Finite Group Databasebeta

← C83×#8417 #35236241

Subgroups — C83×#8417 #35236241

Order 21248.

706 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3523C211yesmaximal, characteristic
2C83×#1986C211yesmaximal, characteristic, complemented
3C83×#1763C211yesmaximal, characteristic, complemented
4C83×#1531C211yesmaximal, characteristic, complemented
5C83×#1524C211yesmaximal, characteristic
6C83×#1509C211yesmaximal, characteristic
7C83×#1491C211yesmaximal, characteristic, complemented
8#557×C83—21no—
9#557×C83C411yescharacteristic, complemented
10#557×C83C411yescharacteristic, complemented
11#556×C83—21no—
12#556×C83C411yescharacteristic, complemented
13#552×C83—22no—
14#552×C83—21no—
15D4×Q8×C83—21no—
16D4×Q8×C83C411yescharacteristic, complemented
17#541×C83—22no—
18#541×C83—21no—
19#533×C83—21no—
20#533×C83C2211yescharacteristic
21#529×C83—21no—
22#529×C83C2211yescharacteristic
23#519×C83—21no—
24C2×C4×Q8×C83C2211yescharacteristic
25#432×C83C2211yescharacteristic
26C2×#103×C83—21no—
27C4×M16×C83C2211yescharacteristic
28C2×#97×C83C2211yescharacteristic
29C4×C4⋊C4×C83C2211yescharacteristic
30#340×C83—21no—
31#339×C83—22no—
32#330×C83—21no—
33#330×C83—22no—
34#329×C83—21no—
35#328×C83—21no—
36#327×C83—21no—
37Q8⋊C8×C83—21no—
38Q8⋊C8×C83—21no—
39C22×Q8×C83—21no—
40C22×Q8×C83D411yescharacteristic
41C2×M16×C83—21no—
42C22×C8×C83—21no—
43C4⋊Q8×C83—22no—
44C4⋊Q8×C83—21no—
45C4⋊Q8×C83—21no—
46C4⋊Q8×C83—21no—
47C4⋊Q8×C83C2×C411yescharacteristic
48C4⋊Q8×C83C2×C411yescharacteristic
49#126×C83—22no—
50#126×C83—22no—
51#126×C83—22no—
52#126×C83—22no—
53#126×C83—21no—
54#125×C83—22no—
55#125×C83—22no—
56#125×C83—22no—
57#125×C83—21no—
58#125×C83—22no—
59#125×C83—22no—
60#125×C83—21no—
61#125×C83D411yescharacteristic
62#125×C83D412yes—
63#124×C83—21no—
64#124×C83D411yescharacteristic
65#123×C83—22no—
66#123×C83—21no—
67C22⋊Q8×C83—22no—
68C22⋊Q8×C83—22no—
69C22⋊Q8×C83—21no—
70C22⋊Q8×C83—21no—
71C22⋊Q8×C83—21no—
72C22⋊Q8×C83—21no—
73C22⋊Q8×C83—22no—
74C22⋊Q8×C83D411yescharacteristic
75#121×C83D411yescharacteristic
76C4×Q8×C83—22no—
77C4×Q8×C83—22no—
78C4×Q8×C83—21no—
79C4×Q8×C83—21no—
80C4×Q8×C83—21no—
81C4×Q8×C83—21no—
82C4×Q8×C83—21no—
83C4×Q8×C83—21no—
84C4×Q8×C83—21no—
85C4×Q8×C83—21no—
86C4×Q8×C83—21no—
87C4×Q8×C83C2×C411yescharacteristic
88C4×Q8×C83C2×C411yescharacteristic
89C4×Q8×C83C2×C411yescharacteristic
90C4×Q8×C83D411yescharacteristic
91C4×Q8×C83D411yescharacteristic
92C4×D4×C83—22no—
93C4×D4×C83—21no—
94C4×D4×C83—21no—
95C4×D4×C83C2×C411yescharacteristic
96#117×C83—22no—
97#117×C83—22no—
98#117×C83—22no—
99#117×C83—21no—
100#117×C83—21no—
101#117×C83D412yes—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83D411yescharacteristic
105C2×C42×C83—21no—
106C2×C42×C83—21no—
107C2×C42×C83C2311yescharacteristic
108C4⋊C8×C83—21no—
109#103×C83—24no—
110C22⋊C8×C83—21no—
111#97×C83—21no—
112#97×C83—21no—
113#97×C83—21no—
114#97×C83—21no—
115C4×C8×C83—21no—
116#95×C83—21no—
117C2×Q8×C83—22no—
118C2×Q8×C83—21no—
119C2×Q8×C83—21no—
120C2×Q8×C83—21no—
121C2×Q8×C83—22no—
122C2×Q8×C83—41no—
123C2×Q8×C83—42no—
124C2×Q8×C83—42no—
125C2×Q8×C83C22⋊C411yescharacteristic
126C2×Q8×C83C22⋊C411yescharacteristic
127C2×Q8×C83C22⋊C411yescharacteristic
128C2×D4×C83—21no—
129C2×D4×C83C22⋊C411yescharacteristic
130C22×C4×C83—22no—
131C22×C4×C83—21no—
132C22×C4×C83—21no—
133C22×C4×C83—21no—
134C22×C4×C83—41no—
135C22×C4×C83C2×D411yescharacteristic
136C22×C4×C83C2×D411yescharacteristic
137C22×C4×C83C2×D411yescharacteristic
138M16×C83—22no—
139M16×C83—41no—
140C2×C8×C83—22no—
141C2×C8×C83—41no—
142C2×C8×C83—41no—
143C2×C8×C83—41no—
144C2×C8×C83—41no—
145C4⋊C4×C83—22no—
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—21no—
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—21no—
