Finite Group Databasebeta

← C83×#14506 #35242330

Subgroups — C83×#14506 #35242330

Order 21248.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C83C211yesmaximal, characteristic
2C2×#388×C83C212yesmaximal, complemented
3C83×#1445C212yesmaximal, complemented
4C83×#1439C212yesmaximal, complemented
5C83×#1438C211yesmaximal, characteristic
6C83×#1438C211yesmaximal, characteristic
7C83×#1437C212yesmaximal, complemented
8C83×C4⋊#95C212yesmaximal, complemented
9C83×C22⋊#95C212yesmaximal, complemented
10C23×D4×C83C2211yescharacteristic
11C24×C4×C83C2211yescharacteristic
12C2×#123×C83—28no—
13C2×#123×C83C418yescomplemented
14C22×C4⋊C4×C83C2212yes—
15C22×C4⋊C4×C83C2212yes—
16C22×C22⋊C4×C83C2212yes—
17C22×C22⋊C4×C83C2212yes—
18C22×C22⋊C4×C83C2211yescharacteristic
19C22×C22⋊C4×C83C2212yes—
20C22×C22⋊C4×C83C2212yes—
21C22×C22⋊C4×C83C2211yescharacteristic
22C22×C22⋊C4×C83C2211yescharacteristic
23C22×C42×C83C2212yes—
24#388×C83—28no—
25C2×#95×C83C2212yes—
26C2×#95×C83C2212yes—
27C2×#95×C83C2212yes—
28C2×#95×C83C2212yes—
29C2×#95×C83C2212yes—
30C2×#95×C83C2212yes—
31C2×#95×C83C2211yescharacteristic
32C2×#95×C83C2212yes—
33C2×#95×C83C2211yescharacteristic
34C25×C83C2311yescharacteristic
35C25×C83C2311yescharacteristic
36C22×D4×C83—24no—
37C22×D4×C83—42no—
38C22×D4×C83—41no—
39C22×D4×C83—41no—
40C22×D4×C83C2×C414yes—
41C23×C4×C83—24no—
42C23×C4×C83—42no—
43C23×C4×C83—41no—
44C23×C4×C83—41no—
45C23×C4×C83C2×C414yes—
46C23×C4×C83C2312yes—
47C23×C4×C83C2312yes—
48C23×C4×C83C2312yes—
49C23×C4×C83C2311yescharacteristic
50C23×C4×C83C2312yes—
51C23×C4×C83C2312yes—
52C23×C4×C83C2312yes—
53#123×C83—232no—
54C2×C4⋊C4×C83—24no—
55C2×C4⋊C4×C83—24no—
56C2×C4⋊C4×C83—24no—
57C2×C4⋊C4×C83—24no—
58C2×C4⋊C4×C83—42no—
59C2×C4⋊C4×C83C2×C418yes—
60C2×C22⋊C4×C83—24no—
61C2×C22⋊C4×C83—24no—
62C2×C22⋊C4×C83—24no—
63C2×C22⋊C4×C83—24no—
64C2×C22⋊C4×C83—24no—
65C2×C22⋊C4×C83—24no—
66C2×C22⋊C4×C83—24no—
67C2×C22⋊C4×C83—24no—
68C2×C22⋊C4×C83—41no—
69C2×C22⋊C4×C83—42no—
70C2×C22⋊C4×C83—42no—
71C2×C22⋊C4×C83—42no—
72C2×C22⋊C4×C83—42no—
73C2×C22⋊C4×C83—41no—
74C2×C22⋊C4×C83—42no—
75C2×C22⋊C4×C83C2×C418yes—
76C2×C22⋊C4×C83C2×C414yes—
77C2×C22⋊C4×C83D418yes—
78C2×C42×C83—24no—
79C2×C42×C83—24no—
80C2×C42×C83—42no—
81#95×C83—24no—
82#95×C83—24no—
83#95×C83—41no—
84#95×C83—42no—
85#95×C83—42no—
86#95×C83—42no—
87#95×C83—42no—
88#95×C83—42no—
89#95×C83—41no—
90C24×C83—22no—
91C24×C83—22no—
92C24×C83—22no—
93C24×C83—22no—
94C24×C83—21no—
95C24×C83—21no—
96C24×C83—22no—
97C24×C83—22no—
98C24×C83—22no—
99C24×C83—22no—
100C24×C83—22no—
101C24×C83—22no—
102C24×C83—21no—
103C24×C83—21no—
104C24×C83—22no—
105C24×C83C22×C412yes—
106C24×C83C22×C412yes—
107C24×C83C2×D414yes—
108C24×C83C2411yescharacteristic
109C2×D4×C83—28no—
110C2×D4×C83—44no—
111C2×D4×C83—44no—
112C2×D4×C83—48no—
113C2×D4×C83—48no—
