Finite Group Databasebeta

← #35423849

Subgroups — #35423849

Order 21248.

740 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916231C211yesmaximal, characteristic
2#24916127C211yesmaximal, characteristic, complemented
3#24916029C211yesmaximal, characteristic, complemented
4#24908174C211yesmaximal, characteristic, complemented
5#24908172C211yesmaximal, characteristic, complemented
6#24908062C211yesmaximal, characteristic, complemented
7#24908060C211yesmaximal, characteristic, complemented
8#24907790C211yesmaximal, characteristic
9#24907788C211yesmaximal, characteristic
10#24907599C211yesmaximal, characteristic, complemented
11#24907482C211yesmaximal, characteristic, complemented
12#24907481C211yesmaximal, characteristic, complemented
13#24907331C211yesmaximal, characteristic, complemented
14#24907310C211yesmaximal, characteristic, complemented
15C83×#1765C211yesmaximal, characteristic, complemented
16C2×#14581812C2211yescharacteristic
17C2×#14581799C2211yescharacteristic
18#41×Dic83C2211yescharacteristic
19#19128169C2211yescharacteristic
20#19127793C411yescharacteristic, complemented
21#19127793C411yescharacteristic, complemented
22#19127792C411yescharacteristic, complemented
23#19127792C411yescharacteristic, complemented
24#19127786C411yescharacteristic, complemented
25#19127786C411yescharacteristic, complemented
26#19127781C411yescharacteristic, complemented
27#19127781C411yescharacteristic, complemented
28#19127763—21no—
29#19127760—21no—
30#19127756—21no—
31#19127752—21no—
32#19127737C2211yescharacteristic
33#19127729—21no—
34#19127729C2211yescharacteristic
35#19127727—21no—
36#19127727C2211yescharacteristic
37#19127724—21no—
38#19127724C2211yescharacteristic
39#19127724C2211yescharacteristic
40C2×Dic83⋊C8—21no—
41C2×Dic83⋊C8C2211yescharacteristic
42C2×#14581720C2211yescharacteristic
43C2×#14581719C2211yescharacteristic
44C2×#14581718C2211yescharacteristic
45#19127698C2211yescharacteristic
46#19127669C2211yescharacteristic
47#19127667C2211yescharacteristic
48#19127657C2211yescharacteristic
49C2×#14581702C2211yescharacteristic
50C2×C8×Dic83C2211yescharacteristic
51C4×Dic83⋊C4C2211yescharacteristic
52#19127466C2211yescharacteristic
53#19127465C2211yescharacteristic
54#19127465C2211yescharacteristic
55#19127462C2211yescharacteristic
56#19127461C2211yescharacteristic
57#19127460C2211yescharacteristic
58#19127456C2211yescharacteristic
59#19127455C2211yescharacteristic
60C2×#131×C83C2211yescharacteristic
61C2×#104×C83C2211yescharacteristic
62#416×C83C2211yescharacteristic
63#405×C83C2211yescharacteristic
64C2×#97×C83C2211yescharacteristic
65#341×C83C2211yescharacteristic
66#339×C83C2211yescharacteristic
67C2×#13050590C2311yescharacteristic
68#14581812—22no—
69#14581812—22no—
