Finite Group Databasebeta

← #35407495

Subgroups — #35407495

Order 21248.

938 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127811C211yesmaximal, characteristic
2#24910107C211yesmaximal, characteristic, complemented
3#24908090C211yesmaximal, characteristic, complemented
4#24908083C211yesmaximal, characteristic, complemented
5#24908048C211yesmaximal, characteristic, complemented
6#24908041C211yesmaximal, characteristic, complemented
7#24907811C211yesmaximal, characteristic
8#24907811C211yesmaximal, characteristic
9#24907724C211yesmaximal, characteristic, complemented
10#24907710C211yesmaximal, characteristic, complemented
11#24907361C211yesmaximal, characteristic, complemented
12C4×#14581712C211yesmaximal, characteristic, complemented
13C4×#14581713C211yesmaximal, characteristic, complemented
14#24907215C211yesmaximal, characteristic, complemented
15C83×#2275C211yesmaximal, characteristic, complemented
16C22×#13050575C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18#19127811C418yescomplemented
19C2×#14581725C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127603C2211yescharacteristic
31#19127603C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127590—21no—
35#19127590—21no—
36#19127563—21no—
37#19127560—21no—
38C4×C4⋊Dic83C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4×D4×C83C2211yescharacteristic
50C2×C4×D4×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52C4×C4⋊C4×C83C2211yescharacteristic
53C4×C4⋊C4×C83C2211yescharacteristic
54C4×C4⋊C4×C83C2211yescharacteristic
55C4×C22⋊C4×C83—21no—
56C4×C22⋊C4×C83—21no—
57C4×C22⋊C4×C83—21no—
58#376×C83C2211yescharacteristic
59C2×#95×C83—21no—
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62C2×#13050575—22no—
63C2×#13050575—22no—
64C2×#13050575C2×C414yes—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C4×D332—22no—
70C4×D332C2×C414yescomplemented
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×(C2×D83)⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×C4×Dic83—21no—
82#14581725—22no—
83#14581723—22no—
84#14581723C2×C414yescomplemented
85#14581713—22no—
86#14581713—22no—
87#14581713C2×C414yes—
88#14581712—22no—
89#14581712—22no—
90#14581712C2×C414yescomplemented
91#14581703C2×C414yescomplemented
92C4×C83⋊C8—22no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C22×D4×C83C2311yescharacteristic
97C23×C4×C83—21no—
98C23×C4×C83—21no—
99C23×C4×C83—21no—
100C4×D4×C83—22no—
101C4×D4×C83—22no—
102C4×D4×C83—22no—
103C4×D4×C83—22no—
104C4×D4×C83—22no—
105C4×D4×C83C2×C414yes—
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C42×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83—21no—
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
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×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332C22×C412yes—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C2×C4×D83—24no—
136C2×C4×D83—81no—
137C2×C4×D83—81no—
138#13050575—28no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8C22×C412yes—
150C2×C83⋊C8C22×C412yes—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83C22×C412yes—
157Dic83⋊C4—81no—
158Dic83⋊C4—81no—
159(C2×D83)⋊C4—24no—
160(C2×D83)⋊C4—81no—
161(C2×D83)⋊C4—81no—
162(C2×D83)⋊C4—81no—
163(C2×D83)⋊C4—81no—
164C4×Dic83—81no—
165C24×C83—21no—
166C2×D4×C83—21no—
167C2×D4×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83C22×C412yes—
170C2×D4×C83#4112yes—
171C2×D4×C83#4112yes—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—24no—
176C22×C4×C83—24no—
177C22×C4×C83—24no—
178C22×C4×C83—81no—
179C22×C4×C83—81no—
180C22×C4×C83—81no—
181C22×C4×C83—81no—
182C22×C4×C83—81no—
183C22×C4×C83—81no—
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2411yescharacteristic
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83C22×C412yes—
195C4⋊C4×C83#4112yes—
196C4⋊C4×C83#4112yes—
197C22⋊C4×C83—24no—
198C22⋊C4×C83—24no—
199C22⋊C4×C83—24no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83—81no—
205C42×C83—81no—
206C42×C83—81no—
207C42×C83C22×C412yes—
208C42×C83#4112yes—
209C23×C83—22no—
210C23×C83—22no—
211C23×C83—22no—
212C23×C83—41no—
213C23×C83—41no—
214C23×C83—41no—
215C23×C83—41no—
216C23×C83C22×D411yescharacteristic
217C22×D83—22no—
218C22×D83—41no—
219C22×D83—41no—