114C22×C4×C83—22no—
115C22×C4×C83—22no—
116C22×C4×C83—22no—
117C22×C4×C83—22no—
118C22×C4×C83—28no—
119C22×C4×C83—22no—
120C22×C4×C83—22no—
121C22×C4×C83—22no—
122C22×C4×C83—22no—
123C22×C4×C83—22no—
124C22×C4×C83—22no—
125C22×C4×C83—22no—
126C22×C4×C83—22no—
127C22×C4×C83—22no—
128C22×C4×C83—22no—
129C22×C4×C83—22no—
130C22×C4×C83—22no—
131C22×C4×C83—22no—
132C22×C4×C83—22no—
133C22×C4×C83—22no—
134C22×C4×C83—22no—
135C22×C4×C83—21no—
136C22×C4×C83—21no—
137C22×C4×C83—22no—
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—22no—
141C22×C4×C83—22no—
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—44no—
156C22×C4×C83—44no—
157C22×C4×C83—48no—
158C22×C4×C83—48no—
159C22×C4×C83C22×C414yes—
160C22×C4×C83C22×C412yes—
161C22×C4×C83C22×C414yes—
162C22×C4×C83C2×D414yes—
163C22×C4×C83C2×D414yes—
164C22×C4×C83#4114yes—
165C22×C4×C83#4114yes—
166C4⋊C4×C83—216no—
167C4⋊C4×C83—48no—
168C22⋊C4×C83—216no—
169C22⋊C4×C83—44no—
170C22⋊C4×C83—48no—
171C22⋊C4×C83—48no—
172C22⋊C4×C83—48no—
173C22⋊C4×C83—44no—
174C22⋊C4×C83C22⋊C4116yescomplemented
175C42×C83—48no—
176C23×C83—24no—
177C23×C83—22no—
178C23×C83—22no—
179C23×C83—24no—
180C23×C83—24no—
181C23×C83—24no—
182C23×C83—24no—
183C23×C83—24no—
184C23×C83—24no—
185C23×C83—24no—
186C23×C83—22no—
187C23×C83—24no—
188C23×C83—22no—
189C23×C83—22no—
190C23×C83—24no—
191C23×C83—42no—
192C23×C83—42no—
193C23×C83—42no—
194C23×C83—42no—
195C23×C83—41no—
196C23×C83—42no—
197C23×C83—41no—
198C23×C83—41no—
199C23×C83—42no—
200C23×C83—41no—
201C23×C83—42no—
202C23×C83—42no—
203C23×C83—42no—
204C23×C83—42no—
205C23×C83—42no—
206C23×C83—42no—
207C23×C83—42no—
208C23×C83—42no—
209C23×C83—42no—
210C23×C83—42no—
211C23×C83—41no—
212C23×C83—42no—
213C23×C83—41no—
214C23×C83—41no—
215C23×C83—42no—
216C23×C83—41no—
217C23×C83C2×C22⋊C414yes—
218C23×C83C23×C411yescharacteristic
219C23×C83C22×D412yes—
220C23×C83C2×#4112yes—
221C23×C83C2×#4112yes—
222C23×C83H(5,𝔽2)12yes—
223C23×C83H(5,𝔽2)11yescharacteristic
224C23×C83H(5,𝔽2)11yescharacteristic
225C23×C83H(5,𝔽2)12yes—
226C23×C832−1+412yes—
227D4×C83—416no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—22no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—22no—
239C2×C4×C83—22no—
240C2×C4×C83—24no—
241C2×C4×C83—24no—
242C2×C4×C83—24no—
243C2×C4×C83—24no—
244C2×C4×C83—24no—
245C2×C4×C83—24no—
246C2×C4×C83—416no—
247C2×C4×C83—42no—
248C2×C4×C83—42no—
249C2×C4×C83—42no—
250C2×C4×C83—42no—
251C2×C4×C83—42no—
252C2×C4×C83—42no—
253C2×C4×C83—42no—
254C2×C4×C83—42no—
255C2×C4×C83—42no—
256C2×C4×C83—42no—
257C2×C4×C83—42no—
258C2×C4×C83—42no—
259C2×C4×C83—42no—
260C2×C4×C83—42no—
261C2×C4×C83—42no—
262C2×C4×C83—42no—
263C2×C4×C83—42no—
264C2×C4×C83—42no—
265C2×C4×C83—42no—
266C2×C4×C83—42no—
267C2×C4×C83—42no—
268C2×C4×C83—42no—
269C2×C4×C83—42no—
270C2×C4×C83—42no—
