Finite Group Databasebeta

← #35426488

Subgroups — #35426488

Order 21248.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917247C211yesmaximal, characteristic
2#24916409C211yesmaximal, characteristic, complemented
3#24916029C211yesmaximal, characteristic
4#24908442C211yesmaximal, characteristic, complemented
5#24908235C211yesmaximal, characteristic, complemented
6#24908214C211yesmaximal, characteristic, complemented
7#24908208C211yesmaximal, characteristic, complemented
8#24908063C211yesmaximal, characteristic, complemented
9#24908060C211yesmaximal, characteristic, complemented
10#24907736C211yesmaximal, characteristic
11#24907319C211yesmaximal, characteristic, complemented
12#24907310C211yesmaximal, characteristic
13#24907298C211yesmaximal, characteristic
14#24907275C211yesmaximal, characteristic
15C83×#1866C211yesmaximal, 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#19127830—21no—
28#19127830—21no—
29#19127828—21no—
30#19127825—21no—
31C2×#14581727C2211yescharacteristic
32#19127739C2211yescharacteristic
33#19127737C2211yescharacteristic
34#19127729—21no—
35#19127729C2211yescharacteristic
36#19127729C2211yescharacteristic
37#19127727—21no—
38#19127727C2211yescharacteristic
39#19127724—21no—
40#19127724C2211yescharacteristic
41C2×Dic83⋊C8—21no—
42C2×Dic83⋊C8C2211yescharacteristic
43C2×Dic83⋊C8C2211yescharacteristic
44C2×#14581719C2211yescharacteristic
45#19127697C2211yescharacteristic
46#19127657C2211yescharacteristic
47#19127656C2211yescharacteristic
48M16×Dic83C2211yescharacteristic
49C2×#14581702C2211yescharacteristic
50C2×C8×Dic83C2211yescharacteristic
51#19127547C2211yescharacteristic
52#19127466C2211yescharacteristic
53#19127465C2211yescharacteristic
54#19127462C2211yescharacteristic
55#19127462C2211yescharacteristic
56#19127461C2211yescharacteristic
57#19127461C2211yescharacteristic
58#19127460C2211yescharacteristic
59#19127456C2211yescharacteristic
60C2×#131×C83C2211yescharacteristic
61C2×#108×C83C2211yescharacteristic
62#424×C83C2211yescharacteristic
63C2×#104×C83C2211yescharacteristic
64#416×C83C2211yescharacteristic
65#339×C83C2211yescharacteristic
66#336×C83C2211yescharacteristic
67#14581799—22no—
68#14581799—41no—
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×C4×Dic83—21no—
93C2×C4×Dic83C2311yescharacteristic
94C2×C4×Dic83C2311yescharacteristic
95#14581727—41no—
96#14581726C2×C411yescharacteristic
97#14581726C2×C411yescharacteristic
98#14581725C2×C411yescharacteristic
99#14581725C2×C411yescharacteristic
100#14581724C2×C411yescharacteristic
101#14581724C2×C411yescharacteristic
102#14581723C2×C411yescharacteristic
103#14581723C2×C411yescharacteristic
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊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
116Dic83⋊C8D411yescharacteristic
117Dic83⋊C8D411yescharacteristic
118Dic83⋊C8D411yescharacteristic
119Dic83⋊C8D411yescharacteristic
120#14581719Q814yescomplemented
121#14581716C2×C411yescharacteristic
122#14581716C2×C411yescharacteristic
123#14581713C2×C411yescharacteristic
124#14581713C2×C411yescharacteristic
125#14581703—21no—
126#14581702—21no—
127#14581702—21no—
128#14581702—41no—
129C4×C83⋊C8—21no—
130C8×Dic83—21no—
131C8×Dic83—21no—
132C8×Dic83—41no—
133C83⋊#95—21no—
134C83⋊#95—21no—
135C2×#41×C83C2311yescharacteristic
136#131×C83—22no—
137#131×C83—41no—
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#108×C83—41no—
