Finite Group Databasebeta

← #35412252

Subgroups — #35412252

Order 21248.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917247C211yesmaximal, characteristic
2#24916401C211yesmaximal, characteristic, complemented
3#24915889C211yesmaximal, characteristic
4#24908405C211yesmaximal, characteristic, complemented
5#24908237C211yesmaximal, characteristic, complemented
6#24908065C211yesmaximal, characteristic, complemented
7#24908061C211yesmaximal, characteristic, complemented
8#24907740C211yesmaximal, characteristic
9#24907736C211yesmaximal, characteristic
10#24907481C211yesmaximal, characteristic
11#24907348C211yesmaximal, characteristic, complemented
12#24907342C211yesmaximal, characteristic, complemented
13#24907337C211yesmaximal, characteristic, complemented
14#24907305C211yesmaximal, characteristic
15C83×#1764C211yesmaximal, characteristic
16C2×#14581799C2211yescharacteristic
17#19128254C2211yescharacteristic
18#41×Dic83C2211yescharacteristic
19#19128010C411yescharacteristic, complemented
20#19128006C411yescharacteristic, complemented
21#19128003C411yescharacteristic, complemented
22#19127999C411yescharacteristic, complemented
23#19127994C411yescharacteristic, complemented
24#19127989C411yescharacteristic, complemented
25#19127984C411yescharacteristic, complemented
26#19127979C411yescharacteristic, complemented
27#19127831—21no—
28#19127831—21no—
29#19127829—21no—
30#19127829—21no—
31C2×#14581728C2211yescharacteristic
32C2×#14581727C2211yescharacteristic
33#19127739C2211yescharacteristic
34#19127737C2211yescharacteristic
35#19127729C2211yescharacteristic
36#19127728C2211yescharacteristic
37#19127724C2211yescharacteristic
38C2×Dic83⋊C8C2211yescharacteristic
39C2×#14581719C2211yescharacteristic
40#19127697C2211yescharacteristic
41#19127659—21no—
42#19127659C2211yescharacteristic
43#19127656—21no—
44#19127656C2211yescharacteristic
45#19127656C2211yescharacteristic
46M16×Dic83—21no—
47M16×Dic83C2211yescharacteristic
48C2×#14581702—21no—
49C2×#14581702C2211yescharacteristic
50C2×#14581702C2211yescharacteristic
51C4×Dic83⋊C4C2211yescharacteristic
52#19127465C2211yescharacteristic
53#19127462C2211yescharacteristic
54#19127462C2211yescharacteristic
55#19127461C2211yescharacteristic
56#19127461C2211yescharacteristic
57#19127460C2211yescharacteristic
58#19127456C2211yescharacteristic
59#19127455C2211yescharacteristic
60C2×#131×C83C2211yescharacteristic
61C2×#104×C83C2211yescharacteristic
62#416×C83C2211yescharacteristic
63#405×C83C2211yescharacteristic
64C2×C4×C8×C83C2211yescharacteristic
65#341×C83C2211yescharacteristic
66#339×C83C2211yescharacteristic
67#14581799—22no—
68#14581799—22no—
69C2×#13050574—21no—
70C2×#13050574C2311yescharacteristic
71C2×#13050574C2311yescharacteristic
72C22×C83⋊C8—21no—
73C22×C83⋊C8C2311yescharacteristic
74C22×C83⋊C8C2311yescharacteristic
75#14581791C2×C411yescharacteristic
76#14581791C2×C411yescharacteristic
77#14581783C2×C411yescharacteristic
78#14581783C2×C411yescharacteristic
79#14581769—21no—
80#14581769—21no—
81Q8×Dic83C2×C411yescharacteristic
82Q8×Dic83C2×C411yescharacteristic
83C4×Dic166—21no—
84C4×Dic166—21no—
