Finite Group Databasebeta

← #35412408

Subgroups — #35412408

Order 21248.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128058C211yesmaximal, characteristic
2#24910174C211yesmaximal, characteristic
3#24908433C211yesmaximal, characteristic, complemented
4#24908277C211yesmaximal, characteristic
5#24908159C211yesmaximal, characteristic
6#24908152C211yesmaximal, characteristic, complemented
7#24908145C211yesmaximal, characteristic
8#24908138C211yesmaximal, characteristic, complemented
9#24907590C211yesmaximal, characteristic, complemented
10#24907296C211yesmaximal, characteristic, complemented
11#24907291C211yesmaximal, characteristic
12C4×#14581715C211yesmaximal, characteristic, complemented
13C4×#14581712C211yesmaximal, characteristic
14#24907150C211yesmaximal, characteristic, complemented
15C83×#2330C211yesmaximal, characteristic, complemented
16C2×#14581780C2211yescharacteristic
17C2×C4×Dic166C2211yescharacteristic
18C2×C4×D332C2211yescharacteristic
19#19128058C418yescomplemented
20C2×#14581715C2211yescharacteristic
21C2×#14581715C2211yescharacteristic
22C2×#14581715C2211yescharacteristic
23C2×#14581715C2211yescharacteristic
24C2×#14581712C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×C4×C83⋊C8C2211yescharacteristic
32C2×C4×C83⋊C8C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34#19127607C2211yescharacteristic
35#19127603C2211yescharacteristic
36#19127593—21no—
37#19127593—21no—
38#19127575—21no—
39#19127575—21no—
40#19127571C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49C2×#125×C83C2211yescharacteristic
50#389×C83—21no—
51#389×C83—21no—
52#389×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#387×C83—21no—
55#387×C83—21no—
56#384×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59#376×C83C2211yescharacteristic
60C22×Dic166C2311yescharacteristic
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581780—22no—
68#14581780C2×C414yes—
69C4×Dic166—22no—
70C4×Dic166—22no—
71C4×D332—22no—
72C4×D332—22no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×C4×Dic83—21no—
84#14581715—22no—
85#14581715—22no—
86#14581715C2×C414yescomplemented
87#14581715C2×C414yescomplemented
88#14581712—22no—
89#14581712—22no—
90#14581712C2×C414yes—
91#14581712C2×C414yes—
92#14581703—22no—
93#14581703—22no—
94#14581703—22no—
95C4×C83⋊C8—22no—
96C4×C83⋊C8—22no—
97C4×C83⋊C8C2×C414yescomplemented
98C83⋊#95—21no—
99C83⋊#95—21no—
100#125×C83—22no—
101#125×C83C2×C414yescomplemented
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C42×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
121#95×C83D411yescharacteristic
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124#95×C83D411yescharacteristic
125C23×D83—21no—
126C2×Dic166—21no—
127C2×Dic166—21no—
128C2×Dic166—21no—
129C2×Dic166—21no—
130C2×Dic166—21no—
131C2×Dic166C22×C412yes—
132C2×D332—21no—
133C2×D332—21no—
134C2×D332—21no—
135C2×D332—21no—
136C2×D332—21no—
137C2×D332C22×C412yes—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—21no—
141C2×C4×D83—42no—
142C2×C4×D83—81no—
143C2×C4×D83—81no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8C22×C412yes—
153C2×C83⋊C8C22×C412yes—
154C2×C83⋊C8#4112yes—
155C2×C83⋊C8#4112yes—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—81no—
161C4⋊Dic83—81no—
162Dic83⋊C4—42no—
163(C2×D83)⋊C4—24no—
164(C2×D83)⋊C4—24no—
165(C2×D83)⋊C4—42no—
166(C2×D83)⋊C4—81no—
167(C2×D83)⋊C4—81no—
168C4×Dic83—42no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2411yescharacteristic
180C4⋊C4×C83—24no—
181C4⋊C4×C83—24no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—81no—
189C4⋊C4×C83—81no—
190C4⋊C4×C83—81no—
191C4⋊C4×C83—81no—
192C4⋊C4×C83C22×C412yes—
193C4⋊C4×C83C22×C412yes—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—81no—
197C42×C83—81no—
198C42×C83C22×C412yes—
