Finite Group Databasebeta

← #35408715

Subgroups — #35408715

Order 21248.

950 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#24909927C211yesmaximal, characteristic, complemented
3#24908338C211yesmaximal, characteristic
4#24908328C211yesmaximal, characteristic
5#24908203C211yesmaximal, characteristic, complemented
6#24908188C211yesmaximal, characteristic, complemented
7#24908163C211yesmaximal, characteristic, complemented
8#24908159C211yesmaximal, characteristic, complemented
9#24908149C211yesmaximal, characteristic, complemented
10#24908145C211yesmaximal, characteristic, complemented
11#24907361C211yesmaximal, characteristic, complemented
12#24907291C211yesmaximal, characteristic, complemented
13#24907288C211yesmaximal, characteristic, complemented
14C4×#14581703C211yesmaximal, characteristic, complemented
15C83×#2293C211yesmaximal, characteristic, complemented
16C22×#13050575C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18C2×#14581749—21no—
19#19127811C418yescomplemented
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×#14581703C2211yescharacteristic
32C2×#14581703C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34C2×C4×C83⋊C8C2211yescharacteristic
35#19127603C2211yescharacteristic
36#19127603C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127592—21no—
40#19127592—21no—
41C4×C4⋊Dic83C2211yescharacteristic
42C4×(C2×D83)⋊C4—21no—
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49C2×C4×D4×C83C2211yescharacteristic
50C2×#117×C83—21no—
51#390×C83C2211yescharacteristic
52#390×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#388×C83—21no—
56#388×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C43×C83C2211yescharacteristic
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×C414yes—
71#14581749—44no—
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×C4×Dic83—21no—
83#14581725—22no—
84#14581723—22no—
85#14581723C2×C414yes—
86#14581713—22no—
87#14581713—22no—
88#14581713C2×C414yes—
89#14581712—22no—
90#14581712—22no—
91#14581712C2×C414yes—
92#14581703—22no—
93#14581703—22no—
94#14581703—22no—
95#14581703C2×C414yes—
96C4×C83⋊C8—22no—
97C4×C83⋊C8—22no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×D4×C83C2311yescharacteristic
102C23×C4×C83—21no—
103C4×D4×C83—22no—
104C4×D4×C83C2×C414yes—
105#117×C83—44no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83—21no—
115C2×C22⋊C4×C83—21no—
116C2×C42×C83—21no—
117C2×C42×C83D411yescharacteristic
118C2×C42×C83D411yescharacteristic
119C2×C42×C83D411yescharacteristic
120C2×C42×C83D411yescharacteristic
121C2×C42×C83C2311yescharacteristic
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127C23×D83—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332—21no—
132C2×D332—21no—
133C2×D332C22×C412yes—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C2×C4×D83—24no—
138C2×C4×D83—42no—
139C2×C4×D83—42no—
140#13050575—28no—
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⋊C8C22×C412yes—
148C2×C83⋊C8C22×C412yes—
149C2×C83⋊C8#4112yes—
150C2×C83⋊C8#4112yes—
151C2×C83⋊C8#4112yes—
152C2×C83⋊C8#4112yes—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83C22×C412yes—
159Dic83⋊C4—42no—
160Dic83⋊C4—42no—
161(C2×D83)⋊C4—24no—
162(C2×D83)⋊C4—42no—
163(C2×D83)⋊C4—42no—
164(C2×D83)⋊C4—42no—
165(C2×D83)⋊C4—42no—
166C4×Dic83—42no—
167C24×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C2×D4×C83—21no—
171C2×D4×C83—21no—
172C2×D4×C83—21no—
173C2×D4×C83C22×C412yes—
174C22×C4×C83—21no—
175C22×C4×C83—24no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—42no—
179C22×C4×C83—42no—
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2411yescharacteristic
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83C22×C412yes—
195C22⋊C4×C83—24no—
196C22⋊C4×C83—42no—
197C22⋊C4×C83—42no—
198C22⋊C4×C83—42no—
199C22⋊C4×C83—42no—
200C42×C83—22no—
201C42×C83—22no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—41no—
207C42×C83—41no—
208C42×C83—42no—
209C42×C83C22×C412yes—
