Finite Group Databasebeta

← C83×#24715 #35252539

Subgroups — C83×#24715 #35252539

Order 21248.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#479×C83C211yesmaximal, characteristic, complemented
2C2×#433×C83C211yesmaximal, characteristic
3C83×#2629C211yesmaximal, characteristic
4C83×#2323C211yesmaximal, characteristic
5C83×#2077C211yesmaximal, characteristic
6C83×#2075C211yesmaximal, characteristic
7C83×#2036C211yesmaximal, characteristic
8C83×#2036C211yesmaximal, characteristic
9C83×#1989C211yesmaximal, characteristic, complemented
10C83×#1947C211yesmaximal, characteristic, complemented
11C83×#1943C211yesmaximal, characteristic, complemented
12C83×#1942C211yesmaximal, characteristic, complemented
13C83×#1935C211yesmaximal, characteristic, complemented
14C83×#1933C211yesmaximal, characteristic, complemented
15C83×#1803C211yesmaximal, characteristic, complemented
16C2×#125×C83C2211yescharacteristic
17C2×C22⋊Q8×C83C2211yescharacteristic
18C2×C4×Q8×C83C2211yescharacteristic
19C2×#117×C83C2211yescharacteristic
20#479×C83—24no—
21#433×C83—24no—
22C2×#107×C83C2211yescharacteristic
23C2×#107×C83C2211yescharacteristic
24C2×#107×C83C2211yescharacteristic
25C2×#106×C83C2211yescharacteristic
26C2×C4⋊C8×C83C2211yescharacteristic
27C2×C4⋊C8×C83C2211yescharacteristic
28C2×#103×C83C2211yescharacteristic
29C2×#103×C83C2211yescharacteristic
30C2×#103×C83C2211yescharacteristic
31C2×#103×C83C2211yescharacteristic
32C2×C22⋊C8×C83C2211yescharacteristic
33C2×C22⋊C8×C83C2211yescharacteristic
34C2×#97×C83C2211yescharacteristic
35C2×C4×C8×C83C2211yescharacteristic
36#398×C83—21no—
37#398×C83—21no—
38#396×C83—21no—
39#396×C83—21no—
40#391×C83C2211yescharacteristic
41#389×C83—21no—
42#389×C83C2211yescharacteristic
43#389×C83C2211yescharacteristic
44#389×C83C2211yescharacteristic
45#388×C83—21no—
46#388×C83—21no—
47#387×C83—21no—
48#384×C83C2211yescharacteristic
49#380×C83C2211yescharacteristic
50#380×C83C2211yescharacteristic
51#380×C83C2211yescharacteristic
52C4×C4⋊C4×C83C2211yescharacteristic
53#340×C83C2211yescharacteristic
54#340×C83C2211yescharacteristic
55#340×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60#340×C83C2211yescharacteristic
61C22×Q8×C83C2311yescharacteristic
62C23×C4×C83C2311yescharacteristic
63C22×C8×C83C2311yescharacteristic
64C22×C8×C83C2311yescharacteristic
65C22×C8×C83C2311yescharacteristic
66C22×C8×C83C2311yescharacteristic
67#125×C83—22no—
68#125×C83—22no—
69C22⋊Q8×C83—22no—
70C22⋊Q8×C83—22no—
71C4×Q8×C83—22no—
72C4×Q8×C83—22no—
73#117×C83—22no—
74#117×C83—22no—
75C2×C4⋊C4×C83—21no—
76C2×C4⋊C4×C83—21no—
77C2×C4⋊C4×C83—21no—
78C2×C4⋊C4×C83—21no—
79C2×C4⋊C4×C83—21no—
80C2×C4⋊C4×C83—21no—
81C2×C4⋊C4×C83—21no—
82C2×C4⋊C4×C83—21no—
83C2×C4⋊C4×C83—21no—
84C2×C4⋊C4×C83D411yescharacteristic
85C2×C4⋊C4×C83D411yescharacteristic
86C2×C4⋊C4×C83C2311yescharacteristic
87C2×C4⋊C4×C83C2311yescharacteristic
88C2×C4⋊C4×C83C2311yescharacteristic
89C2×C4⋊C4×C83C2311yescharacteristic
90C2×C4⋊C4×C83C2311yescharacteristic
91C2×C4⋊C4×C83C2311yescharacteristic
92C2×C4⋊C4×C83C2311yescharacteristic
93C2×C22⋊C4×C83—21no—
94C2×C22⋊C4×C83—21no—
95C2×C22⋊C4×C83—21no—
96C2×C22⋊C4×C83—21no—
