C2×C4×C83—81no—
228C2×C4×C83—82no—
229C2×C4×C83—82no—
230C2×C4×C83—81no—
231C2×C4×C83—81no—
232C2×C4×C83C2×C22⋊C411yescharacteristic
233C2×C4×C83C2×C22⋊C411yescharacteristic
234C2×C4×C83C2×C22⋊C411yescharacteristic
235C2×C4×C83#12011yescharacteristic
236C2×C4×C83#12011yescharacteristic
237C2×C4×C83#12011yescharacteristic
238C8×C83—42no—
239C8×C83—81no—
240C8×C83—81no—
241C22×C83—21no—
242C22×C83—21no—
243C22×C83—21no—
244C22×C83—22no—
245C22×C83—41no—
246C22×C83—41no—
247C22×C83—41no—
248C22×C83—41no—
249C22×C83—41no—
250C22×C83#37911yescharacteristic
251C4×C83—21no—
252C4×C83—21no—
253C4×C83—21no—
254C4×C83—21no—
255C4×C83—21no—
256C4×C83—41no—
257C4×C83—41no—
258C4×C83—42no—
259C4×C83—41no—
260C4×C83—41no—
261C4×C83—41no—
262C4×C83—41no—
263C4×C83—42no—
264C4×C83—82no—
265C4×C83C2×#10411yescharacteristic
266C4×C83#42111yescharacteristic
267#8415C8311yesmaximal, characteristic, direct product factor, Sylow
268C2×C83—41no—
269C2×C83—42no—
270C2×C83—81no—
271C2×C83#178911yescharacteristic
272C2×C83#179411yescharacteristic
273C2×C83#179611yescharacteristic
274#3515C2×C8311yescharacteristic
275#1759C2×C8311yescharacteristic, complemented
276#1529C2×C8311yescharacteristic, complemented
277#1506C2×C8311yescharacteristic, complemented
278#1500C2×C8311yescharacteristic, complemented
279#1488C2×C8311yescharacteristic, complemented
280#1485C2×C8311yescharacteristic, complemented
281C83#841511yescharacteristic, direct product factor, Sylow
282#557C4×C8311yescharacteristic, complemented
283#554C4×C8311yescharacteristic, complemented
284#554C4×C8311yescharacteristic, complemented
285#553—21no—
286#552—21no—
287#552—22no—
288#552—21no—
289#547—21no—
290#547C4×C8311yescharacteristic, complemented
291#541—21no—
292#539—22no—
293#533C22×C8311yescharacteristic
294#529—21no—
295#529C22×C8311yescharacteristic
296C2×#125—21no—
297#520—21no—
298C4×#41C22×C8311yescharacteristic
299#432C22×C8311yescharacteristic
300#431C22×C8311yescharacteristic
301C2×#104—21no—
302C4×M16C22×C8311yescharacteristic
303#376C22×C8311yescharacteristic
304#339—21no—
305#330—21no—
306#329—21no—
307#328—21no—
308#328—22no—
309#327—21no—
310Q8⋊C8—22no—
311Q8⋊C8—21no—
312D4⋊C8—21no—
313C2×#41—21no—
314C2×#41D4×C8311yescharacteristic
315C2×M16—21no—
316C4⋊Q8—21no—
317C4⋊Q8C2×C4×C8311yescharacteristic
318C4⋊Q8C2×C4×C8311yescharacteristic
319#126—22no—
320#126—22no—
321#126—22no—
322#126—21no—
323#125—22no—
324#125—22no—
325#125—22no—
326#125—21no—
327#125—21no—
328#125—22no—
329#125—21no—
330#125—21no—
331#125D4×C8311yescharacteristic
332#124D4×C8311yescharacteristic
333#123—22no—
334#123—22no—
335#123—22no—
336#123—21no—
337#123—22no—
338C22⋊Q8—21no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8D4×C8311yescharacteristic
345C22⋊Q8D4×C8312yes—
346#121—22no—
347#121—21no—
348#121D4×C8311yescharacteristic
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8C2×C4×C8311yescharacteristic
353C4×Q8C2×C4×C8311yescharacteristic
354C4×Q8D4×C8311yescharacteristic
355C4×Q8D4×C8312yes—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C2×C4×C8311yescharacteristic
364C4×D4C2×C4×C8311yescharacteristic