271C2×C4×C83—42no—
272C2×C4×C83—42no—
273C2×C4×C83—42no—
274C2×C4×C83—42no—
275C2×C4×C83C2×C22⋊C418yes—
276C22×C83—24no—
277C22×C83—28no—
278C22×C83—44no—
279C22×C83—42no—
280C22×C83—42no—
281C22×C83—44no—
282C22×C83—44no—
283C22×C83—44no—
284C22×C83—44no—
285C22×C83—42no—
286C22×C83—44no—
287C22×C83—44no—
288C22×C83—42no—
289C22×C83—42no—
290C22×C83—82no—
291C22×C83—82no—
292C22×C83—82no—
293C22×C83—82no—
294C22×C83C22×C22⋊C411yescharacteristic
295C22×C83C4×#4112yes—
296C22×C83#51811yescharacteristic
297C22×C83#51811yescharacteristic
298C22×C83#52012yes—
299C22×C83#53512yes—
300C22×C83#53512yes—
301C22×C83#53812yes—
302C22×C83#53912yes—
303C22×C83#54012yes—
304C22×C83#54212yes—
305C22×C83#54612yes—
306C22×C83#54612yes—
307C22×C83#54712yes—
308C22×C83#54712yes—
309C22×C83#55111yescharacteristic
310C22×C83#55312yes—
311C22×C83#55911yescharacteristic
312C22×C83#56011yescharacteristic
313C22×C83#56212yes—
314C22×C83#56311yescharacteristic
315C4×C83—28no—
316C4×C83—44no—
317C4×C83—42no—
318C4×C83—44no—
319C4×C83—44no—
320C4×C83—82no—
321C4×C83—82no—
322#14506C8311yesmaximal, characteristic, direct product factor, Sylow
323C2×C83—44no—
324C2×C83—82no—
325C2×C83#230911yescharacteristic
326C2×C83#233512yes—
327C2×C83#233912yes—
328C2×C83#237411yescharacteristic
329C2×C83#237611yescharacteristic
330C2×C83#270312yes—
331C2×C83#270612yes—
332C2×C83#271012yes—
333C2×C83#271812yes—
334C22×#123C2×C8311yescharacteristic
335C2×#388C2×C8312yescomplemented
336#1445C2×C8312yescomplemented
337#1439C2×C8312yescomplemented
338#1438C2×C8311yescharacteristic
339#1438C2×C8311yescharacteristic
340#1437C2×C8312yescomplemented
341C4⋊#95C2×C8312yescomplemented
342C22⋊#95C2×C8312yescomplemented
343C83#1450611yescharacteristic, direct product factor, Sylow
344C23×D4C22×C8311yescharacteristic
345C24×C4C22×C8311yescharacteristic
346C2×#123—28no—
347C2×#123C4×C8318yescomplemented
348C22×C4⋊C4C22×C8312yes—
349C22×C4⋊C4C22×C8312yes—
350C22×C22⋊C4C22×C8312yes—
351C22×C22⋊C4C22×C8312yes—
352C22×C22⋊C4C22×C8311yescharacteristic
353C22×C22⋊C4C22×C8312yes—
354C22×C22⋊C4C22×C8312yes—
355C22×C22⋊C4C22×C8311yescharacteristic
356C22×C22⋊C4C22×C8311yescharacteristic
357C22×C42C22×C8312yes—
358#388—28no—
359C2×#95C22×C8312yes—
360C2×#95C22×C8312yes—
361C2×#95C22×C8312yes—
362C2×#95C22×C8312yes—
363C2×#95C22×C8312yes—
364C2×#95C22×C8312yes—
365C2×#95C22×C8311yescharacteristic
366C2×#95C22×C8312yes—
367C2×#95C22×C8311yescharacteristic
368C25C23×C8311yescharacteristic
369C25C23×C8311yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×C4×C8314yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4C2×C4×C8314yes—
380C23×C4C23×C8312yes—
381C23×C4C23×C8312yes—
382C23×C4C23×C8312yes—
383C23×C4C23×C8311yescharacteristic
384C23×C4C23×C8312yes—
385C23×C4C23×C8312yes—