220C22×D83—41no—
221C22×D83—41no—
222C22×D83—41no—
223C22×D83—41no—
224D4×C83—22no—
225D4×C83—22no—
226D4×C83—22no—
227D4×C83#11714yes—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—22no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—21no—
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—84no—
254C2×C4×C83—84no—
255C2×C4×C83—84no—
256C2×C4×C83C4×D412yes—
257C2×C4×C83C4×D412yes—
258C2×C4×C83C23×C411yescharacteristic
259C2×C4×C83C2×#4111yescharacteristic
260C2×C4×C83C2×#4111yescharacteristic
261C2×C4×C83C2×#4111yescharacteristic
262C2×C4×C83C2×#4111yescharacteristic
263C2×C4×C83C2×#4111yescharacteristic
264C2×Dic83—22no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273D332—22no—
274D332—22no—
275D332—41no—
276D332—41no—
277C4×D83—84no—
278C83⋊C8—22no—
279C83⋊C8—22no—
280C83⋊C8—41no—
281C83⋊C8—41no—
282C22×C83—24no—
283C22×C83—42no—
284C22×C83—42no—
285C22×C83—42no—
286C22×C83—42no—
287C22×C83—42no—
288C22×C83—42no—
289C22×C83C2×C4×D411yescharacteristic
290C22×C83#52511yescharacteristic
291C22×C83#52511yescharacteristic
292C22×C83#57611yescharacteristic
293C22×C83#57611yescharacteristic
294C22×C83#57711yescharacteristic
295C22×C83#57711yescharacteristic
296C2×D83—42no—
297C2×D83—42no—
298C2×D83—42no—
299C2×D83—81no—
300C2×D83—81no—
301C2×D83—81no—
302C2×D83—81no—
303C4×C83—22no—
304C4×C83—22no—
305C4×C83—41no—
306C4×C83—42no—
307C4×C83—41no—
308C4×C83—41no—
309C4×C83—42no—
310C4×C83—41no—
311C4×C83—42no—
312C4×C83C2×#11711yescharacteristic
313C4×C83C4×#4111yescharacteristic
314C4×C83#52911yescharacteristic
315C4×C83#52911yescharacteristic
316Dic83—42no—
317Dic83—81no—
318Dic83—81no—
319#21126—831nomaximal, Sylow
320C2×C83—44no—
321C2×C83#229011yescharacteristic
322C2×C83#294611yescharacteristic
323C2×C83#294611yescharacteristic
324C2×C83#311811yescharacteristic
325C2×C83#311911yescharacteristic
326C2×C83#312211yescharacteristic
327C2×C83#312511yescharacteristic
328D83—82no—
329C2×#442—831no—
330#2319—831no—
331#2275D8311yescharacteristic, complemented
332#1926—831no—
333#1925—831no—
334#1920—831no—
335#1919—831no—
336#1879—831no—
337#1879—831no—
338#1864—831no—
339#1862—831no—
340#1776—831no—
341C4×#102—831no—
342C4×#102—831no—
343#1746—831no—
344C83#2112611yescharacteristic, complemented, Sylow
345C22×D8—831no—
346C2×C4×D4—831no—
347C2×C4×D4C2×D8311yescharacteristic
348C2×C4×D4C2×D8311yescharacteristic
349C2×C4×D4C2×D8311yescharacteristic
350#442—838no—
351C2×#107—831no—
352C2×#106—831no—
353C2×#106—831no—
354C2×#102—831no—
355C2×#102—831no—
356C2×#102—831no—
357C2×#102—831no—
358C2×#102—831no—
359C2×#102—831no—
360C2×#97—831no—
361C2×C4×C8—831no—
362#390—831no—
363#390—831no—
364#389—831no—
365#389—831no—
366#388—1661no—
367#388—1661no—
368#381—1661no—
369#380—1661no—
370C4×C4⋊C4—831no—
371C4×C4⋊C4C2×D8311yescharacteristic
372C4×C4⋊C4C2×D8311yescharacteristic
373C4×C4⋊C4C2×D8311yescharacteristic
374C4×C22⋊C4—21no—
375C4×C22⋊C4—21no—
376C4×C22⋊C4—21no—
377#376C2×D8311yescharacteristic
378C2×#95—21no—
379#340—831no—
380#340—831no—
381#340—831no—
382#340—831no—
383#340—831no—
384#340—831no—
385#336—831no—
386#336—831no—
387#336—831no—
388#336—831no—
389C22×D4—831no—
390C22×D4C22×D8311yescharacteristic
391C23×C4—21no—
392C23×C4—21no—
393C23×C4—21no—
394C23×C4—1661no—
395C2×D8—834no—
396C2×D8—1662no—
397C2×D8—1662no—
398C22×C8—831no—
399C22×C8—831no—
400C22×C8—831no—
401C22×C8—831no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—834no—
408C4×D4—1662no—
409C4×D4C4×D8314yes—
410C2×C4⋊C4—831no—
411C2×C4⋊C4—831no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—1661no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4C22×D8311yescharacteristic
416C2×C4⋊C4C22×D8311yescharacteristic
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1661no—
422C2×C22⋊C4—1661no—
423C2×C22⋊C4—1661no—
424C2×C22⋊C4—1661no—
425C2×C22⋊C4—1661no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—1661no—
430C2×C42C22×D8311yescharacteristic
431C2×C42C22×D8311yescharacteristic
432C2×C42C22×D8311yescharacteristic
433#107—1662no—
434#106—834no—
435#106—1662no—
436#102—834no—
437#102—834no—
438#102—1662no—
439#102—1662no—
440#102—1662no—
441#102—1662no—
442#97—834no—
443C4×C8—1662no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1661no—