210C42×C83#4112yes—
211C23×C83—22no—
212C23×C83—41no—
213C23×C83—41no—
214C23×C83—41no—
215C23×C83—41no—
216C23×C83—41no—
217C23×C83—41no—
218C23×C83C22×D411yescharacteristic
219C22×D83—22no—
220C22×D83—41no—
221C22×D83—41no—
222C22×D83—41no—
223C22×D83—41no—
224C22×D83—41no—
225C22×D83—41no—
226D4×C83—22no—
227D4×C83—22no—
228D4×C83—41no—
229D4×C83—41no—
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—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—48no—
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×C83C4×D412yes—
255C2×C4×C83C4×D412yes—
256C2×C4×C83AGL(1,ℤ/8)12yes—
257C2×C4×C83AGL(1,ℤ/8)12yes—
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—48no—
278C83⋊C8—22no—
279C83⋊C8—22no—
280C83⋊C8—22no—
281C83⋊C8#11714yescomplemented
282C22×C83—42no—
283C22×C83—42no—
284C22×C83—42no—
285C22×C83—81no—
286C22×C83—81no—
287C22×C83—81no—
288C22×C83—81no—
289C22×C83C2×C4×D411yescharacteristic
290C22×C83#53211yescharacteristic
291C22×C83#53211yescharacteristic
292C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C83#57511yescharacteristic
295C22×C83#57511yescharacteristic
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—22no—
306C4×C83—22no—
307C4×C83—41no—
308C4×C83—41no—
309C4×C83—42no—
310C4×C83—81no—
311C4×C83—81no—
312C4×C83C2×#11711yescharacteristic
313C4×C83C4×#4111yescharacteristic
314C4×C83#52911yescharacteristic
315C4×C83#52911yescharacteristic
316Dic83—42no—
317Dic83—81no—
318Dic83—81no—
319#21286—831nomaximal, Sylow
320C2×C83—82no—
321C2×C83#229311yescharacteristic
322C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C83#318011yescharacteristic
325C2×C83#318011yescharacteristic
326C2×C83#318311yescharacteristic
327C2×C83#318411yescharacteristic
328D83—82no—
329C2×#442—831no—
330#2293—831no—
331#2293D8311yescharacteristic, complemented
332#1972—831no—
333#1970—831no—
334#1944—831no—
335#1941—831no—
336#1936—831no—
337#1936—831no—
338#1934—831no—
339#1934—831no—
340#1776—831no—
341#1762—831no—
342#1762—831no—
343C4×#97—831no—
344C83#2128611yescharacteristic, complemented, Sylow
345C22×D8—831no—
346C2×C4×D4—831no—
347C2×C4×D4C2×D8311yescharacteristic
348C2×#117—21no—
349C2×#117—1661no—
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×#97—831no—
362C2×#97—831no—
363C2×#97—831no—
364C2×C4×C8—831no—
365C2×C4×C8—831no—
366#390—831no—
367#390—831no—
368#390C2×D8311yescharacteristic
369#390C2×D8311yescharacteristic
370#389—831no—
371#389—831no—
372#389C2×D8311yescharacteristic
373#389C2×D8311yescharacteristic
374#388—21no—
375#388—21no—
376#388—1661no—
377#388—1661no—
378C4×C4⋊C4—831no—
379C4×C4⋊C4C2×D8311yescharacteristic
380C4×C22⋊C4—21no—
381C4×C22⋊C4—1661no—
382C43C2×D8311yescharacteristic
383#340—831no—
384#340—831no—
385#340—831no—
386#340—831no—
387#340—831no—
388#340—831no—
389C22×D4—831no—
390C22×D4C22×D8311yescharacteristic
391C23×C4—21no—
392C23×C4—1661no—
393C2×D8—834no—
394C2×D8—1662no—
395C2×D8—1662no—
396C22×C8—831no—
397C22×C8—831no—
398C22×C8—831no—
399C22×C8—831no—
400C4×D4—22no—
401C4×D4—834no—
402C4×D4—1662no—
403C4×D4C4×D8314yes—
404#117—44no—
405#117—3324no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—831no—
409C2×C4⋊C4—831no—
410C2×C4⋊C4—831no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4C22×D8311yescharacteristic
414C2×C4⋊C4C22×D8311yescharacteristic
415C2×C4⋊C4C22×D8311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
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—1661no—
428C2×C42C22⋊D8311yescharacteristic
429C2×C42C22⋊D8311yescharacteristic
430C2×C42C22⋊D8311yescharacteristic
431C2×C42C22⋊D8311yescharacteristic
432C2×C42C22×D8311yescharacteristic
433C2×C42C22×D8311yescharacteristic
434C2×C42C22×D8311yescharacteristic
435#107—1662no—
436#106—834no—
437#106—1662no—
438#102—834no—
439#102—834no—
440#102—1662no—
441#102—1662no—
442#102—1662no—
443#102—1662no—
444#97—834no—
445#97—1662no—
446#97—1662no—
447#97—1662no—
448C4×C8—1662no—
449C4×C8—1662no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1661no—