365C4×D4D4×C8311yescharacteristic
366#117—22no—
367#117—22no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117D4×C8311yescharacteristic
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42C23×C8311yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382#104—24no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385#97—21no—
386#97—21no—
387C4×C8—21no—
388C4×C8—21no—
389#95—21no—
390#95—21no—
391#41—22no—
392#41—41no—
393#41—42no—
394#41—42no—
395C2×Q8—21no—
396C2×Q8C22⋊C4×C8311yescharacteristic
397C2×Q8C22⋊C4×C8311yescharacteristic
398C2×Q8C22⋊C4×C8312yes—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4C22⋊C4×C8311yescharacteristic
403C2×D4C22⋊C4×C8311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4C2×D4×C8311yescharacteristic
413C22×C4C2×D4×C8311yescharacteristic
414C22×C4C2×D4×C8311yescharacteristic
415M16—22no—
416M16—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4C22⋊C4×C8311yescharacteristic
440C4⋊C4C22⋊C4×C8311yescharacteristic
441C4⋊C4C22⋊C4×C8311yescharacteristic
442C4⋊C4C22⋊C4×C8311yescharacteristic
443C4⋊C4C22⋊C4×C8312yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42C22×C4×C8311yescharacteristic
465C42C2×D4×C8311yescharacteristic
466C42C2×D4×C8311yescharacteristic
467C42C2×D4×C8311yescharacteristic
468C23—21no—
469C23—21no—
470C23#120×C8311yescharacteristic
471Q8—21no—
472Q8—21no—
473Q8—42no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477D4—21no—
478D4—41no—
479D4—42no—
480D4—44no—
481D4—81no—
482D4#104×C8312yes—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4C2×C22⋊C4×C8311yescharacteristic
507C2×C4C2×C22⋊C4×C8311yescharacteristic
508C2×C4C2×C22⋊C4×C8311yescharacteristic
509C2×C4#120×C8311yescharacteristic
510C2×C4#120×C8311yescharacteristic
511C2×C4#120×C8311yescharacteristic
512C8—42no—
513C8—81no—
514C8—81no—
515C22—21no—
516C22—22no—
517C22—21no—
518C22—21no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22#379×C8311yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—41no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—82no—
539C4C2×#104×C8311yescharacteristic
540C4#421×C8311yescharacteristic
541C2—41no—
542C2—42no—
543C2—81no—
544C2C83×#178911yescharacteristic
545C2C83×#179411yescharacteristic
546C2C83×#179611yescharacteristic
subgroupG/H
center, socleC22×C83#379
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC83×#3515C2
LCSG ▷ C42 ▷ 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×#1759C2semidirect
3. C83×#1529C2semidirect
4. C83×#1506C2semidirect
5. C83×#1500C2semidirect
6. C83×#1488C2semidirect
7. C83×#1485C2semidirect
8. #557×C83C4semidirect
9. #554×C83C4semidirect
10. #554×C83C4semidirect
16. #547×C83C4semidirect
267. #8415C83direct (N × H)
275. #1759C2×C83semidirect
276. #1529C2×C83semidirect
277. #1506C2×C83semidirect
278. #1500C2×C83semidirect
279. #1488C2×C83semidirect
280. #1485C2×C83semidirect
281. C83#8415direct (N × H)
282. #557C4×C83semidirect
283. #554C4×C83semidirect
284. #554C4×C83semidirect
290. #547C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35152
C83×#17592
C83×#15292
C83×#15062
C83×#15002
C83×#14882
C83×#14852
#841583