97C2×C22⋊C4×C83D411yescharacteristic
98C2×C22⋊C4×C83D411yescharacteristic
99C2×C42×C83—21no—
100C2×C42×C83—21no—
101C2×C42×C83C2311yescharacteristic
102C2×C42×C83C2311yescharacteristic
103#107×C83—22no—
104#107×C83—22no—
105#107×C83—22no—
106#106×C83—22no—
107C4⋊C8×C83—22no—
108C4⋊C8×C83D414yes—
109#103×C83—22no—
110#103×C83—22no—
111#103×C83—22no—
112#103×C83—22no—
113C22⋊C8×C83—22no—
114C22⋊C8×C83—22no—
115#97×C83—22no—
116C4×C8×C83—22no—
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125#95×C83—21no—
126C24×C83C2×D411yescharacteristic
127C2×Q8×C83—21no—
128C2×Q8×C83—21no—
129C2×Q8×C83—21no—
130C2×Q8×C83—21no—
131C2×Q8×C83—21no—
132C2×Q8×C83#4112yes—
133C22×C4×C83—21no—
134C22×C4×C83—21no—
135C22×C4×C83—21no—
136C22×C4×C83—21no—
137C22×C4×C83—21no—
138C22×C4×C83—21no—
139C22×C4×C83—21no—
140C22×C4×C83—21no—
141C22×C4×C83—21no—
142C22×C4×C83—21no—
143C22×C4×C83—21no—
144C22×C4×C83—21no—
145C22×C4×C83—21no—
146C22×C4×C83—21no—
147C22×C4×C83C2×D411yescharacteristic
148C22×C4×C83C2×D411yescharacteristic
149C22×C4×C83C2×D411yescharacteristic
150C22×C4×C83C2×D411yescharacteristic
151C22×C4×C83C2×D411yescharacteristic
152C22×C4×C83C2411yescharacteristic
153C2×C8×C83—21no—
154C2×C8×C83—21no—
155C2×C8×C83—21no—
156C2×C8×C83—21no—
157C2×C8×C83—21no—
158C2×C8×C83—21no—
159C2×C8×C83—21no—
160C2×C8×C83—21no—
161C2×C8×C83—21no—
162C2×C8×C83—21no—
163C2×C8×C83C2×D412yes—
164C2×C8×C83C2×D412yes—
165C4⋊C4×C83—24no—
166C4⋊C4×C83—22no—
167C4⋊C4×C83—22no—
168C4⋊C4×C83—21no—
169C4⋊C4×C83—21no—
170C4⋊C4×C83—21no—
171C4⋊C4×C83—21no—
172C4⋊C4×C83—21no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—42no—
182C4⋊C4×C83—42no—
183C4⋊C4×C83—42no—
184C4⋊C4×C83—81no—
185C4⋊C4×C83—81no—
186C4⋊C4×C83—81no—
187C4⋊C4×C83—81no—
188C4⋊C4×C83—81no—
189C4⋊C4×C83#4112yes—
190C22⋊C4×C83—24no—
191C22⋊C4×C83—22no—
192C22⋊C4×C83—22no—
193C22⋊C4×C83—81no—
194C22⋊C4×C83—81no—
195C22⋊C4×C83—81no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—42no—
199C42×C83—81no—
200C42×C83C2×D412yes—
201C42×C83#4112yes—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83C22×D411yescharacteristic
210Q8×C83—22no—
211Q8×C83—22no—
212Q8×C83—41no—
213Q8×C83—41no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83#12112yes—
256C2×C4×C83#12112yes—
257C2×C4×C83C22×D411yescharacteristic
258C2×C4×C83C2×#4111yescharacteristic
259C2×C4×C83C2×#4111yescharacteristic
260C2×C4×C83C2×#4111yescharacteristic
261C2×C4×C832−1+411yescharacteristic
262C2×C4×C832−1+411yescharacteristic
263C8×C83—22no—
264C8×C83—22no—
265C8×C83—41no—
266C8×C83—41no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83C2×#12111yescharacteristic
275C22×C83#52511yescharacteristic
276C22×C83#53611yescharacteristic
277C22×C83#57611yescharacteristic
278C22×C83#57711yescharacteristic
279C22×C83#57811yescharacteristic
280C22×C83#57811yescharacteristic
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—42no—
284C4×C83—41no—
285C4×C83—42no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83—81no—
289C4×C83—81no—
290C4×C83—81no—
291C4×C83—81no—
292C4×C83—81no—
293C4×C83—81no—
294C4×C83#54711yescharacteristic
295C4×C83#54811yescharacteristic
296C4×C83#55211yescharacteristic