199C42×C83#4112yes—
200C42×C83#4112yes—
201C42×C83#4112yes—
202C23×C83C22×D411yescharacteristic
203C22×D83—22no—
204C22×D83—41no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209C22×D83—41no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
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—22no—
219C2×C4×C83—22no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
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×C83C4×D412yes—
237C2×C4×C83C4×D412yes—
238C2×C4×C83#12412yes—
239C2×C4×C83#12412yes—
240C2×C4×C83C23×C411yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C2×Dic83—22no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255D332—22no—
256D332—22no—
257D332—41no—
258D332—41no—
259C4×D83—84no—
260Dic166—22no—
261Dic166—22no—
262Dic166—41no—
263Dic166—41no—
264C83⋊C8—22no—
265C83⋊C8—22no—
266C83⋊C8—22no—
267C83⋊C8—22no—
268C22×C83C2×C4×D411yescharacteristic
269C22×C83C2×#12411yescharacteristic
270C22×C83#53211yescharacteristic
271C22×C83#57611yescharacteristic
272C22×C83#57711yescharacteristic
273C22×C83#57711yescharacteristic
274C22×C83#57811yescharacteristic
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—42no—
278C2×D83—81no—
279C2×D83—81no—
280C2×D83—81no—
281C2×D83—81no—
282C4×C83—22no—
283C4×C83—22no—
284C4×C83—22no—
285C4×C83—22no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83—42no—
289C4×C83—42no—
290C4×C83—81no—
291C4×C83—81no—
292C4×C83C4×#4111yescharacteristic
293C4×C83C4×#4111yescharacteristic
294C4×C83#52911yescharacteristic
295C4×C83#52911yescharacteristic
296Dic83—42no—
297Dic83—81no—
298Dic83—81no—
299#21643—831nomaximal, Sylow
300C2×C83C4×#12411yescharacteristic
301C2×C83#295111yescharacteristic
302C2×C83#295211yescharacteristic
303C2×C83#313611yescharacteristic
304C2×C83#313711yescharacteristic
305C2×C83#317811yescharacteristic
306C2×C83#317911yescharacteristic
307D83—82no—
308C2×#488—831no—
309#2330D8311yescharacteristic, complemented
310#2326—831no—
311#1992—831no—
312#1959—831no—
313#1936—831no—
314#1935—831no—
315#1934—831no—
316#1933—831no—
317#1834—831no—
318#1763—831no—
319#1762—831no—
320C4×#103—831no—
321C4×#102—831no—
322#1726—831no—
323C83#2164311yescharacteristic, complemented, Sylow
324C2×#125C2×D8311yescharacteristic
325C2×#124—831no—
326C2×C4×Q8—831no—
327C2×C4×D4—831no—
328#488—838no—
329C2×#103—831no—
330C2×#103—831no—
331C2×#103—831no—
332C2×#103—831no—
333C2×#102—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#97—831no—
338C2×#97—831no—
339C2×#97—831no—
340C2×C4×C8—831no—
341C2×C4×C8—831no—
342C2×C4×C8—831no—
343#391—831no—
344#390—831no—
345#389—21no—
346#389—21no—
347#389C2×D8311yescharacteristic
348#389C2×D8311yescharacteristic
349#388—1661no—
350#388—1661no—
351#387—21no—
352#387—21no—
353#385—1661no—
354#385—1661no—
355#384—831no—
356#384C2×D8311yescharacteristic
357C4×C4⋊C4C2×D8311yescharacteristic
358C4×C4⋊C4C2×D8311yescharacteristic
359#376C2×D8311yescharacteristic
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368C22×Q8—831no—
369C22×D4—831no—
370C23×C4—1661no—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8—831no—
375#125—22no—
376#125C4×D8314yescomplemented
377#124—834no—
378#124—1662no—
379C4×Q8—1662no—
380C4×Q8—1662no—
381C4×D4—1662no—
382C4×D4—1662no—
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—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4C22×D8311yescharacteristic
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—1661no—
406C2×C42C22×D8311yescharacteristic
407C2×C42C22×D8311yescharacteristic
408C2×C42C22×D8311yescharacteristic
409#103—834no—
410#103—834no—
411#103—1662no—
412#103—1662no—
413#102—834no—
414#102—834no—
415#102—1662no—
416#102—1662no—
417#97—1662no—
418#97—1662no—
419#97—1662no—
420C4×C8—834no—
421C4×C8—1662no—