70C2×#13050580C2×C411yescharacteristic
71C2×#13050578C2×C411yescharacteristic
72C2×#13050577C2×C411yescharacteristic
73C2×#13050575C2×C411yescharacteristic
74#14581799—22no—
75#14581799—41no—
76C2×#13050574—21no—
77C2×#13050574C2311yescharacteristic
78C2×#13050574C2311yescharacteristic
79C22×C83⋊C8—21no—
80C22×C83⋊C8C2311yescharacteristic
81C22×C83⋊C8C2311yescharacteristic
82Q8×Dic83C2×C411yescharacteristic
83Q8×Dic83C2×C411yescharacteristic
84#14581757C2×C411yescharacteristic
85#14581757C2×C411yescharacteristic
86D4×Dic83—21no—
87D4×Dic83—21no—
88D4×Dic83C2×C411yescharacteristic
89D4×Dic83C2×C411yescharacteristic
90#14581754C2×C411yescharacteristic
91#14581754C2×C411yescharacteristic
92(C22⋊D83)⋊C4—21no—
93(C22⋊D83)⋊C4—21no—
94#14581751—21no—
95#14581751C2311yescharacteristic
96#14581749—21no—
97C2×Dic83⋊C4—21no—
98C2×C4×Dic83—21no—
99C2×C4×Dic83C2311yescharacteristic
100C2×C4×Dic83C2311yescharacteristic
101C42×D83—21no—
102#14581725C2×C411yescharacteristic
103#14581725C2×C411yescharacteristic
104#14581723C2×C411yescharacteristic
105#14581723C2×C411yescharacteristic
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110Dic83⋊C8—21no—
111Dic83⋊C8—21no—
112Dic83⋊C8—21no—
113Dic83⋊C8—21no—
114Dic83⋊C8D411yescharacteristic
115Dic83⋊C8D411yescharacteristic
116Dic83⋊C8D411yescharacteristic
117Dic83⋊C8D411yescharacteristic
118#14581720—41no—
119#14581719—22no—
120#14581718—41no—
121#14581717C2×C411yescharacteristic
122#14581717C2×C411yescharacteristic
123#14581714C2×C411yescharacteristic
124#14581714C2×C411yescharacteristic
125(C2×D83)⋊C8—21no—
126(C2×D83)⋊C8—21no—
127(C2×D83)⋊C8—21no—
128(C2×D83)⋊C8—21no—
129#14581703—21no—
130#14581702—41no—
131C4×C83⋊C8—21no—
132C8×Dic83C2×C411yescharacteristic
133C8×Dic83C2×C411yescharacteristic
134C8×Dic83C2×C411yescharacteristic
135C8×Dic83C2×C411yescharacteristic
136C83⋊#95—21no—
137C2×#41×C83C2311yescharacteristic
138#131×C83—22no—
139#131×C83—41no—
140C2×M16×C83—21no—
141C2×M16×C83C2311yescharacteristic
142C2×M16×C83C2311yescharacteristic
143C22×C8×C83—21no—
144C22×C8×C83C2311yescharacteristic
145C22×C8×C83C2311yescharacteristic
146#117×C83C2311yescharacteristic
147C2×C42×C83C2311yescharacteristic
148#104×C83—22no—
149#103×C83C2×C411yescharacteristic
150#103×C83C2×C411yescharacteristic
151#102×C83C2×C411yescharacteristic
152#102×C83C2×C411yescharacteristic
153#97×C83—21no—
154#97×C83—41no—
155C4×C8×C83—21no—
156#13050590—21no—
157#13050590—21no—
158#13050590—21no—
159#13050590—21no—
160C2×Dic166C22×C411yescharacteristic
161C2×D332C22×C411yescharacteristic
162C2×C22⋊D83—21no—
163C22×Dic83—21no—
164C22×Dic83—21no—
165C22×Dic83—21no—
166C22×Dic83#4111yescharacteristic
167C22×Dic83#4111yescharacteristic
168C2×C4×D83—21no—
169C2×C4×D83—21no—