297C4×C83#55611yescharacteristic
298#24715C8311yesmaximal, characteristic, direct product factor, Sylow
299C2×C83—81no—
300C2×C83#241511yescharacteristic
301C2×C83#306111yescharacteristic
302C2×C83#306311yescharacteristic
303C2×C83#311911yescharacteristic
304C2×C83#312711yescharacteristic
305C2×C83#324211yescharacteristic
306C2×C83#324311yescharacteristic
307C2×#479C2×C8311yescharacteristic, complemented
308C2×#433C2×C8311yescharacteristic
309#2629C2×C8311yescharacteristic
310#2323C2×C8311yescharacteristic
311#2077C2×C8311yescharacteristic
312#2075C2×C8311yescharacteristic
313#2036C2×C8311yescharacteristic
314#2036C2×C8311yescharacteristic
315#1989C2×C8311yescharacteristic, complemented
316#1947C2×C8311yescharacteristic, complemented
317#1943C2×C8311yescharacteristic, complemented
318#1942C2×C8311yescharacteristic, complemented
319#1935C2×C8311yescharacteristic, complemented
320#1933C2×C8311yescharacteristic, complemented
321#1803C2×C8311yescharacteristic, complemented
322C83#2471511yescharacteristic, direct product factor, Sylow
323C2×#125C22×C8311yescharacteristic
324C2×C22⋊Q8C22×C8311yescharacteristic
325C2×C4×Q8C22×C8311yescharacteristic
326C2×#117C22×C8311yescharacteristic
327#479—24no—
328#433—24no—
329C2×#107C22×C8311yescharacteristic
330C2×#107C22×C8311yescharacteristic
331C2×#107C22×C8311yescharacteristic
332C2×#106C22×C8311yescharacteristic
333C2×C4⋊C8C22×C8311yescharacteristic
334C2×C4⋊C8C22×C8311yescharacteristic
335C2×#103C22×C8311yescharacteristic
336C2×#103C22×C8311yescharacteristic
337C2×#103C22×C8311yescharacteristic
338C2×#103C22×C8311yescharacteristic
339C2×C22⋊C8C22×C8311yescharacteristic
340C2×C22⋊C8C22×C8311yescharacteristic
341C2×#97C22×C8311yescharacteristic
342C2×C4×C8C22×C8311yescharacteristic
343#398—21no—
344#398—21no—
345#396—21no—
346#396—21no—
347#391C22×C8311yescharacteristic
348#389—21no—
349#389C22×C8311yescharacteristic
350#389C22×C8311yescharacteristic
351#389C22×C8311yescharacteristic
352#388—21no—
353#388—21no—
354#387—21no—
355#384C22×C8311yescharacteristic
356#380C22×C8311yescharacteristic
357#380C22×C8311yescharacteristic
358#380C22×C8311yescharacteristic
359C4×C4⋊C4C22×C8311yescharacteristic
360#340C22×C8311yescharacteristic
361#340C22×C8311yescharacteristic
362#340C22×C8311yescharacteristic
363#340C22×C8311yescharacteristic
364#340C22×C8311yescharacteristic
365#340C22×C8311yescharacteristic
366#340C22×C8311yescharacteristic
367#340C22×C8311yescharacteristic
368C22×Q8C23×C8311yescharacteristic
369C23×C4C23×C8311yescharacteristic
370C22×C8C23×C8311yescharacteristic
371C22×C8C23×C8311yescharacteristic
372C22×C8C23×C8311yescharacteristic
373C22×C8C23×C8311yescharacteristic
374#125—22no—
375#125—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C4×Q8—22no—
379C4×Q8—22no—
380#117—22no—
381#117—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4D4×C8311yescharacteristic
392C2×C4⋊C4D4×C8311yescharacteristic
393C2×C4⋊C4C23×C8311yescharacteristic
394C2×C4⋊C4C23×C8311yescharacteristic
395C2×C4⋊C4C23×C8311yescharacteristic
396C2×C4⋊C4C23×C8311yescharacteristic
397C2×C4⋊C4C23×C8311yescharacteristic
398C2×C4⋊C4C23×C8311yescharacteristic