147C4⋊C8×C83—21no—
148C4⋊C8×C83—21no—
149#104×C83—22no—
150#103×C83C2×C411yescharacteristic
151#103×C83C2×C411yescharacteristic
152#102×C83C2×C411yescharacteristic
153#102×C83C2×C411yescharacteristic
154C2×Dic166—21no—
155C2×Dic166—21no—
156C22×Dic83—21no—
157C22×Dic83—21no—
158C22×Dic83—21no—
159C22×Dic83#4111yescharacteristic
160C22×Dic83#4111yescharacteristic
161#13050574—21no—
162#13050574—21no—
163#13050574—21no—
164#13050574—42no—
165#13050574C2×Q812yes—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—42no—
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—
189C22⋊Dic83—21no—
190C22⋊Dic83—21no—
191C22⋊Dic83—21no—
192C22⋊Dic83—21no—
193C4×Dic83—21no—
194C4×Dic83—21no—
195C4×Dic83—21no—
196C4×Dic83—21no—
197C4×Dic83—21no—
198C4×Dic83C22×C411yescharacteristic
199C4×Dic83C22×C411yescharacteristic
200C4×Dic83C2×D411yescharacteristic
201C4×Dic83C2×D411yescharacteristic
202C4×Dic83C2×Q812yes—
203#41×C83—21no—
204#41×C83—21no—
205#41×C83C2×Q812yes—
206#41×C83#4112yes—
207C2×Q8×C83C22×C411yescharacteristic
208C2×D4×C83—21no—
209C2×D4×C83C22×C411yescharacteristic
210C22×C4×C83—21no—
211C22×C4×C83—21no—
212C22×C4×C83C2411yescharacteristic
213M16×C83—21no—
214M16×C83—21no—
215M16×C83—21no—
216M16×C83—21no—
217M16×C83—42no—
218C2×C8×C83—21no—
219C2×C8×C83—21no—
220C2×C8×C83—21no—
221C2×C8×C83—21no—
222C2×C8×C83—21no—
223C2×C8×C83—21no—
224C2×C8×C83—42no—
225C2×C8×C83C22×C411yescharacteristic
226C2×C8×C83C22×C411yescharacteristic
227C2×C8×C83C2×D411yescharacteristic
228C2×C8×C83C2×D411yescharacteristic
229C4⋊C4×C83C22×C411yescharacteristic
230C4⋊C4×C83C22×C411yescharacteristic
231C22⋊C4×C83—21no—
232C42×C83—21no—
233C42×C83—21no—
234C42×C83—21no—
235C23×C83—21no—
236C23×C83C2×#4111yescharacteristic
237Q8×C83—21no—
238Q8×C83C4×Q812yes—
239D4×C83—22no—
240D4×C83—22no—
241D4×C83—21no—
242D4×C83C4×Q812yes—
243C2×C4×C83—21no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—21no—
247C2×C4×C83—21no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83C23×C411yescharacteristic
252C2×C4×C83C22×D411yescharacteristic
253C2×C4×C83C22×Q811yescharacteristic
254C2×C4×C83C2×#4111yescharacteristic
255C2×C4×C83H(5,𝔽2)11yescharacteristic
256C2×C4×C832−1+411yescharacteristic
257C2×Dic83—21no—
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—21no—
261C2×Dic83—22no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—42no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—81no—
268C2×Dic83C4×D411yescharacteristic
269C2×Dic83C4×D411yescharacteristic
270C2×Dic83C4×D411yescharacteristic
271C2×Dic83C4×D411yescharacteristic
272Dic166—42no—
273Dic166—81no—
274C8×C83—21no—
275C8×C83—21no—
276C8×C83—21no—
277C8×C83—21no—
278C8×C83—41no—
279C8×C83—41no—
280C83⋊C8—21no—
281C83⋊C8—21no—
282C83⋊C8—21no—
283C83⋊C8—21no—
284C83⋊C8—22no—
285C83⋊C8—41no—
286C22×C83—21no—
287C22×C83—21no—
288C22×C83—22no—
289C22×C83—41no—
290C22×C83—41no—
291C22×C83C2×C4×D411yescharacteristic
292C22×C83#55211yescharacteristic
293C22×C83#55411yescharacteristic
294C4×C83—22no—
295C4×C83—41no—
296C4×C83—41no—
297C4×C83—81no—
298C4×C83C2×C4×Q811yescharacteristic
299C4×C83#52011yescharacteristic
300C4×C83#54611yescharacteristic
301C4×C83D4×Q811yescharacteristic
302Dic83—21no—
303Dic83—21no—
304Dic83—21no—
305Dic83—21no—
306Dic83—42no—
307Dic83—41no—
308Dic83—41no—
309Dic83—41no—
310Dic83—41no—