422C4×C8—1662no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1661no—
428#95—1661no—
429#95C22⋊D8311yescharacteristic
430#95C22⋊D8311yescharacteristic
431#95C22⋊D8311yescharacteristic
432#95C22⋊D8311yescharacteristic
433C24—1661no—
434C2×Q8—832no—
435C2×Q8—1661no—
436C2×Q8—1661no—
437C2×Q8—1661no—
438C2×Q8—1661no—
439C2×Q8—1661no—
440C2×D4—832no—
441C2×D4—1661no—
442C2×D4—1661no—
443C2×D4—1661no—
444C2×D4—1661no—
445C2×D4—1661no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—3322no—
451C22×C4—6641no—
452C22×C4—6641no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C2×C22⋊D8311yescharacteristic
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4C2×C22⋊D8311yescharacteristic
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C23×D8311yescharacteristic
463C2×C8—832no—
464C2×C8—832no—
465C2×C8—832no—
466C2×C8—832no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—3322no—
488C4⋊C4—6641no—
489C4⋊C4—6641no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4C2×C4×D8312yes—
495C4⋊C4C2×C4×D8312yes—
496C22⋊C4—3322no—
497C22⋊C4—6641no—
498C22⋊C4—6641no—
499C22⋊C4—1664no—
500C22⋊C4—1664no—
501C42—21no—
502C42—21no—
503C42—81no—
504C42—81no—
505C42—3322no—
506C42C2×C4×D8312yes—
507C42#1305059012yes—
508C42#1305059112yes—
509C42#1305059112yes—
510C23—3321no—
511C23—3321no—
512C23—3321no—
513C23—3321no—
514C23—3321no—
515C23—3321no—
516C23—1662no—
517C23C22×C22⋊D8311yescharacteristic
518Q8—3321no—
519Q8—3321no—
520Q8—1662no—
521Q8—1662no—
522D4—3321no—
523D4—3321no—
524D4—1662no—
525D4—1662no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
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—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—6644no—
561C2×C4—1662no—
562C2×C4C4×C22⋊D8312yes—
563C2×C4C4×C22⋊D8312yes—
564C2×C4#1458178112yes—
565C2×C4#1458178112yes—
566C2×C4C22×C4×D8311yescharacteristic
567C2×C4#41×D8311yescharacteristic
568C2×C4#41×D8311yescharacteristic
569C2×C4C2×#1305059011yescharacteristic
570C2×C4C2×#1305059111yescharacteristic
571C2×C4C2×#1305059111yescharacteristic
572C8—1662no—
573C8—1662no—
574C8—1662no—
575C8—1662no—
576C22—3322no—
577C22—3322no—
578C22—3322no—
579C22—6641no—
580C22—6641no—
581C22—6641no—
582C22—6641no—
583C22C2×C4×C22⋊D8311yescharacteristic
584C22C2×#1458178111yescharacteristic
585C22#1912824711yescharacteristic
586C22#1912850411yescharacteristic
587C22#1912850911yescharacteristic
588C22#1912851211yescharacteristic
589C22#1912851611yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—3322no—
601C4—6641no—
602C4—6641no—
603C4C4×#1305059111yescharacteristic
604C4C4×#1305059111yescharacteristic
605C4#1912823811yescharacteristic
606C4#1912823811yescharacteristic
607C2—6642no—
608C2C4×#1458178111yescharacteristic
609C2#2491633611yescharacteristic
610C2#2491634411yescharacteristic
611C2#2491782211yescharacteristic
612C2#2491783311yescharacteristic
613C2#2491813311yescharacteristic
614C2#2491814811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2330C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#2330D83
hypocenterC83#21643
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
3. #24908433C2semidirect
6. #24908152C2semidirect
8. #24908138C2semidirect
9. #24907590C2semidirect
10. #24907296C2semidirect
12. C4×#14581715C2semidirect
14. #24907150C2semidirect
15. C83×#2330C2semidirect
19. #19128058C4semidirect
86. #14581715C2×C4semidirect
87. #14581715C2×C4semidirect
97. C4×C83⋊C8C2×C4semidirect
101. #125×C83C2×C4semidirect
309. #2330D83semidirect
323. C83#21643semidirect
376. #125C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191280582
#249101742
#249084332
#249082772
#249081592
#249081522
#249081452
#249081382
#249075902
#249072962
#249072912
C4×#145817152
C4×#145817122
#249071502
C83×#23302
#2164383