85D4×Dic83—21no—
86D4×Dic83—21no—
87D4×Dic83C2×C411yescharacteristic
88D4×Dic83C2×C411yescharacteristic
89#14581751—21no—
90#14581751C2311yescharacteristic
91#14581751C2311yescharacteristic
92C2×Dic83⋊C4—21no—
93C2×C4×Dic83—21no—
94C2×C4×Dic83C2311yescharacteristic
95C2×C4×Dic83C2311yescharacteristic
96#14581728—22no—
97#14581727—22no—
98#14581726C2×C411yescharacteristic
99#14581726C2×C411yescharacteristic
100#14581725C2×C411yescharacteristic
101#14581725C2×C411yescharacteristic
102#14581724C2×C411yescharacteristic
103#14581724C2×C411yescharacteristic
104#14581723C2×C411yescharacteristic
105#14581723C2×C411yescharacteristic
106C4⋊(C83⋊C8)—21no—
107C4⋊(C83⋊C8)—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110Dic83⋊C8—21no—
111Dic83⋊C8—21no—
112Dic83⋊C8C2×C411yescharacteristic
113Dic83⋊C8C2×C411yescharacteristic
114Dic83⋊C8C2×C411yescharacteristic
115Dic83⋊C8C2×C411yescharacteristic
116#14581719—22no—
117#14581716C2×C411yescharacteristic
118#14581716C2×C411yescharacteristic
119#14581713C2×C411yescharacteristic
120#14581713C2×C411yescharacteristic
121#14581702—21no—
122#14581702—21no—
123#14581702—21no—
124#14581702—21no—
125#14581702—22no—
126#14581702D411yescharacteristic
127#14581702D411yescharacteristic
128#14581702D411yescharacteristic
129#14581702D411yescharacteristic
130C8×Dic83—21no—
131C8×Dic83—21no—
132C8×Dic83—21no—
133C8×Dic83—21no—
134C83⋊#95—21no—
135C2×#41×C83C2311yescharacteristic
136#131×C83—22no—
137#131×C83Q814yescomplemented
138C2×M16×C83—21no—
139C2×M16×C83C2311yescharacteristic
140C2×M16×C83C2311yescharacteristic
141C22×C8×C83—21no—
142C22×C8×C83C2311yescharacteristic
143C22×C8×C83C2311yescharacteristic
144#117×C83C2311yescharacteristic
145C2×C42×C83C2311yescharacteristic
146#104×C83—22no—
147#103×C83C2×C411yescharacteristic
148#103×C83C2×C411yescharacteristic
149#102×C83C2×C411yescharacteristic
150#102×C83C2×C411yescharacteristic
151#97×C83—21no—
152C4×C8×C83—21no—
153C4×C8×C83—22no—
154C2×Dic166—21no—
155C2×Dic166—21no—
156C22×Dic83—21no—
157C22×Dic83—21no—
158C22×Dic83—21no—
159C22×Dic83#4111yescharacteristic
160C22×Dic83#4111yescharacteristic
161#13050574—24no—
162#13050574—21no—
163#13050574—21no—
164#13050574—21no—
165#13050574#4112yes—
166C2×C83⋊C8—24no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8C22×C411yescharacteristic
174C2×C83⋊C8C22×C411yescharacteristic
175C2×C83⋊C8C2×D411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C4⋊Dic83—21no—
178C4⋊Dic83—21no—
179C4⋊Dic83—21no—
180C4⋊Dic83—21no—
181C4⋊Dic83C22×C411yescharacteristic
182C4⋊Dic83C22×C411yescharacteristic
183C4⋊Dic83C22×C411yescharacteristic
184C4⋊Dic83C22×C411yescharacteristic
185Dic83⋊C4—21no—
186Dic83⋊C4—21no—
187Dic83⋊C4—21no—
188Dic83⋊C4—21no—
189Dic83⋊C4—42no—
190C22⋊Dic83—21no—
191C22⋊Dic83—21no—
192C22⋊Dic83—21no—
193C22⋊Dic83—21no—
194C4×Dic83—21no—
195C4×Dic83—21no—
196C4×Dic83—21no—
197C4×Dic83—21no—
198C4×Dic83—21no—
199C4×Dic83—21no—
200C4×Dic83C22×C411yescharacteristic
201C4×Dic83C22×C411yescharacteristic
202C4×Dic83C2×D411yescharacteristic
203C4×Dic83C2×D411yescharacteristic
204#41×C83—21no—
205#41×C83—21no—