163C4⋊C4×C83—21no—
164C4⋊C4×C83—21no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83—22no—
167C4⋊C4×C83—41no—
168C4⋊C4×C83C22⋊C411yescharacteristic
169C4⋊C4×C83C22⋊C411yescharacteristic
170C4⋊C4×C83C22⋊C412yes—
171C4⋊C4×C83C22⋊C412yes—
172C4⋊C4×C83C22⋊C411yescharacteristic
173C4⋊C4×C83C22⋊C411yescharacteristic
174C22⋊C4×C83—22no—
175C22⋊C4×C83—22no—
176C22⋊C4×C83—22no—
177C22⋊C4×C83—21no—
178C22⋊C4×C83—22no—
179C22⋊C4×C83—21no—
180C22⋊C4×C83—21no—
181C22⋊C4×C83—21no—
182C22⋊C4×C83—21no—
183C42×C83—22no—
184C42×C83—22no—
185C42×C83—22no—
186C42×C83—21no—
187C42×C83—21no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—22no—
192C42×C83—42no—
193C42×C83C22×C411yescharacteristic
194C42×C83C2×D411yescharacteristic
195C42×C83C2×D411yescharacteristic
196C42×C83C2×D411yescharacteristic
197C23×C83—21no—
198C23×C83#12011yescharacteristic
199Q8×C83—21no—
200Q8×C83—21no—
201Q8×C83—21no—
202Q8×C83—42no—
203Q8×C83—41no—
204Q8×C83—42no—
205Q8×C83—44no—
206Q8×C83—41no—
207Q8×C83—41no—
208Q8×C83#10412yes—
209D4×C83—41no—
210D4×C83—41no—
211D4×C83—41no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—22no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—22no—
225C2×C4×C83—41no—
226C2×C4×C83—42no—
227C2×C4×C83—44no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—42no—
233C2×C4×C83—81no—
234C2×C4×C83—81no—
235C2×C4×C83C2×C22⋊C411yescharacteristic
236C2×C4×C83C2×C22⋊C411yescharacteristic
237C2×C4×C83C2×C22⋊C411yescharacteristic
238C2×C4×C83#12011yescharacteristic
239C2×C4×C83#12011yescharacteristic
240C2×C4×C83#12011yescharacteristic
241C8×C83—42no—
242C8×C83—42no—
243C8×C83—81no—
244C22×C83—21no—
245C22×C83—21no—
246C22×C83—41no—
247C22×C83—41no—
248C22×C83—41no—
249C22×C83#37911yescharacteristic
250C22×C83#45411yescharacteristic
251C22×C83#45611yescharacteristic
252C4×C83—21no—
253C4×C83—21no—
254C4×C83—21no—
255C4×C83—21no—
256C4×C83—21no—
257C4×C83—22no—
258C4×C83—41no—
259C4×C83—41no—
260C4×C83—41no—
261C4×C83—41no—
262C4×C83—42no—
263C4×C83—41no—
264C4×C83—42no—
265C4×C83—41no—
266C4×C83—82no—
267C4×C83C2×#10411yescharacteristic
268C4×C83#42111yescharacteristic
269#8417C8311yesmaximal, characteristic, direct product factor, Sylow
270C2×C83—21no—
271C2×C83—21no—
272C2×C83—81no—
273C2×C83#178911yescharacteristic
274C2×C83#179311yescharacteristic
275C2×C83#179611yescharacteristic
276#3523C2×C8311yescharacteristic
277#1986C2×C8311yescharacteristic, complemented
278#1763C2×C8311yescharacteristic, complemented
279#1531C2×C8311yescharacteristic, complemented
280#1524C2×C8311yescharacteristic
281#1509C2×C8311yescharacteristic
282#1491C2×C8311yescharacteristic, complemented
283C83#841711yescharacteristic, direct product factor, Sylow
284#557—21no—
285#557C4×C8311yescharacteristic, complemented
286#557C4×C8311yescharacteristic, complemented
287#556—21no—
288#556C4×C8311yescharacteristic, complemented
289#552—22no—
290#552—21no—
291D4×Q8—21no—
292D4×Q8C4×C8311yescharacteristic, complemented
293#541—22no—
294#541—21no—
295#533—21no—
296#533C22×C8311yescharacteristic
297#529—21no—
298#529C22×C8311yescharacteristic
299#519—21no—
300C2×C4×Q8C22×C8311yescharacteristic
301#432C22×C8311yescharacteristic
302C2×#103—21no—
303C4×M16C22×C8311yescharacteristic
304C2×#97C22×C8311yescharacteristic
305C4×C4⋊C4C22×C8311yescharacteristic
306#340—21no—
307#339—22no—
308#330—21no—
309#330—22no—
310#329—21no—
311#328—21no—
312#327—21no—
313Q8⋊C8—21no—
314Q8⋊C8—21no—
315C22×Q8—21no—
316C22×Q8D4×C8311yescharacteristic
317C2×M16—21no—
318C22×C8—21no—
319C4⋊Q8—22no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8C2×C4×C8311yescharacteristic
324C4⋊Q8C2×C4×C8311yescharacteristic
325#126—22no—
326#126—22no—
327#126—22no—
328#126—22no—
329#126—21no—
330#125—22no—
331#125—22no—