454#95—1661no—
455#95—1661no—
456C24—21no—
457C24—1661no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—832no—
464C2×D4—1661no—
465C2×D4—1661no—
466C2×D4—1661no—
467C2×D4—1661no—
468C2×D4—1661no—
469C2×D4C2×C4×D8312yes—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—3322no—
477C22×C4—3322no—
478C22×C4—1661no—
479C22×C4—1661no—
480C22×C4—1661no—
481C22×C4—1664no—
482C22×C4C2×C22⋊D8311yescharacteristic
483C22×C4C2×C22⋊D8311yescharacteristic
484C22×C4C2×C22⋊D8311yescharacteristic
485C22×C4C2×C22⋊D8311yescharacteristic
486C22×C4C2×C22⋊D8311yescharacteristic
487C22×C4C2×C22⋊D8311yescharacteristic
488C22×C4C23×D8311yescharacteristic
489D8—1668no—
490C2×C8—832no—
491C2×C8—832no—
492C2×C8—832no—
493C2×C8—832no—
494C2×C8—832no—
495C2×C8—832no—
496C2×C8—1661no—
497C2×C8—1661no—
498C2×C8—1661no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—3322no—
510C4⋊C4—3322no—
511C4⋊C4—832no—
512C4⋊C4—1661no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4C2×C4×D8312yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—3322no—
524C22⋊C4—3322no—
525C22⋊C4—3322no—
526C22⋊C4—3322no—
527C22⋊C4—1664no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—41no—
535C42—41no—
536C42—42no—
537C42—3322no—
538C42C2×C4×D8312yes—
539C42#1305059012yes—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—3321no—
548C23—3321no—
549C23—3321no—
550C23—3321no—
551C23—3321no—
552C23—3321no—
553C23—1662no—
554C23C22×C22⋊D8311yescharacteristic
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—3321no—
560D4—3321no—
561D4—1662no—
562D4—1662no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—3328no—
588C2×C4—3321no—
589C2×C4—3321no—
590C2×C4—3321no—
591C2×C4—3321no—
592C2×C4—3321no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—1662no—
597C2×C4C4×C22⋊D8312yes—
598C2×C4C4×C22⋊D8312yes—
599C2×C4#1458181712yes—
600C2×C4#1458181812yes—
601C2×C4C22×C4×D8311yescharacteristic
602C2×C4#41×D8311yescharacteristic
603C2×C4#41×D8311yescharacteristic
604C2×C4#41×D8311yescharacteristic
605C2×C4#41×D8311yescharacteristic
606C2×C4C2×#1305059011yescharacteristic
607C8—834no—
608C8—1662no—
609C8—1662no—
610C8—1662no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—3322no—
619C22—3322no—
620C22—3322no—
621C22—6641no—
622C22—6641no—
623C22—6641no—
624C22—6641no—
625C22C2×C4×C22⋊D8311yescharacteristic
626C22#1912824711yescharacteristic
627C22#1912824711yescharacteristic
628C22C2×#1458181711yescharacteristic
629C22C2×#1458181811yescharacteristic
630C22#1912849911yescharacteristic
631C22#1912849911yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—3322no—
642C4—6641no—
643C4—6641no—
644C4#117×D8311yescharacteristic
645C4#1912816911yescharacteristic
646C4#1912822911yescharacteristic
647C4#1912822911yescharacteristic
648C2—82no—
649C2—6642no—
650C2#2490992211yescharacteristic
651C2C4×#1458181711yescharacteristic
652C2C4×#1458181811yescharacteristic
653C2#2491815511yescharacteristic
654C2#2491816211yescharacteristic
655C2#2491818611yescharacteristic
656C2#2491819711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2293C2
socleC23×C83C22×D4
prime radicalC22×#13050575C22
cosocleC22×C4×C83C24
hypercenter#2293D83
hypocenterC83#21286
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #24909927C2semidirect
5. #24908203C2semidirect
6. #24908188C2semidirect
7. #24908163C2semidirect
8. #24908159C2semidirect
9. #24908149C2semidirect
10. #24908145C2semidirect
11. #24907361C2semidirect
12. #24907291C2semidirect
13. #24907288C2semidirect
14. C4×#14581703C2semidirect
15. C83×#2293C2semidirect
19. #19127811C4semidirect
281. C83⋊C8#117semidirect
331. #2293D83semidirect
344. C83#21286semidirect

Maximal subgroups

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

subgroupindex
C2×#191278112
#249099272
#249083382
#249083282
#249082032
#249081882
#249081632
#249081592
#249081492
#249081452
#249073612
#249072912
#249072882
C4×#145817032
C83×#22932
#2128683