206#41×C83C2×Q812yes—
207#41×C83#4112yes—
208C2×Q8×C83C22×C411yescharacteristic
209C2×D4×C83—21no—
210C2×D4×C83C22×C411yescharacteristic
211C22×C4×C83—21no—
212C22×C4×C83—21no—
213C22×C4×C83C2411yescharacteristic
214M16×C83—21no—
215M16×C83—24no—
216M16×C83—21no—
217M16×C83—21no—
218M16×C83C2×Q812yes—
219C2×C8×C83—21no—
220C2×C8×C83—21no—
221C2×C8×C83—21no—
222C2×C8×C83—24no—
223C2×C8×C83—21no—
224C2×C8×C83C22×C411yescharacteristic
225C2×C8×C83C22×C411yescharacteristic
226C2×C8×C83C2×D411yescharacteristic
227C2×C8×C83C2×D411yescharacteristic
228C2×C8×C83C2×Q812yes—
229C2×C8×C83#4112yes—
230C4⋊C4×C83C22×C411yescharacteristic
231C4⋊C4×C83C22×C411yescharacteristic
232C22⋊C4×C83—21no—
233C42×C83—21no—
234C42×C83—21no—
235C42×C83—21no—
236C23×C83—21no—
237C23×C83C2×#4111yescharacteristic
238Q8×C83—21no—
239Q8×C83C4×Q812yes—
240D4×C83—22no—
241D4×C83—22no—
242D4×C83—21no—
243D4×C83C4×Q812yes—
244C2×C4×C83—21no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—21no—
249C2×C4×C83—21no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83C23×C411yescharacteristic
253C2×C4×C83C22×D411yescharacteristic
254C2×C4×C83C22×Q811yescharacteristic
255C2×C4×C83C2×#4111yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C2×C4×C83C2×#4111yescharacteristic
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—21no—
261C2×Dic83—21no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—81no—
268C2×Dic83—81no—
269C2×Dic83C4×D411yescharacteristic
270C2×Dic83C4×D411yescharacteristic
271C2×Dic83C4×D411yescharacteristic
272C2×Dic83C4×D411yescharacteristic
273Dic166—81no—
274Dic166—81no—
275C8×C83—21no—
276C8×C83—21no—
277C8×C83—22no—
278C8×C83—41no—
279C8×C83C22⋊Q812yes—
280C8×C83C22⋊Q812yes—
281C83⋊C8—21no—
282C83⋊C8—21no—
283C83⋊C8—21no—
284C83⋊C8—22no—
285C83⋊C8—21no—
286C83⋊C8—41no—
287C22×C83—21no—
288C22×C83—21no—
289C22×C83—22no—
290C22×C83—41no—
291C22×C83—41no—
292C22×C83C2×C4×D411yescharacteristic
293C22×C83#52911yescharacteristic
294C22×C83#53311yescharacteristic
295C4×C83—22no—
296C4×C83—42no—
297C4×C83—41no—
298C4×C83—41no—
299C4×C83C2×C4×Q811yescharacteristic
300C4×C83C4×#4111yescharacteristic
301C4×C83C2×C22⋊Q811yescharacteristic
302C4×C83#52511yescharacteristic
303Dic83—21no—
304Dic83—21no—
305Dic83—21no—
306Dic83—21no—
307Dic83—41no—
308Dic83—41no—
309Dic83—41no—
310Dic83—41no—
311Dic83—81no—
312#21629—831nomaximal, Sylow
313C2×C83—21no—
314C2×C83—21no—
315C2×C83—42no—
316C2×C83C4×C22⋊Q811yescharacteristic
317C2×C83#295611yescharacteristic
318C2×C83#295711yescharacteristic
319#3076—831no—
320#2961—831no—
321#2875—831no—
322#1986—831no—
323#1950—831no—
324#1922—831no—
325#1922—831no—
326#1866—831no—
327#1866—831no—
328#1806—831no—
329#1773—831no—
330#1772—831no—
331#1771—831no—
332#1765—831no—
333#1764D8311yescharacteristic
334C83#2162911yescharacteristic, complemented, Sylow
335C2×#131—831no—
336C2×#131C2×D8311yescharacteristic
337#533—831no—
338C4×#41—831no—
339#479—831no—
340#479—831no—
341#478—831no—
342#478—831no—
343#477—831no—
344#476—831no—
345#475—831no—
346#474—831no—
347#446—1661no—
348#446—1661no—
349#446—1661no—
350#446—1661no—
351C2×#108—831no—