448#95—1661no—
449#95—1661no—
450#95C22⋊D8311yescharacteristic
451#95C22⋊D8311yescharacteristic
452#95C22⋊D8311yescharacteristic
453#95C22⋊D8311yescharacteristic
454C24—21no—
455C24—1661no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—832no—
460C2×D4—1661no—
461C2×D4—1661no—
462C2×D4—1661no—
463C2×D4—1661no—
464C2×D4—1661no—
465C2×D4C2×C4×D8312yes—
466C2×D4#1305059012yes—
467C2×D4#1305059012yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—6641no—
481C22×C4—6641no—
482C22×C4—1661no—
483C22×C4—1661no—
484C22×C4—1661no—
485C22×C4—1664no—
486C22×C4C2×C22⋊D8311yescharacteristic
487C22×C4C2×C22⋊D8311yescharacteristic
488C22×C4C2×C22⋊D8311yescharacteristic
489C22×C4C2×C22⋊D8311yescharacteristic
490C22×C4C2×C22⋊D8311yescharacteristic
491C22×C4C2×C22⋊D8311yescharacteristic
492C22×C4C23×D8311yescharacteristic
493D8—1668no—
494C2×C8—832no—
495C2×C8—832no—
496C2×C8—1661no—
497C2×C8—1661no—
498C2×C8—1661no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C2×C8—1661no—
503C2×C8—1661no—
504C2×C8—1661no—
505C2×C8—1661no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—832no—
510C4⋊C4—6641no—
511C4⋊C4—6641no—
512C4⋊C4—1661no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4C2×C4×D8312yes—
518C4⋊C4#1305059012yes—
519C4⋊C4#1305059012yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—6641no—
524C22⋊C4—6641no—
525C22⋊C4—6641no—
526C22⋊C4—6641no—
527C22⋊C4—1664no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—81no—
533C42—81no—
534C42—81no—
535C42—6641no—
536C42C2×C4×D8312yes—
537C42#1305059012yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—3321no—
546C23—3321no—
547C23—3321no—
548C23—3321no—
549C23—3321no—
550C23—3321no—
551C23—1662no—
552C23C22×C22⋊D8311yescharacteristic
553D4—22no—
554D4—22no—
555D4—22no—
556D4—3321no—
557D4—3321no—
558D4—1662no—
559D4—1662no—
560D4#1458174914yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—3321no—
590C2×C4—3321no—
591C2×C4—3321no—
592C2×C4—3321no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—3321no—
597C2×C4—6644no—
598C2×C4—1662no—
599C2×C4C4×C22⋊D8312yes—
600C2×C4C4×C22⋊D8312yes—
601C2×C4C22×C4×D8311yescharacteristic
602C2×C4C2×#1305059011yescharacteristic
603C2×C4C2×#1305059011yescharacteristic
604C2×C4C2×#1305059011yescharacteristic
605C2×C4C2×#1305059011yescharacteristic
606C2×C4C2×#1305059011yescharacteristic
607C8—3321no—
608C8—3321no—
609C8—1662no—
610C8—1662no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—3322no—
619C22—3322no—
620C22—3322no—
621C22—6641no—
622C22—6641no—
623C22—6641no—
624C22—6641no—
625C22C2×C4×C22⋊D8311yescharacteristic
626C22#1912821211yescharacteristic
627C22#1912821211yescharacteristic
628C22#1912850411yescharacteristic
629C22#1912850411yescharacteristic
630C22#1912850911yescharacteristic
631C22#1912851211yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—3322no—
642C4—6641no—
643C4—6641no—
644C4C2×#1458174911yescharacteristic
645C4C4×#1305059011yescharacteristic
646C4#1912823511yescharacteristic
647C4#1912823511yescharacteristic
648C2—44no—
649C2—6642no—
650C2#2490990611yescharacteristic
651C2#2491629111yescharacteristic
652C2#2491629511yescharacteristic
653C2#2491764711yescharacteristic
654C2#2491766211yescharacteristic
655C2#2491770611yescharacteristic
656C2#2491774011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2275C2
socleC23×C83C22×D4
prime radicalC22×#13050575C22
cosocleC22×C4×C83C24
hypercenter#2275D83
hypocenterC83#21126
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #24910107C2semidirect
3. #24908090C2semidirect
4. #24908083C2semidirect
5. #24908048C2semidirect
6. #24908041C2semidirect
9. #24907724C2semidirect
10. #24907710C2semidirect
11. #24907361C2semidirect
12. C4×#14581712C2semidirect
13. C4×#14581713C2semidirect
14. #24907215C2semidirect
15. C83×#2275C2semidirect
18. #19127811C4semidirect
70. C4×D332C2×C4semidirect
84. #14581723C2×C4semidirect
90. #14581712C2×C4semidirect
91. #14581703C2×C4semidirect
331. #2275D83semidirect
344. C83#21126semidirect

Maximal subgroups

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

subgroupindex
C2×#191278112
#249101072
#249080902
#249080832
#249080482
#249080412
#249078112
#249077242
#249077102
#249073612
C4×#145817122
C4×#145817132
#249072152
C83×#22752
#2112683