Finite Group Databasebeta

← #35426375

Subgroups — #35426375

Order 21248.

878 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×#19127990C211yesmaximal, characteristic
2#24910522C211yesmaximal, characteristic, complemented
3#24910372C211yesmaximal, characteristic
4#24908434C211yesmaximal, characteristic, complemented
5#24908230C211yesmaximal, characteristic, complemented
6#24908203C211yesmaximal, characteristic, complemented
7#24908192C211yesmaximal, characteristic, complemented
8#24908051C211yesmaximal, characteristic, complemented
9#24907727C211yesmaximal, characteristic
10#24907711C211yesmaximal, characteristic
11#24907336C211yesmaximal, characteristic, complemented
12#24907291C211yesmaximal, characteristic
13#24907287C211yesmaximal, characteristic
14#24907277C211yesmaximal, characteristic
15C83×#1919C211yesmaximal, characteristic, complemented
16C2×#14581791C2211yescharacteristic
17C2×C4×Dic166—21no—
18C2×C4×D332C2211yescharacteristic
19C2×#14581754C2211yescharacteristic
20C2×#14581754C2211yescharacteristic
21#19127990C418yescomplemented
22C2×#14581725C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×Dic83⋊C8C2211yescharacteristic
26C2×#14581714C2211yescharacteristic
27C2×#14581714C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29C2×#14581712C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×#14581702C2211yescharacteristic
32C2×C8×Dic83C2211yescharacteristic
33#19127605—21no—
34#19127600C2211yescharacteristic
35#19127600C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127588—21no—
38#19127585—21no—
39#19127575—21no—
40#19127572—21no—
41#19127566C2211yescharacteristic
42#19127558—21no—
43C4⋊C4×Dic83C2211yescharacteristic
44C4⋊C4×Dic83C2211yescharacteristic
45C4×Dic83⋊C4C2211yescharacteristic
46#95×D83—21no—
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#106×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55C4×C4⋊C4×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×Dic166—21no—
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C22×C4×D83—21no—
64C22×C4×D83—21no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581791—22no—
68#14581791C2×C414yescomplemented
69C4×Dic166—44no—
70C4×D332—22no—
71C4×D332—41no—
72#14581754—22no—
73#14581754—22no—
74#14581754—41no—
75#14581754C2×C414yes—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4D411yescharacteristic
84C2×Dic83⋊C4D411yescharacteristic
85C2×(C2×D83)⋊C4—21no—
86C2×(C2×D83)⋊C4—21no—
87C2×(C2×D83)⋊C4—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90C2×C4×Dic83C2311yescharacteristic
91C2×C4×Dic83C2311yescharacteristic
92#14581725—41no—
93#14581724—41no—
94#14581723C2×C414yescomplemented
95Dic83⋊C8C2×C414yescomplemented
96#14581714—22no—
97#14581714C2×C414yes—
98#14581712C2×C414yes—
99#14581712Q814yescomplemented
100#14581703—41no—
101#14581702—22no—
102C8×Dic83—41no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95D411yescharacteristic
111C83⋊#95D411yescharacteristic
112C22×C8×C83C2311yescharacteristic
113C22×C8×C83C2311yescharacteristic
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C42×C83—21no—
120C2×C42×C83C2311yescharacteristic
121#106×C83C2×C414yescomplemented
122#95×C83—21no—
123#95×C83—21no—
124C23×D83—21no—
125C2×Dic166—24no—
126C2×Dic166—42no—
127C2×Dic166—42no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332C22×C412yes—
132C2×D332C2×Q812yes—
133C2×D332#4112yes—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83C2×D411yescharacteristic
138C22×Dic83C2×D411yescharacteristic
139C22×Dic83C2×D411yescharacteristic
140C22×Dic83C2×D411yescharacteristic
141C2×C4×D83—24no—
142C2×C4×D83—42no—
143C2×C4×D83—42no—
144C2×C4×D83—42no—
145C2×C4×D83—42no—
146C2×C4×D83—81no—
147C2×C4×D83—81no—
148C2×C4×D83—81no—
149C2×C4×D83—81no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8C22×C412yes—
155C2×C83⋊C8C2×Q812yes—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—42no—
162C4⋊Dic83—81no—
163C4⋊Dic83C22×C412yes—
164Dic83⋊C4—24no—
165Dic83⋊C4—22no—
166Dic83⋊C4—41no—
