Finite Group Databasebeta

← C83×#29542 #35257366

Subgroups — C83×#29542 #35257366

Order 21248.

746 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2319C211yesmaximal, characteristic, complemented
2C83×#2319C211yesmaximal, characteristic, complemented
3C83×#2088C211yesmaximal, characteristic, complemented
4C83×#2067C211yesmaximal, characteristic, complemented
5C83×#2065C211yesmaximal, characteristic, complemented
6C83×#2054C211yesmaximal, characteristic
7C83×#1926C211yesmaximal, characteristic, complemented
8C83×#1926C211yesmaximal, characteristic, complemented
9C83×#1920C211yesmaximal, characteristic, complemented
10C83×#1918C211yesmaximal, characteristic, complemented
11C83×#1879C211yesmaximal, characteristic
12C83×#1879C211yesmaximal, characteristic
13C83×#1864C211yesmaximal, characteristic, complemented
14C83×#1862C211yesmaximal, characteristic, complemented
15C83×#1832C211yesmaximal, characteristic, complemented
16C22×D8×C83C2211yescharacteristic
17C2×C4×D4×C83C2211yescharacteristic
18C2×C4×D4×C83C2211yescharacteristic
19C2×#107×C83C2211yescharacteristic
20C2×#106×C83C2211yescharacteristic
21C2×#106×C83C2211yescharacteristic
22C2×C4⋊C8×C83C2211yescharacteristic
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×#102×C83C2211yescharacteristic
25C2×#102×C83C2211yescharacteristic
26C2×#102×C83C2211yescharacteristic
27C2×#102×C83C2211yescharacteristic
28C2×#102×C83C2211yescharacteristic
29C2×#102×C83C2211yescharacteristic
30#390×C83C2211yescharacteristic
31#390×C83C2211yescharacteristic
32#390×C83C2211yescharacteristic
33#390×C83C2211yescharacteristic
34#389×C83C2211yescharacteristic
35#389×C83C2211yescharacteristic
36#389×C83C2211yescharacteristic
37#389×C83C2211yescharacteristic
38#388×C83—21no—
39#388×C83—21no—
40#388×C83—21no—
41#388×C83—21no—
42#381×C83—21no—
43#381×C83—21no—
44#380×C83—21no—
45#380×C83—21no—
46C4×C4⋊C4×C83C2211yescharacteristic
47C4×C4⋊C4×C83C2211yescharacteristic
48#376×C83C2211yescharacteristic
49#340×C83C2211yescharacteristic
50#340×C83C2211yescharacteristic
51#340×C83C2211yescharacteristic
52#340×C83C2211yescharacteristic
53#340×C83C2211yescharacteristic
54#340×C83C2211yescharacteristic
55#336×C83C2211yescharacteristic
56#336×C83C2211yescharacteristic
57#336×C83C2211yescharacteristic
58#336×C83C2211yescharacteristic
59C22×D4×C83C2311yescharacteristic
60C22×D4×C83C2311yescharacteristic
61C23×C4×C83—21no—
62C23×C4×C83—21no—
63C2×D8×C83—22no—
64C2×D8×C83—22no—
65C2×D8×C83—41no—
66C22×C8×C83C2311yescharacteristic
67C22×C8×C83C2311yescharacteristic
68C22×C8×C83C2311yescharacteristic
69C22×C8×C83C2311yescharacteristic
70C4×D4×C83—22no—
71C4×D4×C83—22no—
72C4×D4×C83—22no—
73C4×D4×C83—22no—
74C2×C4⋊C4×C83—21no—
75C2×C4⋊C4×C83—21no—
76C2×C4⋊C4×C83—21no—
77C2×C4⋊C4×C83—21no—
78C2×C4⋊C4×C83C2311yescharacteristic
79C2×C4⋊C4×C83C2311yescharacteristic
80C2×C4⋊C4×C83C2311yescharacteristic
81C2×C4⋊C4×C83C2311yescharacteristic
82C2×C4⋊C4×C83C2311yescharacteristic
83C2×C4⋊C4×C83C2311yescharacteristic
84C2×C22⋊C4×C83—21no—
85C2×C22⋊C4×C83—21no—
86C2×C22⋊C4×C83—21no—
87C2×C22⋊C4×C83—21no—
88C2×C22⋊C4×C83—21no—
89C2×C22⋊C4×C83—21no—
90C2×C22⋊C4×C83—21no—
91C2×C22⋊C4×C83—21no—
92C2×C22⋊C4×C83—21no—
93C2×C22⋊C4×C83—21no—
94C2×C42×C83—21no—
95C2×C42×C83—21no—
96C2×C42×C83C2311yescharacteristic