352C2×#108—831no—
353#427—831no—
354#427—831no—
355#424—831no—
356#424—831no—
357#423—831no—
358C2×C4⋊C8—831no—
359C2×#104—831no—
360C2×#104C2×D8311yescharacteristic
361#416—831no—
362#416C2×D8311yescharacteristic
363#405—831no—
364#405—831no—
365#405—831no—
366#405—1661no—
367#405—1661no—
368#405C2×D8311yescharacteristic
369C4×M16—831no—
370C4×M16—1661no—
371C2×#97—831no—
372C2×#97—831no—
373C2×#97—1661no—
374C2×C4×C8C2×D8311yescharacteristic
375C4×C4⋊C4—831no—
376#341—831no—
377#341C2×D8311yescharacteristic
378#339—831no—
379#339—831no—
380#339—831no—
381#339—831no—
382#339—831no—
383#339C2×D8311yescharacteristic
384#336—831no—
385#336—831no—
386C2×#41C22×D8311yescharacteristic
387#131—22no—
388#131—1662no—
389#131—1662no—
390#131Dic16614yescomplemented
391C2×M16—21no—
392C2×M16—831no—
393C2×M16—831no—
394C2×M16—1661no—
395C2×M16C22×D8311yescharacteristic
396C2×M16C22×D8311yescharacteristic
397C22×C8—21no—
398C22×C8—831no—
399C22×C8—831no—
400C22×C8—1661no—
401C22×C8C22×D8311yescharacteristic
402C22×C8C22×D8311yescharacteristic
403C4⋊Q8—831no—
404C4⋊Q8—831no—
405#125—831no—
406#125—831no—
407C22⋊Q8—1661no—
408C22⋊Q8—1661no—
409C4×Q8—831no—
410C4×Q8—831no—
411C4×Q8—1661no—
412C4×Q8—1661no—
413C4×D4—831no—
414C4×D4—831no—
415C4×D4—1661no—
416C4×D4—1661no—
417#117—831no—
418#117—831no—
419#117—1661no—
420#117C22×D8311yescharacteristic
421C2×C4⋊C4—1661no—
422C2×C42—831no—
423C2×C42—831no—
424C2×C42—1661no—
425C2×C42C22×D8311yescharacteristic
426#108—1662no—
427#108—1662no—
428#107—831no—
429#107—831no—
430#107—831no—
431#107—831no—
432#106—831no—
433#106—831no—
434#106—831no—
435#106—831no—
436C4⋊C8—831no—
437C4⋊C8—831no—
438C4⋊C8—831no—
439C4⋊C8—831no—
440C4⋊C8—1661no—
441C4⋊C8—1661no—
442C4⋊C8—1661no—
443C4⋊C8—1661no—
444C4⋊C8—1661no—
445C4⋊C8—1661no—
446#104—22no—
447#104—1662no—
448#103—831no—
449#103—831no—
450#103C4×D8311yescharacteristic
451#103C4×D8311yescharacteristic
452#102—831no—
453#102—831no—
454#102C4×D8311yescharacteristic
455#102C4×D8311yescharacteristic
456#97—21no—
457#97—831no—
458#97—831no—
459#97—831no—
460#97—831no—
461#97—1661no—
462#97—1661no—
463#97—1661no—
464#97—1661no—
465#97—1662no—
466C4×C8—21no—
467C4×C8—22no—
468C4×C8—1661no—
469C4×C8—1661no—
470C4×C8—1661no—
471C4×C8—1661no—
472#95—1661no—
473#41—21no—
474#41—21no—
475#41C2×Dic16612yes—
476#41#1305059012yes—
477C2×Q8—1661no—
478C2×Q8—1661no—
479C2×Q8C2×C4×D8311yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D8311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—831no—
485C22×C4—831no—
486C22×C4—1661no—
487C22×C4—1661no—
488C22×C4—1661no—
489C22×C4C23×D8311yescharacteristic
490M16—21no—
491M16—24no—
492M16—21no—
493M16—21no—
494M16—832no—
495M16—1661no—
496M16—1661no—
497M16—1664no—
498M16—1661no—
499M16C2×Dic16612yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—24no—
504C2×C8—21no—
505C2×C8—831no—
506C2×C8—831no—
507C2×C8—831no—
508C2×C8—831no—
509C2×C8—1661no—
510C2×C8—1664no—
511C2×C8—1661no—
512C2×C8—1661no—
513C2×C8—1661no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8C2×C4×D8311yescharacteristic
517C2×C8C2×C4×D8311yescharacteristic