332#125—22no—
333#125—21no—
334#125—22no—
335#125—22no—
336#125—21no—
337#125D4×C8311yescharacteristic
338#125D4×C8312yes—
339#124—21no—
340#124D4×C8311yescharacteristic
341#123—22no—
342#123—21no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—21no—
346C22⋊Q8—21no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8D4×C8311yescharacteristic
351#121D4×C8311yescharacteristic
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8C2×C4×C8311yescharacteristic
364C4×Q8C2×C4×C8311yescharacteristic
365C4×Q8C2×C4×C8311yescharacteristic
366C4×Q8D4×C8311yescharacteristic
367C4×Q8D4×C8311yescharacteristic
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4C2×C4×C8311yescharacteristic
372#117—22no—
373#117—22no—
374#117—22no—
375#117—21no—
376#117—21no—
377#117D4×C8312yes—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4D4×C8311yescharacteristic
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C23×C8311yescharacteristic
384C4⋊C8—21no—
385#103—24no—
386C22⋊C8—21no—
387#97—21no—
388#97—21no—
389#97—21no—
390#97—21no—
391C4×C8—21no—
392#95—21no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—41no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8C22⋊C4×C8311yescharacteristic
402C2×Q8C22⋊C4×C8311yescharacteristic
403C2×Q8C22⋊C4×C8311yescharacteristic
404C2×D4—21no—
405C2×D4C22⋊C4×C8311yescharacteristic
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4C2×D4×C8311yescharacteristic
412C22×C4C2×D4×C8311yescharacteristic
413C22×C4C2×D4×C8311yescharacteristic
414M16—22no—
415M16—41no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—41no—
444C4⋊C4C22⋊C4×C8311yescharacteristic
445C4⋊C4C22⋊C4×C8311yescharacteristic
446C4⋊C4C22⋊C4×C8312yes—
447C4⋊C4C22⋊C4×C8312yes—
448C4⋊C4C22⋊C4×C8311yescharacteristic
449C4⋊C4C22⋊C4×C8311yescharacteristic
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—42no—
469C42C22×C4×C8311yescharacteristic
470C42C2×D4×C8311yescharacteristic
471C42C2×D4×C8311yescharacteristic
472C42C2×D4×C8311yescharacteristic
473C23—21no—
474C23#120×C8311yescharacteristic
475Q8—21no—
476Q8—21no—
477Q8—21no—
478Q8—42no—
479Q8—41no—
480Q8—44no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8#104×C8312yes—
485D4—41no—
486D4—41no—
487D4—41no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4C2×C22⋊C4×C8311yescharacteristic
512C2×C4C2×C22⋊C4×C8311yescharacteristic
513C2×C4C2×C22⋊C4×C8311yescharacteristic
514C2×C4#120×C8311yescharacteristic
515C2×C4#120×C8311yescharacteristic
516C2×C4#120×C8311yescharacteristic
517C8—42no—
518C8—42no—
519C8—81no—
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22#379×C8311yescharacteristic
526C22#454×C8311yescharacteristic
527C22#456×C8311yescharacteristic
528C4—21no—
529C4—22no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—42no—
541C4—41no—
542C4—82no—
543C4C2×#104×C8311yescharacteristic
544C4#421×C8311yescharacteristic
545C2—21no—
546C2—21no—
547C2—81no—
548C2C83×#178911yescharacteristic
549C2C83×#179311yescharacteristic
550C2C83×#179611yescharacteristic
subgroupG/H
center, socleC22×C83#379
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radical#121×C83D4
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×#1986C2semidirect
3. C83×#1763C2semidirect
4. C83×#1531C2semidirect
7. C83×#1491C2semidirect
9. #557×C83C4semidirect
10. #557×C83C4semidirect
12. #556×C83C4semidirect
16. D4×Q8×C83C4semidirect
269. #8417C83direct (N × H)
277. #1986C2×C83semidirect
278. #1763C2×C83semidirect
279. #1531C2×C83semidirect
282. #1491C2×C83semidirect
283. C83#8417direct (N × H)
285. #557C4×C83semidirect
286. #557C4×C83semidirect
288. #556C4×C83semidirect
292. D4×Q8C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35232
C83×#19862
C83×#17632
C83×#15312
C83×#15242
C83×#15092
C83×#14912
#841783