97C2×C42×C83C2311yescharacteristic
98C2×C42×C83C2311yescharacteristic
99#107×C83—22no—
100#106×C83—22no—
101#106×C83—41no—
102C4⋊C8×C83—41no—
103C4⋊C8×C83—41no—
104#102×C83—22no—
105#102×C83—22no—
106#102×C83—22no—
107#102×C83—22no—
108#102×C83—41no—
109#102×C83—41no—
110#95×C83—21no—
111#95×C83—21no—
112#95×C83—21no—
113#95×C83—21no—
114#95×C83—21no—
115#95×C83—21no—
116#95×C83D411yescharacteristic
117#95×C83D411yescharacteristic
118#95×C83D411yescharacteristic
119#95×C83D411yescharacteristic
120C24×C83—21no—
121C24×C83—21no—
122C2×D4×C83—21no—
123C2×D4×C83—21no—
124C2×D4×C83—21no—
125C2×D4×C83—21no—
126C2×D4×C83—21no—
127C2×D4×C83—21no—
128C2×D4×C83—21no—
129C2×D4×C83—21no—
130C2×D4×C83—21no—
131C2×D4×C83—21no—
132C2×D4×C83#4112yes—
133C2×D4×C83#4112yes—
134C22×C4×C83—21no—
135C22×C4×C83—21no—
136C22×C4×C83—24no—
137C22×C4×C83—24no—
138C22×C4×C83—21no—
139C22×C4×C83—21no—
140C22×C4×C83—21no—
141C22×C4×C83—21no—
142C22×C4×C83—81no—
143C22×C4×C83—81no—
144C22×C4×C83—81no—
145C22×C4×C83—81no—
146C22×C4×C83C2×D411yescharacteristic
147C22×C4×C83C2×D411yescharacteristic
148C22×C4×C83C2×D411yescharacteristic
149C22×C4×C83C2×D411yescharacteristic
150C22×C4×C83C2×D411yescharacteristic
151C22×C4×C83C2×D411yescharacteristic
152C22×C4×C83C2411yescharacteristic
153D8×C83—44no—
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×C83—21no—
164C2×C8×C83—21no—
165C2×C8×C83—21no—
166C4⋊C4×C83—21no—
167C4⋊C4×C83—21no—
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—81no—
177C4⋊C4×C83—81no—
178C4⋊C4×C83—81no—
179C4⋊C4×C83—81no—
180C4⋊C4×C83#4112yes—
181C4⋊C4×C83#4112yes—
182C22⋊C4×C83—24no—
183C22⋊C4×C83—24no—
184C22⋊C4×C83—81no—
185C22⋊C4×C83—81no—
186C22⋊C4×C83—81no—
187C22⋊C4×C83—81no—
188C22⋊C4×C83—81no—
189C22⋊C4×C83—81no—
190C22⋊C4×C83—81no—
191C22⋊C4×C83—81no—
192C42×C83—21no—
193C42×C83—21no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—81no—
199C42×C83—81no—
200C23×C83—22no—
201C23×C83—22no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83—41no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83—41no—
214C23×C83C22×D411yescharacteristic
215D4×C83—22no—
216D4×C83—22no—
217D4×C83—22no—
218D4×C83—22no—
219D4×C83—41no—
220D4×C83—41no—
221D4×C83—41no—
222D4×C83—41no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
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—41no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83—84no—
260C2×C4×C83—84no—
261C2×C4×C83C2×#4111yescharacteristic
262C2×C4×C83C2×#4111yescharacteristic
263C2×C4×C83C2×#4111yescharacteristic
264C2×C4×C83C2×#4111yescharacteristic
265C2×C4×C83H(5,𝔽2)11yescharacteristic
266C2×C4×C832−1+411yescharacteristic
267C8×C83—41no—
268C8×C83—41no—
269C8×C83—41no—
270C8×C83—41no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—81no—
278C22×C83—81no—
279C22×C83—81no—
280C22×C83—81no—
281C22×C83—81no—
282C22×C83—81no—
283C22×C83—81no—
284C22×C83—81no—
285C22×C83#52511yescharacteristic
286C22×C83#52511yescharacteristic
287C22×C83#53711yescharacteristic
288C22×C83#57611yescharacteristic
289C22×C83#57611yescharacteristic
290C22×C83#57711yescharacteristic
291C22×C83#57711yescharacteristic
292C4×C83—41no—
293C4×C83—42no—
294C4×C83—41no—
295C4×C83—41no—
296C4×C83—42no—
297C4×C83—41no—
298C4×C83—41no—
299C4×C83—41no—