311#22984—831nomaximal, Sylow
312C2×C83—21no—
313C2×C83—21no—
314C2×C83—42no—
315C2×C83#236011yescharacteristic
316C2×C83#295511yescharacteristic
317C2×C83#295711yescharacteristic
318#3076—831no—
319#2962—831no—
320#2906—831no—
321#1995—831no—
322#1950—831no—
323#1946—831no—
324#1945—831no—
325#1922—831no—
326#1922—831no—
327#1866—831no—
328#1866D8311yescharacteristic
329#1768—831no—
330#1765—831no—
331#1764—831no—
332#1759—831no—
333C83#2298411yescharacteristic, complemented, Sylow
334C2×#131—831no—
335C2×#131C2×D8311yescharacteristic
336#533—831no—
337C4×#41—831no—
338#479—831no—
339#479—831no—
340#478—831no—
341#478—831no—
342#477—831no—
343#476—831no—
344#475—831no—
345#474—831no—
346#446—1661no—
347#446—1661no—
348#446—1661no—
349C8×Q8—1661no—
350C2×#108—831no—
351C2×#108C2×D8311yescharacteristic
352#427—831no—
353#427—831no—
354#424—831no—
355#424—831no—
356#424—831no—
357#424—1661no—
358#424—1661no—
359#424C2×D8311yescharacteristic
360#423—831no—
361#423—1661no—
362C2×C4⋊C8—831no—
363C2×C4⋊C8—831no—
364C2×C4⋊C8—1661no—
365C2×#104—831no—
366C2×#104C2×D8311yescharacteristic
367#416—831no—
368#416C2×D8311yescharacteristic
369#405—831no—
370#405—831no—
371C4×M16—831no—
372C2×#97—831no—
373C2×C4×C8—831no—
374#376—831no—
375#341—831no—
376#341—831no—
377#339—831no—
378#339—831no—
379#339—831no—
380#339—831no—
381#339—831no—
382#339C2×D8311yescharacteristic
383#336—831no—
384#336C2×D8311yescharacteristic
385C2×#41C22×D8311yescharacteristic
386#131—22no—
387#131—41no—
388#131—3321no—
389#131—1662no—
390C2×M16—21no—
391C2×M16—831no—
392C2×M16—831no—
393C2×M16—1661no—
394C2×M16C22×D8311yescharacteristic
395C2×M16C22×D8311yescharacteristic
396C22×C8—21no—
397C22×C8—831no—
398C22×C8—831no—
399C22×C8—1661no—
400C22×C8C22×D8311yescharacteristic
401C22×C8C22×D8311yescharacteristic
402C4⋊Q8—831no—
403C4⋊Q8—831no—
404#125—831no—
405#125—831no—
406C22⋊Q8—1661no—
407C22⋊Q8—1661no—
408C4×Q8—831no—
409C4×Q8—831no—
410C4×Q8—1661no—
411C4×Q8—1661no—
412C4×D4—831no—
413C4×D4—831no—
414C4×D4—1661no—
415C4×D4—1661no—
416#117—831no—
417#117—831no—
418#117—1661no—
419#117C22×D8311yescharacteristic
420C2×C42—831no—
421C2×C42—831no—
422C2×C42—1661no—
423C2×C42C22×D8311yescharacteristic
424#108—41no—
425#108—3321no—
426#107—831no—
427#107—831no—
428#107—831no—
429#107—831no—
430#106—831no—
431#106—831no—
432#106—831no—
433#106—831no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—831no—
437C4⋊C8—831no—
438C4⋊C8—831no—
439C4⋊C8—831no—
440C4⋊C8—831no—
441C4⋊C8—831no—
442C4⋊C8—831no—
443C4⋊C8—831no—
444C4⋊C8—1661no—
445C4⋊C8—1661no—
446C4⋊C8—1661no—
447C4⋊C8—1661no—
448C4⋊C8—1661no—
449C4⋊C8—1661no—
450C4⋊C8—1661no—
451C4⋊C8—1661no—
452#104—22no—
453#104—834no—
454#103—831no—
455#103—831no—
456#103C4×D8311yescharacteristic
457#103C4×D8311yescharacteristic
458#102—831no—
459#102—831no—
460#102C4×D8311yescharacteristic
461#102C4×D8311yescharacteristic
462#97—3321no—
463#97—1661no—
464#97—1661no—
465#97—1661no—
466C4×C8—3321no—
467C4×C8—1661no—
468C4×C8—1661no—
469C4×C8—1661no—
470#95—1661no—
471#95—1661no—
472#41—21no—
473#41—21no—
474#41Q8×D8312yes—
475#41#1305058912yes—
476C2×Q8—1661no—
477C2×Q8—1661no—
478C2×Q8C2×C4×D8311yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D8311yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—831no—
484C22×C4—831no—
485C22×C4—1661no—
486C22×C4—1661no—
487C22×C4—1661no—