170C2×C4×D83—21no—
171#13050580—22no—
172#13050578—22no—
173#13050577—22no—
174#13050575—22no—
175#13050574—21no—
176#13050574—21no—
177#13050574—21no—
178#13050574—21no—
179#13050574—42no—
180C2×C83⋊C8—21no—
181C2×C83⋊C8—21no—
182C2×C83⋊C8—21no—
183C2×C83⋊C8—21no—
184C2×C83⋊C8—21no—
185C2×C83⋊C8—21no—
186C2×C83⋊C8—42no—
187C2×C83⋊C8C22×C411yescharacteristic
188C2×C83⋊C8C22×C411yescharacteristic
189C2×C83⋊C8C2×D411yescharacteristic
190C2×C83⋊C8C2×D411yescharacteristic
191C4⋊Dic83—21no—
192C4⋊Dic83—21no—
193C4⋊Dic83C22×C411yescharacteristic
194C4⋊Dic83C22×C411yescharacteristic
195Dic83⋊C4—21no—
196Dic83⋊C4—21no—
197Dic83⋊C4—42no—
198C22⋊Dic83—21no—
199C22⋊Dic83—21no—
200C22⋊Dic83—21no—
201(C2×D83)⋊C4—21no—
202(C2×D83)⋊C4—21no—
203C4×Dic83—21no—
204C4×Dic83—21no—
205C4×Dic83—21no—
206C4×Dic83—21no—
207C4×Dic83—21no—
208C4×Dic83—21no—
209C4×Dic83—21no—
210C4×Dic83C22×C411yescharacteristic
211C4×Dic83C22×C411yescharacteristic
212C4×Dic83C2×D411yescharacteristic
213C4×Dic83C2×D411yescharacteristic
214#41×C83—21no—
215#41×C83—21no—
216#41×C83#4112yes—
217#41×C83#4112yes—
218C2×Q8×C83C22×C411yescharacteristic
219C2×D4×C83—21no—
220C2×D4×C83C22×C411yescharacteristic
221C22×C4×C83—21no—
222C22×C4×C83—21no—
223C22×C4×C83C2411yescharacteristic
224M16×C83—21no—
225M16×C83—21no—
226M16×C83—21no—
227M16×C83—21no—
228M16×C83—42no—
229C2×C8×C83—21no—
230C2×C8×C83—21no—
231C2×C8×C83—21no—
232C2×C8×C83—21no—
233C2×C8×C83—21no—
234C2×C8×C83—21no—
235C2×C8×C83—42no—
236C2×C8×C83C22×C411yescharacteristic
237C2×C8×C83C22×C411yescharacteristic
238C2×C8×C83C2×D411yescharacteristic
239C2×C8×C83C2×D411yescharacteristic
240C4⋊C4×C83C22×C411yescharacteristic
241C4⋊C4×C83C22×C411yescharacteristic
242C22⋊C4×C83—21no—
243C42×C83—21no—
244C42×C83—21no—
245C42×C83—21no—
246C23×C83—21no—
247C23×C83C2×#4111yescharacteristic
248C22×D83—21no—
249Q8×C83—21no—
250Q8×C83#11712yes—
251D4×C83—22no—
252D4×C83—22no—
253D4×C83—21no—
254D4×C83#11712yes—
255C2×C4×C83—21no—
256C2×C4×C83—22no—
257C2×C4×C83—22no—
258C2×C4×C83—21no—
259C2×C4×C83—21no—
260C2×C4×C83—41no—
261C2×C4×C83—41no—
262C2×C4×C83—41no—
263C2×C4×C83C23×C411yescharacteristic
264C2×C4×C83C22×D411yescharacteristic
265C2×C4×C83C2×#4111yescharacteristic
266C2×C4×C83C2×#4111yescharacteristic
267C2×C4×C83H(5,𝔽2)11yescharacteristic
268C2×C4×C832−1+411yescharacteristic
269C22⋊D83—41no—
270C22⋊D83—41no—
271C2×Dic83—22no—
272C2×Dic83—21no—
273C2×Dic83—21no—
274C2×Dic83—21no—
275C2×Dic83—41no—
276C2×Dic83—42no—
277C2×Dic83—41no—
278C2×Dic83—41no—
279C2×Dic83—41no—
280C2×Dic83—81no—
281C2×Dic83C4×D411yescharacteristic
282C2×Dic83C4×D411yescharacteristic
283C2×Dic83C4×D411yescharacteristic
284C2×Dic83C4×D411yescharacteristic
285D332—21no—
286D332—21no—
287C4×D83—41no—
288C4×D83—41no—