300C4×C83—81no—
301C4×C83—81no—
302C4×C83—81no—
303C4×C83—81no—
304C4×C83#55111yescharacteristic
305C4×C83#55211yescharacteristic
306C4×C83#55211yescharacteristic
307C4×C83#55511yescharacteristic
308#29542C8311yesmaximal, characteristic, direct product factor, Sylow
309C2×C83—82no—
310C2×C83—82no—
311C2×C83#260911yescharacteristic
312C2×C83#311811yescharacteristic
313C2×C83#311911yescharacteristic
314C2×C83#312211yescharacteristic
315C2×C83#312511yescharacteristic
316C2×C83#325711yescharacteristic
317C2×C83#325711yescharacteristic
318#2319C2×C8311yescharacteristic, complemented
319#2319C2×C8311yescharacteristic, complemented
320#2088C2×C8311yescharacteristic, complemented
321#2067C2×C8311yescharacteristic, complemented
322#2065C2×C8311yescharacteristic, complemented
323#2054C2×C8311yescharacteristic
324#1926C2×C8311yescharacteristic, complemented
325#1926C2×C8311yescharacteristic, complemented
326#1920C2×C8311yescharacteristic, complemented
327#1918C2×C8311yescharacteristic, complemented
328#1879C2×C8311yescharacteristic
329#1879C2×C8311yescharacteristic
330#1864C2×C8311yescharacteristic, complemented
331#1862C2×C8311yescharacteristic, complemented
332#1832C2×C8311yescharacteristic, complemented
333C83#2954211yescharacteristic, direct product factor, Sylow
334C22×D8C22×C8311yescharacteristic
335C2×C4×D4C22×C8311yescharacteristic
336C2×C4×D4C22×C8311yescharacteristic
337C2×#107C22×C8311yescharacteristic
338C2×#106C22×C8311yescharacteristic
339C2×#106C22×C8311yescharacteristic
340C2×C4⋊C8C22×C8311yescharacteristic
341C2×C4⋊C8C22×C8311yescharacteristic
342C2×#102C22×C8311yescharacteristic
343C2×#102C22×C8311yescharacteristic
344C2×#102C22×C8311yescharacteristic
345C2×#102C22×C8311yescharacteristic
346C2×#102C22×C8311yescharacteristic
347C2×#102C22×C8311yescharacteristic
348#390C22×C8311yescharacteristic
349#390C22×C8311yescharacteristic
350#390C22×C8311yescharacteristic
351#390C22×C8311yescharacteristic
352#389C22×C8311yescharacteristic
353#389C22×C8311yescharacteristic
354#389C22×C8311yescharacteristic
355#389C22×C8311yescharacteristic
356#388—21no—
357#388—21no—
358#388—21no—
359#388—21no—
360#381—21no—
361#381—21no—
362#380—21no—
363#380—21no—
364C4×C4⋊C4C22×C8311yescharacteristic
365C4×C4⋊C4C22×C8311yescharacteristic
366#376C22×C8311yescharacteristic
367#340C22×C8311yescharacteristic
368#340C22×C8311yescharacteristic
369#340C22×C8311yescharacteristic
370#340C22×C8311yescharacteristic
371#340C22×C8311yescharacteristic
372#340C22×C8311yescharacteristic
373#336C22×C8311yescharacteristic
374#336C22×C8311yescharacteristic
375#336C22×C8311yescharacteristic
376#336C22×C8311yescharacteristic
377C22×D4C23×C8311yescharacteristic
378C22×D4C23×C8311yescharacteristic
379C23×C4—21no—
380C23×C4—21no—
381C2×D8—22no—
382C2×D8—22no—
383C2×D8—41no—
384C22×C8C23×C8311yescharacteristic
385C22×C8C23×C8311yescharacteristic
386C22×C8C23×C8311yescharacteristic
387C22×C8C23×C8311yescharacteristic
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4C23×C8311yescharacteristic
397C2×C4⋊C4C23×C8311yescharacteristic
398C2×C4⋊C4C23×C8311yescharacteristic
399C2×C4⋊C4C23×C8311yescharacteristic
400C2×C4⋊C4C23×C8311yescharacteristic
401C2×C4⋊C4C23×C8311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42C23×C8311yescharacteristic