518C2×C8D4×D8311yescharacteristic
519C2×C8D4×D8311yescharacteristic
520C2×C8C2×Dic16612yes—
521C2×C8#1305058912yes—
522C4⋊C4—3322no—
523C4⋊C4—831no—
524C4⋊C4—831no—
525C4⋊C4—831no—
526C4⋊C4—831no—
527C4⋊C4—1661no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4—1661no—
534C4⋊C4—1661no—
535C4⋊C4C2×C4×D8311yescharacteristic
536C4⋊C4C2×C4×D8311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—1661no—
539C22⋊C4—1661no—
540C22⋊C4—1661no—
541C22⋊C4—1661no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—831no—
546C42—831no—
547C42—831no—
548C42—831no—
549C42—1661no—
550C42—1661no—
551C42—1661no—
552C42—1661no—
553C42—1661no—
554C42—1661no—
555C23—21no—
556C23#41×D8311yescharacteristic
557Q8—21no—
558Q8—6641no—
559Q8—6641no—
560Q8C4×Dic16612yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4C4×Dic16612yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—831no—
579C2×C4—831no—
580C2×C4—831no—
581C2×C4—831no—
582C2×C4—6641no—
583C2×C4—6641no—
584C2×C4—1661no—
585C2×C4—1661no—
586C2×C4—1661no—
587C2×C4—1661no—
588C2×C4C22×C4×D8311yescharacteristic
589C2×C4C2×D4×D8311yescharacteristic
590C2×C4C22×Dic16611yescharacteristic
591C2×C4#41×D8311yescharacteristic
592C2×C4C2×#1305058911yescharacteristic
593C2×C4C2×#1305059011yescharacteristic
594C8—21no—
595C8—21no—
596C8—22no—
597C8—41no—
598C8—3321no—
599C8—1661no—
600C8—1661no—
601C8—1661no—
602C8—1661no—
603C8—1662no—
604C8#1458176912yes—
605C8#1458176912yes—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C4×D4×D8311yescharacteristic
612C22#1912823111yescharacteristic
613C22#1912825411yescharacteristic
614C4—22no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—3321no—
619C4—3321no—
620C4—3321no—
621C4—3321no—
622C4—6641no—
623C4—1661no—
624C4—1661no—
625C4—1661no—
626C4—1661no—
627C4C2×C4×Dic16611yescharacteristic
628C4C4×#1305058911yescharacteristic
629C4C2×#1458176911yescharacteristic
630C4#1912821011yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C4×#1458176911yescharacteristic
635C2#2491636811yescharacteristic
636C2#2491637511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1764C2
socleC22×C83C2×C4×D4
prime radicalC2×#41×C83C23
cosocleC22×C4×C83C24
hypercenter#1764D83
hypocenterC83#21629
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1764

Complements

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

NH ≅ G/Ntype
2. #24916401C2semidirect
4. #24908405C2semidirect
5. #24908237C2semidirect
6. #24908065C2semidirect
7. #24908061C2semidirect
11. #24907348C2semidirect
12. #24907342C2semidirect
13. #24907337C2semidirect
19. #19128010C4semidirect
20. #19128006C4semidirect
21. #19128003C4semidirect
22. #19127999C4semidirect
23. #19127994C4semidirect
24. #19127989C4semidirect
25. #19127984C4semidirect
26. #19127979C4semidirect
137. #131×C83Q8semidirect
334. C83#21629semidirect
390. #131Dic166semidirect

Maximal subgroups

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

subgroupindex
#249172472
#249164012
#249158892
#249084052
#249082372
#249080652
#249080612
#249077402
#249077362
#249074812
#249073482
#249073422
#249073372
#249073052
C83×#17642
#2162983