289C4×D83—42no—
290C4×D83—81no—
291Dic166—21no—
292Dic166—21no—
293C8×C83—21no—
294C8×C83—21no—
295C8×C83—21no—
296C8×C83—21no—
297C8×C83—41no—
298C8×C83—41no—
299C83⋊C8—21no—
300C83⋊C8—21no—
301C83⋊C8—21no—
302C83⋊C8—21no—
303C83⋊C8—41no—
304C83⋊C8—41no—
305C22×C83—21no—
306C22×C83—21no—
307C22×C83—22no—
308C22×C83—41no—
309C22×C83—41no—
310C22×C83C2×C4×D411yescharacteristic
311C22×C83#55211yescharacteristic
312C22×C83#55311yescharacteristic
313C2×D83—41no—
314C2×D83—41no—
315C2×D83—41no—
316C4×C83—22no—
317C4×C83—41no—
318C4×C83—41no—
319C4×C83—81no—
320C4×C83C2×#11711yescharacteristic
321C4×C83#52011yescharacteristic
322C4×C83#54611yescharacteristic
323C4×C83#54811yescharacteristic
324Dic83—21no—
325Dic83—21no—
326Dic83—21no—
327Dic83—41no—
328Dic83—41no—
329Dic83—41no—
330Dic83—81no—
331Dic83#44412yes—
332#22782—831nomaximal, Sylow
333C2×C83—21no—
334C2×C83—21no—
335C2×C83—42no—
336C2×C83#235211yescharacteristic
337C2×C83C2×#44411yescharacteristic
338C2×C83#295711yescharacteristic
339D83—81no—
340C4×#135—831no—
341#2925—831no—
342#2906—831no—
343#1938—831no—
344#1938—831no—
345#1922—831no—
346#1922—831no—
347#1876—831no—
348#1876—831no—
349#1837—831no—
350#1806—831no—
351#1806—831no—
352#1770—831no—
353#1765—831no—
354#1765D8311yescharacteristic, complemented
355C83#2278211yescharacteristic, complemented, Sylow
356C2×#135—831no—
357C2×#131—831no—
358C2×#131C2×D8311yescharacteristic
359C4×#41—831no—
360C4×#41—831no—
361C4×Q16—831no—
362C4×Q16—831no—
363C4×SD16—831no—
364C4×SD16—831no—
365C4×SD16—831no—
366C4×SD16—831no—
367C4×D8—831no—
368C4×D8—831no—
369#433—1661no—
370#433—1661no—
371#432—1661no—
372#431—1661no—
373#427—831no—
374#424—831no—
375#424—831no—
376#424—1661no—
377#424—1661no—
378#423—831no—
379#423—831no—
380#423—1661no—
381C2×C4⋊C8—831no—
382C2×C4⋊C8—1661no—
383C2×#104—831no—
384C2×#104—831no—
385C2×#104—831no—
386C2×#104C2×D8311yescharacteristic
387#416—831no—
388#416C2×D8311yescharacteristic
389#408—831no—
390#408—831no—
391#405—831no—
392#405C2×D8311yescharacteristic
393C2×#97—831no—
394C2×#97C2×D8311yescharacteristic
395C2×C4×C8—831no—
396C4×C4⋊C4—831no—
397#341—831no—
398#341—831no—
399#341—831no—
400#341C2×D8311yescharacteristic
401#339—831no—
402#339—831no—
403#339—831no—
404#339C2×D8311yescharacteristic
405#336—831no—
406#336—831no—
407C2×#41—831no—
408C2×#41C22×D8311yescharacteristic
409#135—1662no—
410#135—1662no—
411C2×Q16—831no—
412C2×SD16—831no—
413C2×SD16—831no—
414C2×D8—831no—
415#131—22no—
416#131—41no—
417#131—3321no—
418#131—1662no—
419C2×M16—21no—
420C2×M16—831no—
421C2×M16—831no—
422C2×M16—1661no—
423C2×M16C22×D8311yescharacteristic
424C2×M16C22×D8311yescharacteristic
425C22×C8—21no—
426C22×C8—831no—
427C22×C8—831no—
428C22×C8—1661no—