488C22×C4C23×D8311yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—42no—
494M16—3322no—
495M16—832no—
496M16—1661no—
497M16—1661no—
498M16—1661no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—42no—
506C2×C8—3322no—
507C2×C8—831no—
508C2×C8—831no—
509C2×C8—831no—
510C2×C8—831no—
511C2×C8—1661no—
512C2×C8—1661no—
513C2×C8—1661no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8C2×C4×D8311yescharacteristic
518C2×C8C2×C4×D8311yescharacteristic
519C2×C8D4×D8311yescharacteristic
520C2×C8D4×D8311yescharacteristic
521C4⋊C4—831no—
522C4⋊C4—831no—
523C4⋊C4—831no—
524C4⋊C4—831no—
525C4⋊C4—1661no—
526C4⋊C4—1661no—
527C4⋊C4—1661no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4C2×C4×D8311yescharacteristic
534C4⋊C4C2×C4×D8311yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—1661no—
537C22⋊C4—1661no—
538C22⋊C4—1661no—
539C22⋊C4—1661no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—831no—
544C42—831no—
545C42—831no—
546C42—831no—
547C42—832no—
548C42—1661no—
549C42—1661no—
550C42—1661no—
551C42—1661no—
552C42—1661no—
553C23—21no—
554C23#41×D8311yescharacteristic
555Q8—21no—
556Q8—3322no—
557Q8—6641no—
558Q8#1458175712yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4#1458175712yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3322no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—831no—
577C2×C4—831no—
578C2×C4—831no—
579C2×C4—831no—
580C2×C4—6641no—
581C2×C4—1662no—
582C2×C4—1661no—
583C2×C4—1661no—
584C2×C4—1661no—
585C2×C4—1661no—
586C2×C4C22×C4×D8311yescharacteristic
587C2×C4C2×D4×D8311yescharacteristic
588C2×C4C2×Q8×D8311yescharacteristic
589C2×C4C2×#1305058911yescharacteristic
590C2×C4#1458183711yescharacteristic
591C2×C4#1458183911yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—3321no—
599C8—1661no—
600C8—1661no—
601C8—1662no—
602C8—1661no—
603C8—1661no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C4×D4×D8311yescharacteristic
610C22#1912838811yescharacteristic
611C22#1912840511yescharacteristic
612C4—22no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—3322no—
617C4—3321no—
618C4—3321no—
619C4—3321no—
620C4—3321no—
621C4—1661no—
622C4—1661no—
623C4—1661no—
624C4—1661no—
625C4C2×#1458175711yescharacteristic
626C4#1912818011yescharacteristic
627C4#1912833811yescharacteristic
628C4#1912836411yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2491046311yescharacteristic
633C2#2491636111yescharacteristic
634C2#2491637711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1866C2
socleC22×C83C2×C4×D4
prime radicalC2×#41×C83C23
cosocleC22×C4×C83C24
hypercenter#1866D83
hypocenterC83#22984
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #24916409C2semidirect
4. #24908442C2semidirect
5. #24908235C2semidirect
6. #24908214C2semidirect
7. #24908208C2semidirect
8. #24908063C2semidirect
9. #24908060C2semidirect
11. #24907319C2semidirect
19. #19128010C4semidirect
20. #19128006C4semidirect
21. #19128003C4semidirect
22. #19127999C4semidirect
23. #19127994C4semidirect
24. #19127989C4semidirect
25. #19127984C4semidirect
26. #19127979C4semidirect
120. #14581719Q8semidirect
333. C83#22984semidirect

Maximal subgroups

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

subgroupindex
#249172472
#249164092
#249160292
#249084422
#249082352
#249082142
#249082082
#249080632
#249080602
#249077362
#249073192
#249073102
#249072982
#249072752
C83×#18662
#2298483