386C23×C4C23×C8312yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4C2×C4×C8318yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—41no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4C2×C4×C8318yes—
410C2×C22⋊C4C2×C4×C8314yes—
411C2×C22⋊C4D4×C8318yes—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—42no—
415#95—24no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—41no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C8312yes—
440C24C22×C4×C8312yes—
441C24C2×D4×C8314yes—
442C24C24×C8311yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C8314yes—
494C22×C4C22×C4×C8312yes—
495C22×C4C22×C4×C8314yes—
496C22×C4C2×D4×C8314yes—
497C22×C4C2×D4×C8314yes—
498C22×C4#41×C8314yes—
499C22×C4#41×C8314yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C83116yescomplemented
509C42—48no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C8314yes—
552C23C23×C4×C8311yescharacteristic
553C23C22×D4×C8312yes—
554C23C2×#41×C8312yes—
555C23C2×#41×C8312yes—
556C23H(5,𝔽2)×C8312yes—
557C23H(5,𝔽2)×C8311yescharacteristic
558C23H(5,𝔽2)×C8311yescharacteristic
559C23H(5,𝔽2)×C8312yes—
560C232−1+4×C8312yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C8318yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C8311yescharacteristic
629C22C4×#41×C8312yes—
630C22#518×C8311yescharacteristic
631C22#518×C8311yescharacteristic
632C22#520×C8312yes—
633C22#535×C8312yes—
634C22#535×C8312yes—
635C22#538×C8312yes—
636C22#539×C8312yes—
637C22#540×C8312yes—
638C22#542×C8312yes—
639C22#546×C8312yes—
640C22#546×C8312yes—
641C22#547×C8312yes—
642C22#547×C8312yes—
643C22#551×C8311yescharacteristic
644C22#553×C8312yes—
645C22#559×C8311yescharacteristic
646C22#560×C8311yescharacteristic
647C22#562×C8312yes—
648C22#563×C8311yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C83×#230911yescharacteristic
659C2C83×#233512yes—
660C2C83×#233912yes—
661C2C83×#237411yescharacteristic
662C2C83×#237611yescharacteristic
663C2C83×#270312yes—
664C2C83×#270612yes—
665C2C83×#271012yes—
666C2C83×#271812yes—
subgroupG/H
center, socleC24×C83C24
commutatorC23C23×C4×C83
Frattini, cosocleC24C24×C83
prime radicalC23×D4×C83C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#388×C83C2semidirect
3. C83×#1445C2semidirect
4. C83×#1439C2semidirect
7. C83×#1437C2semidirect
8. C83×C4⋊#95C2semidirect
9. C83×C22⋊#95C2semidirect
13. C2×#123×C83C4semidirect
174. C22⋊C4×C83C22⋊C4semidirect
322. #14506C83direct (N × H)
335. C2×#388C2×C83semidirect
336. #1445C2×C83semidirect
337. #1439C2×C83semidirect
340. #1437C2×C83semidirect
341. C4⋊#95C2×C83semidirect
342. C22⋊#95C2×C83semidirect
343. C83#14506direct (N × H)
347. C2×#123C4×C83semidirect
508. C22⋊C4C22⋊C4×C83semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C832
C2×#388×C832
C83×#14452
C83×#14392
C83×#14382
C83×#14372
C83×C4⋊#952
C83×C22⋊#952
#1450683