167Dic83⋊C4—42no—
168(C2×D83)⋊C4—24no—
169(C2×D83)⋊C4—42no—
170(C2×D83)⋊C4—42no—
171C4×Dic83—21no—
172C4×Dic83—21no—
173C4×Dic83—42no—
174C4×Dic83—42no—
175C4×Dic83C22×C412yes—
176C4×Dic83#4112yes—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83C22×C412yes—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—81no—
196C4⋊C4×C83C22×C412yes—
197C4⋊C4×C83C22×C412yes—
198C4⋊C4×C83C2×Q812yes—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—42no—
202C23×C83C22×D411yescharacteristic
203C22×D83—22no—
204C22×D83—22no—
205C22×D83—22no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209C22×D83—41no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—22no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83C23×C411yescharacteristic
229C2×C4×C83C22×Q811yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C2×C4×C83H(5,𝔽2)11yescharacteristic
233C2×C4×C832−1+411yescharacteristic
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—22no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83C4×D412yes—
253C2×Dic83C4×D412yes—
254C2×Dic83AGL(1,ℤ/8)12yes—
255D332—22no—
256D332—22no—
257D332—22no—
258D332C4×Q814yescomplemented
259C4×D83—84no—
260C4×D83—84no—
261C4×D83—84no—
262Dic166—48no—
263C8×C83—22no—
264C8×C83—41no—
265C83⋊C8—22no—
266C83⋊C8—22no—
267C22×C83C2×C4×D411yescharacteristic
268C22×C83#54611yescharacteristic
269C22×C83D4×Q811yescharacteristic
270C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C83#57511yescharacteristic
272C22×C83#57711yescharacteristic
273C22×C83#57711yescharacteristic
274C2×D83—24no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—42no—
278C2×D83—42no—
279C2×D83—42no—
280C2×D83—42no—
281C4×C83—42no—
282C4×C83—42no—
283C4×C83—42no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—81no—
287C4×C83C2×C4×Q811yescharacteristic
288C4×C83#52011yescharacteristic
289C4×C83#55211yescharacteristic
290C4×C83#55411yescharacteristic
291Dic83—22no—
292Dic83—22no—
293Dic83—22no—
294Dic83—42no—
295Dic83—41no—
296Dic83—81no—
297Dic83—81no—
298#22975—831nomaximal, Sylow
299C2×C83#236011yescharacteristic
300C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C83#294911yescharacteristic
302C2×C83#330611yescharacteristic
303C2×C83#331111yescharacteristic
304C2×C83#338411yescharacteristic
305C2×C83#338811yescharacteristic
306D83—44no—
307C2×#476—831no—
308#2365—831no—
309#2349—831no—
310#1993—831no—
311#1949—831no—
312#1944—831no—
313#1941—831no—
314#1920—831no—
315#1919D8311yescharacteristic, complemented
316#1864—831no—
317#1862—831no—
318#1771—831no—
319#1762—831no—
320#1762—831no—
321C4×#102—831no—
322C83#2297511yescharacteristic, complemented, Sylow
323C2×C4⋊Q8—831no—
324C2×C4×Q8—1661no—
325C2×C4×D4—831no—
326C2×C4×D4—831no—
327C2×C4×D4—831no—
328#476—838no—
329C2×#107—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×#106C2×D8311yescharacteristic
333C2×C4⋊C8—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#102—831no—
338C2×#97—831no—
339C2×#97—831no—
340C2×C4×C8—831no—
341#391—1661no—
342#389—831no—
343#389—831no—
344#389—831no—
345#389C2×D8311yescharacteristic
346#387—1661no—
347#387—1661no—
348#385—1661no—
349#385—1661no—
350#382—831no—
351#380—1661no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4C2×D8311yescharacteristic
356C2×#95—1661no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340C2×D8311yescharacteristic
363#340C2×D8311yescharacteristic
364#340C2×D8311yescharacteristic
365#336—831no—
366#336C2×D8311yescharacteristic
367C22×Q8—1661no—
368C22×D4—831no—
369C23×C4—1661no—
370C23×C4—1661no—
371C23×C4—1661no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8C22×D8311yescharacteristic
375C22×C8C22×D8311yescharacteristic
376C4⋊Q8—834no—
377C4⋊Q8—1662no—
378C4×Q8—3324no—
379C4×D4—3321no—
380C4×D4—3321no—
381C4×D4—834no—
382C4×D4—1662no—
383C4×D4—1662no—
384C4×D4—1662no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C42—21no—
403C2×C42—831no—
404C2×C42—831no—
405C2×C42—1661no—
406C2×C42—1661no—
407C2×C42C22×D8311yescharacteristic
408#107—3321no—