415C2×C42C23×C8311yescharacteristic
416C2×C42C23×C8311yescharacteristic
417#107—22no—
418#106—22no—
419#106—41no—
420C4⋊C8—41no—
421C4⋊C8—41no—
422#102—22no—
423#102—22no—
424#102—22no—
425#102—22no—
426#102—41no—
427#102—41no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95D4×C8311yescharacteristic
435#95D4×C8311yescharacteristic
436#95D4×C8311yescharacteristic
437#95D4×C8311yescharacteristic
438C24—21no—
439C24—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4#41×C8312yes—
451C2×D4#41×C8312yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4C2×D4×C8311yescharacteristic
465C22×C4C2×D4×C8311yescharacteristic
466C22×C4C2×D4×C8311yescharacteristic
467C22×C4C2×D4×C8311yescharacteristic
468C22×C4C2×D4×C8311yescharacteristic
469C22×C4C2×D4×C8311yescharacteristic
470C22×C4C24×C8311yescharacteristic
471D8—44no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4#41×C8312yes—
499C4⋊C4#41×C8312yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—81no—
517C42—81no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×D4×C8311yescharacteristic
533D4—22no—
534D4—22no—
535D4—22no—
536D4—22no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4C2×#41×C8311yescharacteristic
580C2×C4C2×#41×C8311yescharacteristic
581C2×C4C2×#41×C8311yescharacteristic
582C2×C4C2×#41×C8311yescharacteristic
583C2×C4H(5,𝔽2)×C8311yescharacteristic
584C2×C42−1+4×C8311yescharacteristic
585C8—41no—
586C8—41no—
587C8—41no—
588C8—41no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#525×C8311yescharacteristic
604C22#525×C8311yescharacteristic
605C22#537×C8311yescharacteristic
606C22#576×C8311yescharacteristic
607C22#576×C8311yescharacteristic
608C22#577×C8311yescharacteristic
609C22#577×C8311yescharacteristic
610C4—41no—
611C4—42no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4#551×C8311yescharacteristic
623C4#552×C8311yescharacteristic
624C4#552×C8311yescharacteristic
625C4#555×C8311yescharacteristic
626C2—82no—
627C2—82no—
628C2C83×#260911yescharacteristic
629C2C83×#311811yescharacteristic
630C2C83×#311911yescharacteristic
631C2C83×#312211yescharacteristic
632C2C83×#312511yescharacteristic
633C2C83×#325711yescharacteristic
634C2C83×#325711yescharacteristic
subgroupG/H
center, socleC23×C83C22×D4
commutator, Frattini, cosocleC22×C4C24×C83
prime radicalC22×D8×C83C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C83 ◁ #376×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#2319C2semidirect
2. C83×#2319C2semidirect
3. C83×#2088C2semidirect
4. C83×#2067C2semidirect
5. C83×#2065C2semidirect
7. C83×#1926C2semidirect
8. C83×#1926C2semidirect
9. C83×#1920C2semidirect
10. C83×#1918C2semidirect
13. C83×#1864C2semidirect
14. C83×#1862C2semidirect
15. C83×#1832C2semidirect
308. #29542C83direct (N × H)
318. #2319C2×C83semidirect
319. #2319C2×C83semidirect
320. #2088C2×C83semidirect
321. #2067C2×C83semidirect
322. #2065C2×C83semidirect
324. #1926C2×C83semidirect
325. #1926C2×C83semidirect
326. #1920C2×C83semidirect
327. #1918C2×C83semidirect
330. #1864C2×C83semidirect
331. #1862C2×C83semidirect
332. #1832C2×C83semidirect
333. C83#29542direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#23192
C83×#20882
C83×#20672
C83×#20652
C83×#20542
C83×#19262
C83×#19202
C83×#19182
C83×#18792
C83×#18642
C83×#18622
C83×#18322
#2954283