Finite Group Databasebeta

← #35423875

Subgroups — #35423875

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#24916264C211yesmaximal, characteristic
2#24916125C211yesmaximal, characteristic, complemented
3#24916029C211yesmaximal, characteristic, complemented
4#24908177C211yesmaximal, characteristic, complemented
5#24908164C211yesmaximal, characteristic, complemented
6#24908063C211yesmaximal, characteristic, complemented
7#24908061C211yesmaximal, characteristic, complemented
8#24907792C211yesmaximal, characteristic
9#24907602C211yesmaximal, characteristic, complemented
10#24907481C211yesmaximal, characteristic, complemented
11#24907481C211yesmaximal, characteristic, complemented
12#24907334C211yesmaximal, characteristic, complemented
13#24907305C211yesmaximal, characteristic, complemented
14#24907301C211yesmaximal, characteristic, complemented
15C83×#1876C211yesmaximal, characteristic
16C2×#14581799C2211yescharacteristic
17#41×Dic83C2211yescharacteristic
18#41×Dic83C2211yescharacteristic
19#19127813C411yescharacteristic, complemented
20#19127813C411yescharacteristic, complemented
21#19127808C411yescharacteristic, complemented
22#19127808C411yescharacteristic, complemented
23#19127802C411yescharacteristic, complemented
24#19127802C411yescharacteristic, complemented
25#19127802C411yescharacteristic, complemented
26#19127802C411yescharacteristic, complemented
27#19127759—21no—
28#19127759—21no—
29#19127755—21no—
30#19127751—21no—
31#19127739C2211yescharacteristic
32#19127729—21no—
33#19127729C2211yescharacteristic
34#19127727—21no—
35#19127727C2211yescharacteristic
36#19127724—21no—
37#19127724C2211yescharacteristic
38#19127724C2211yescharacteristic
39C2×Dic83⋊C8—21no—
40C2×Dic83⋊C8C2211yescharacteristic
41C2×#14581719C2211yescharacteristic
42C2×#14581719C2211yescharacteristic
43#19127697C2211yescharacteristic
44#19127697C2211yescharacteristic
45#19127668C2211yescharacteristic
46#19127659C2211yescharacteristic
47#19127656C2211yescharacteristic
48C2×#14581702C2211yescharacteristic
49C2×#14581702C2211yescharacteristic
50C2×C4×C83⋊C8C2211yescharacteristic
51C4×Dic83⋊C4C2211yescharacteristic
52#19127466C2211yescharacteristic
53#19127465C2211yescharacteristic
54#19127465C2211yescharacteristic
55#19127462C2211yescharacteristic
56#19127462C2211yescharacteristic
57#19127461C2211yescharacteristic
58#19127461C2211yescharacteristic
59#19127455C2211yescharacteristic
60C2×#135×C83C2211yescharacteristic
61C2×#131×C83C2211yescharacteristic
62C2×#104×C83C2211yescharacteristic
63C2×#104×C83C2211yescharacteristic
64#408×C83C2211yescharacteristic
65#341×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
75Q8×Dic83C2×C411yescharacteristic
76Q8×Dic83C2×C411yescharacteristic
77Q8×Dic83C2×C411yescharacteristic
78Q8×Dic83C2×C411yescharacteristic
79D4×Dic83—21no—
80D4×Dic83—21no—
81D4×Dic83—21no—
82D4×Dic83—21no—
83D4×Dic83C2×C411yescharacteristic
84D4×Dic83C2×C411yescharacteristic
85D4×Dic83C2×C411yescharacteristic
86D4×Dic83C2×C411yescharacteristic
87#14581751—21no—
88#14581751—21no—
89#14581751C2311yescharacteristic
90#14581751C2311yescharacteristic
91C2×Dic83⋊C4—21no—
92C2×C4×Dic83—21no—
93C2×C4×Dic83—21no—
94C2×C4×Dic83C2311yescharacteristic
95C2×C4×Dic83C2311yescharacteristic
96#14581726C2×C411yescharacteristic
97#14581726C2×C411yescharacteristic