409#106—3321no—
410#106—834no—
411#106C4×D8314yescomplemented
412C4⋊C8—834no—
413#102—834no—
414#102—834no—
415#102—834no—
416#102—1662no—
417#97—3321no—
418#97—1662no—
419C4×C8—3321no—
420#95—21no—
421#95—21no—
422#95—831no—
423#95—831no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431C24—1661no—
432C2×Q8—3322no—
433C2×Q8—3322no—
434C2×Q8—1664no—
435C2×D4—832no—
436C2×D4—832no—
437C2×D4—832no—
438C2×D4—1661no—
439C2×D4—1661no—
440C2×D4—1661no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—3322no—
448C22×C4—3322no—
449C22×C4—831no—
450C22×C4—831no—
451C22×C4—831no—
452C22×C4—831no—
453C22×C4—6641no—
454C22×C4—6641no—
455C22×C4—6641no—
456C22×C4—6641no—
457C22×C4—1664no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4D4×D8311yescharacteristic
462C22×C4D4×D8311yescharacteristic
463C22×C4C23×D8311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—832no—
470C2×C8—832no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8C2×C4×D8312yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—81no—
482C4⋊C4—3322no—
483C4⋊C4—3321no—
484C4⋊C4—3322no—
485C4⋊C4—832no—
486C4⋊C4—6641no—
487C4⋊C4—1662no—
488C4⋊C4—1664no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4C2×C4×D8312yes—
495C4⋊C4C2×C4×D8312yes—
496C4⋊C4Q8×D8312yes—
497C22⋊C4—3322no—
498C22⋊C4—3322no—
499C22⋊C4—1664no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—3322no—
504C42—3322no—
505C42—832no—
506C42—832no—
507C42—1661no—
508C42—1661no—
509C23—3321no—
510C23—3321no—
511C23—3321no—
512C23—3321no—
513C23—1662no—
514C23—1662no—
515C23—1662no—
516C23C2×D4×D8311yescharacteristic
517Q8—3328no—
518D4—834no—
519D4—1662no—
520D4—1662no—
521D4—1662no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—832no—
549C2×C4—832no—
550C2×C4—832no—
551C2×C4—6644no—
552C2×C4—6644no—
553C2×C4—6644no—
554C2×C4—1661no—
555C2×C4—1662no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4—1661no—
561C2×C4—1661no—
562C2×C4—1661no—
563C2×C4—1661no—
564C2×C4C22×C4×D8311yescharacteristic
565C2×C4C2×Q8×D8311yescharacteristic
566C2×C4#41×D8311yescharacteristic
567C2×C4#41×D8311yescharacteristic
568C2×C4#1458183811yescharacteristic
569C2×C4#1458183911yescharacteristic
570C8—22no—
571C8—41no—
572C8—1662no—
573C8—1662no—
574C22—3322no—
575C22—3322no—
576C22—3322no—
577C22—3322no—
578C22—3322no—
579C22—3322no—
580C22—1664no—
581C22C4×D4×D8311yescharacteristic
582C22#1912834311yescharacteristic
583C22#1912836411yescharacteristic
584C22AGL(1,ℤ/8)×D8311yescharacteristic
585C22#1912850011yescharacteristic
586C22#1912850711yescharacteristic
587C22#1912850811yescharacteristic
588C4—42no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—3322no—
595C4—3321no—
596C4—6641no—
597C4—6641no—
598C4—1662no—
599C4—1662no—
600C4—1662no—
601C4C4×Q8×D8311yescharacteristic
602C4#1912818111yescharacteristic
603C4#1912838711yescharacteristic
604C4#1912840711yescharacteristic
605C2—3324no—
606C2#2491047111yescharacteristic
607C2#2491626811yescharacteristic
608C2#2491631711yescharacteristic
609C2#2491942711yescharacteristic
610C2#2491950211yescharacteristic
611C2#2492036511yescharacteristic
612C2#2492041511yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1919C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#1919D83
hypocenterC83#22975
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
2. #24910522C2semidirect
4. #24908434C2semidirect
5. #24908230C2semidirect
6. #24908203C2semidirect
7. #24908192C2semidirect
8. #24908051C2semidirect
11. #24907336C2semidirect
15. C83×#1919C2semidirect
21. #19127990C4semidirect
68. #14581791C2×C4semidirect
94. #14581723C2×C4semidirect
95. Dic83⋊C8C2×C4semidirect
99. #14581712Q8semidirect
121. #106×C83C2×C4semidirect
258. D332C4×Q8semidirect
315. #1919D83semidirect
322. C83#22975semidirect
411. #106C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191279902
#249105222
#249103722
#249084342
#249082302
#249082032
#249081922
#249080512
#249077272
#249077112
#249073362
#249072912
#249072872
#249072772
C83×#19192
#2297583