399C2×C4⋊C4C23×C8311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4D4×C8311yescharacteristic
405C2×C22⋊C4D4×C8311yescharacteristic
406C2×C42—21no—
407C2×C42—21no—
408C2×C42C23×C8311yescharacteristic
409C2×C42C23×C8311yescharacteristic
410#107—22no—
411#107—22no—
412#107—22no—
413#106—22no—
414C4⋊C8—22no—
415C4⋊C8D4×C8314yes—
416#103—22no—
417#103—22no—
418#103—22no—
419#103—22no—
420C22⋊C8—22no—
421C22⋊C8—22no—
422#97—22no—
423C4×C8—22no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433C24C2×D4×C8311yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8#41×C8312yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4C2×D4×C8311yescharacteristic
455C22×C4C2×D4×C8311yescharacteristic
456C22×C4C2×D4×C8311yescharacteristic
457C22×C4C2×D4×C8311yescharacteristic
458C22×C4C2×D4×C8311yescharacteristic
459C22×C4C24×C8311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8C2×D4×C8312yes—
471C2×C8C2×D4×C8312yes—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4#41×C8312yes—
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—81no—
507C42C2×D4×C8312yes—
508C42#41×C8312yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×D4×C8311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4#121×C8312yes—
563C2×C4#121×C8312yes—
564C2×C4C22×D4×C8311yescharacteristic
565C2×C4C2×#41×C8311yescharacteristic
566C2×C4C2×#41×C8311yescharacteristic
567C2×C4C2×#41×C8311yescharacteristic
568C2×C42−1+4×C8311yescharacteristic
569C2×C42−1+4×C8311yescharacteristic
570C8—22no—
571C8—22no—
572C8—41no—
573C8—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#121×C8311yescharacteristic
582C22#525×C8311yescharacteristic
583C22#536×C8311yescharacteristic
584C22#576×C8311yescharacteristic
585C22#577×C8311yescharacteristic
586C22#578×C8311yescharacteristic
587C22#578×C8311yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4#547×C8311yescharacteristic
602C4#548×C8311yescharacteristic
603C4#552×C8311yescharacteristic
604C4#556×C8311yescharacteristic
605C2—81no—
606C2C83×#241511yescharacteristic
607C2C83×#306111yescharacteristic
608C2C83×#306311yescharacteristic
609C2C83×#311911yescharacteristic
610C2C83×#312711yescharacteristic
611C2C83×#324211yescharacteristic
612C2C83×#324311yescharacteristic
subgroupG/H
center, socleC23×C83C22×D4
commutator, Frattini, cosocleC22×C4C24×C83
prime radicalC24×C83C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C83 ◁ C2×#117×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#479×C83C2semidirect
9. C83×#1989C2semidirect
10. C83×#1947C2semidirect
11. C83×#1943C2semidirect
12. C83×#1942C2semidirect
13. C83×#1935C2semidirect
14. C83×#1933C2semidirect
15. C83×#1803C2semidirect
298. #24715C83direct (N × H)
307. C2×#479C2×C83semidirect
315. #1989C2×C83semidirect
316. #1947C2×C83semidirect
317. #1943C2×C83semidirect
318. #1942C2×C83semidirect
319. #1935C2×C83semidirect
320. #1933C2×C83semidirect
321. #1803C2×C83semidirect
322. C83#24715direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#479×C832
C2×#433×C832
C83×#26292
C83×#23232
C83×#20772
C83×#20752
C83×#20362
C83×#19892
C83×#19472
C83×#19432
C83×#19422
C83×#19352
C83×#19332
C83×#18032
#2471583