429C22×C8C22×D8311yescharacteristic
430C22×C8C22×D8311yescharacteristic
431C4×Q8—831no—
432C4×Q8—831no—
433C4×Q8—831no—
434C4×Q8—831no—
435C4×D4—831no—
436C4×D4—831no—
437C4×D4—831no—
438C4×D4—831no—
439C4×D4—1661no—
440C4×D4—1661no—
441C4×D4—1661no—
442C4×D4—1661no—
443#117—831no—
444#117—1661no—
445#117—1661no—
446#117C22×D8311yescharacteristic
447C2×C4⋊C4—1661no—
448C2×C42—831no—
449C2×C42—831no—
450C2×C42—1661no—
451C2×C42—1661no—
452C2×C42C22×D8311yescharacteristic
453#107—831no—
454#107—831no—
455#106—831no—
456#106—831no—
457C4⋊C8—831no—
458C4⋊C8—831no—
459C4⋊C8—831no—
460C4⋊C8—831no—
461C4⋊C8—1661no—
462C4⋊C8—1661no—
463C4⋊C8—1661no—
464C4⋊C8—1661no—
465C4⋊C8—1661no—
466C4⋊C8—1661no—
467C4⋊C8—1661no—
468C4⋊C8—1661no—
469#104—22no—
470#104—3321no—
471#104—3321no—
472#104—1662no—
473#103—831no—
474#103—831no—
475#103C4×D8311yescharacteristic
476#103C4×D8311yescharacteristic
477#102—831no—
478#102—831no—
479#102C4×D8311yescharacteristic
480#102C4×D8311yescharacteristic
481C22⋊C8—1661no—
482C22⋊C8—1661no—
483C22⋊C8—1661no—
484C22⋊C8—1661no—
485#97—21no—
486#97—41no—
487#97—3321no—
488#97—1661no—
489C4×C8—21no—
490C4×C8—831no—
491C4×C8—831no—
492C4×C8—831no—
493C4×C8—831no—
494C4×C8—1661no—
495#95—1661no—
496#41—21no—
497#41—21no—
498#41—1661no—
499#41—1661no—
500#41—1661no—
501#41—1661no—
502#41#1305059012yes—
503#41#1305059012yes—
504C2×Q8—831no—
505C2×Q8C2×C4×D8311yescharacteristic
506C2×D4—21no—
507C2×D4—831no—
508C2×D4—1661no—
509C2×D4C2×C4×D8311yescharacteristic
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—831no—
513C22×C4—831no—
514C22×C4—1661no—
515C22×C4—1661no—
516C22×C4—1661no—
517C22×C4—1661no—
518C22×C4—1661no—
519C22×C4—1661no—
520C22×C4C23×D8311yescharacteristic
521Q16—1662no—
522SD16—1662no—
523SD16—1662no—
524D8—1662no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—21no—
529M16—42no—
530M16—3322no—
531M16—1661no—
532M16—1661no—
533M16—1661no—
534M16—1661no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—42no—
542C2×C8—3322no—
543C2×C8—831no—
544C2×C8—831no—
545C2×C8—831no—
546C2×C8—831no—
547C2×C8—1661no—
548C2×C8—1661no—
549C2×C8—1661no—
550C2×C8—1661no—
551C2×C8—1661no—
552C2×C8—1661no—
553C2×C8C2×C4×D8311yescharacteristic
554C2×C8C2×C4×D8311yescharacteristic
555C2×C8D4×D8311yescharacteristic
556C2×C8D4×D8311yescharacteristic
557C4⋊C4—3322no—
558C4⋊C4—831no—
559C4⋊C4—831no—
560C4⋊C4—1661no—
561C4⋊C4—1661no—
562C4⋊C4—1661no—
563C4⋊C4—1661no—
564C4⋊C4C2×C4×D8311yescharacteristic
565C4⋊C4C2×C4×D8311yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—1661no—
568C22⋊C4—1661no—
569C22⋊C4—1661no—
570C22⋊C4—1661no—
571C22⋊C4—1661no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—831no—
576C42—831no—
577C42—831no—
578C42—831no—
579C42—1661no—
580C42—1661no—
581C42—1661no—
582C42—1661no—
583C42—1661no—
584C42—1661no—
585C42—1661no—
586C23—21no—
587C23—1661no—
588C23#41×D8311yescharacteristic
589Q8—21no—
590Q8—1661no—
591Q8—1661no—
592Q8#1458174912yes—
593D4—22no—
594D4—22no—
595D4—21no—
596D4—3321no—
597D4—3321no—
598D4—1661no—
599D4—1661no—
600D4#1458174912yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—3322no—
610C2×C4—3322no—
611C2×C4—3321no—
612C2×C4—3321no—
613C2×C4—3321no—
614C2×C4—3321no—
615C2×C4—3321no—
616C2×C4—3321no—
617C2×C4—831no—
618C2×C4—831no—
619C2×C4—831no—
620C2×C4—831no—
621C2×C4—6641no—
622C2×C4—6641no—
623C2×C4—1661no—
624C2×C4—1661no—
625C2×C4—1661no—
626C2×C4—1662no—
627C2×C4C22×C4×D8311yescharacteristic
628C2×C4C2×D4×D8311yescharacteristic
629C2×C4C2×#1305059011yescharacteristic
630C2×C4C2×#1305059011yescharacteristic
631C2×C4#1458183711yescharacteristic
632C2×C4#1458183911yescharacteristic
633C8—21no—
634C8—21no—
635C8—21no—
636C8—21no—
637C8—41no—
638C8—41no—
639C8—3321no—
640C8—3321no—
641C8—1661no—
642C8—1661no—
643C8—1661no—
644C8—1661no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—3321no—
651C22—3321no—
652C22—3321no—
653C22C4×D4×D8311yescharacteristic
654C22#1912838411yescharacteristic
655C22#1912840011yescharacteristic
656C4—22no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—3321no—
661C4—3321no—
662C4—3321no—
663C4—832no—
664C4—6641no—
665C4—1661no—
666C4—1661no—
667C4—1661no—
668C4C2×#1458174911yescharacteristic
669C4#1912818011yescharacteristic
670C4#1912834111yescharacteristic
671C4#1912836011yescharacteristic
672C2—21no—
673C2—21no—
674C2—42no—
675C2—6641no—
676C2#2491038611yescharacteristic
677C2#444×D8311yescharacteristic
678C2#2491637711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1765C2
socleC22×C83C2×C4×D4
prime radicalC2×#14581812C22
cosocleC22×C4×C83C24
hypercenter#1765D83
hypocenterC83#22782
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #24916127C2semidirect
3. #24916029C2semidirect
4. #24908174C2semidirect
5. #24908172C2semidirect
6. #24908062C2semidirect
7. #24908060C2semidirect
10. #24907599C2semidirect
11. #24907482C2semidirect
12. #24907481C2semidirect
13. #24907331C2semidirect
14. #24907310C2semidirect
15. C83×#1765C2semidirect
20. #19127793C4semidirect
21. #19127793C4semidirect
22. #19127792C4semidirect
23. #19127792C4semidirect
24. #19127786C4semidirect
25. #19127786C4semidirect
26. #19127781C4semidirect
27. #19127781C4semidirect
354. #1765D83semidirect
355. C83#22782semidirect

Maximal subgroups

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

subgroupindex
#249162312
#249161272
#249160292
#249081742
#249081722
#249080622
#249080602
#249077902
#249077882
#249075992
#249074822
#249074812
#249073312
#249073102
C83×#17652
#2278283