98#14581724C2×C411yescharacteristic
99#14581724C2×C411yescharacteristic
100Dic83⋊C8—21no—
101Dic83⋊C8—21no—
102Dic83⋊C8—21no—
103Dic83⋊C8—21no—
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8D411yescharacteristic
109Dic83⋊C8D411yescharacteristic
110Dic83⋊C8D411yescharacteristic
111Dic83⋊C8D411yescharacteristic
112#14581719—22no—
113#14581719—41no—
114#14581716C2×C411yescharacteristic
115#14581716C2×C411yescharacteristic
116#14581716C2×C411yescharacteristic
117#14581716C2×C411yescharacteristic
118#14581713C2×C411yescharacteristic
119#14581713C2×C411yescharacteristic
120#14581713C2×C411yescharacteristic
121#14581713C2×C411yescharacteristic
122C22⋊(C83⋊C8)—21no—
123C22⋊(C83⋊C8)—21no—
124#14581702—21no—
125#14581702—21no—
126#14581702—41no—
127#14581702C2×C411yescharacteristic
128#14581702C2×C411yescharacteristic
129#14581702C2×C411yescharacteristic
130#14581702C2×C411yescharacteristic
131C4×C83⋊C8—41no—
132C8×Dic83—21no—
133C8×Dic83—21no—
134C83⋊#95—21no—
135C2×#41×C83C2311yescharacteristic
136C2×#41×C83C2311yescharacteristic
137#135×C83—22no—
138#135×C83—22no—
139C2×Q16×C83C2×C411yescharacteristic
140C2×SD16×C83C2×C411yescharacteristic
141C2×SD16×C83C2×C411yescharacteristic
142C2×D8×C83C2×C411yescharacteristic
143#131×C83—22no—
144#131×C83—41no—
145C2×M16×C83—21no—
146C2×M16×C83C2311yescharacteristic
147C2×M16×C83C2311yescharacteristic
148C22×C8×C83—21no—
149C22×C8×C83C2311yescharacteristic
150C22×C8×C83C2311yescharacteristic
151C2×C42×C83C2311yescharacteristic
152#104×C83—22no—
153#104×C83—41no—
154C22⋊C8×C83—21no—
155C22⋊C8×C83—21no—
156C22×Dic83—21no—
157C22×Dic83—21no—
158C22×Dic83—21no—
159C22×Dic83—21no—
160C22×Dic83—21no—
161C22×Dic83#4111yescharacteristic
162C22×Dic83#4111yescharacteristic
163#13050574—21no—
164#13050574—21no—
165#13050574—21no—
166#13050574—21no—
167#13050574—42no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8—21no—
174C2×C83⋊C8—42no—
175C2×C83⋊C8C22×C411yescharacteristic
176C2×C83⋊C8C22×C411yescharacteristic
177C2×C83⋊C8C2×D411yescharacteristic
178C2×C83⋊C8C2×D411yescharacteristic
179C4⋊Dic83—21no—
180C4⋊Dic83—21no—
181C4⋊Dic83—21no—
182C4⋊Dic83—21no—
183C4⋊Dic83C22×C411yescharacteristic
184C4⋊Dic83C22×C411yescharacteristic
185C4⋊Dic83C22×C411yescharacteristic
186C4⋊Dic83C22×C411yescharacteristic
187Dic83⋊C4—42no—
188C22⋊Dic83—21no—
189C22⋊Dic83—21no—
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×Dic83—21no—
201C4×Dic83—21no—
202C4×Dic83C22×C411yescharacteristic
203C4×Dic83C22×C411yescharacteristic
204C4×Dic83C2×D411yescharacteristic
205C4×Dic83C2×D411yescharacteristic
206#41×C83—21no—
207#41×C83—21no—
208#41×C83—21no—
209#41×C83—21no—
210#41×C83—21no—
211#41×C83—21no—
212#41×C83#4112yes—
213#41×C83#4112yes—
214C2×Q8×C83C22×C411yescharacteristic
215C2×Q8×C83C22×C411yescharacteristic
216C2×D4×C83—21no—
217C2×D4×C83—21no—
218C2×D4×C83C22×C411yescharacteristic
219C2×D4×C83C22×C411yescharacteristic
220C22×C4×C83—21no—
221C22×C4×C83—21no—
222C22×C4×C83—21no—
223C22×C4×C83C2411yescharacteristic
224Q16×C83—22no—
225SD16×C83—22no—
226SD16×C83—22no—
227D8×C83—22no—
228M16×C83—21no—
229M16×C83—21no—
230M16×C83—21no—
231M16×C83—21no—
232M16×C83—42no—
233C2×C8×C83—21no—
234C2×C8×C83—21no—
235C2×C8×C83—21no—
236C2×C8×C83—21no—
237C2×C8×C83—21no—
238C2×C8×C83—21no—
239C2×C8×C83—42no—
240C2×C8×C83C22×C411yescharacteristic
241C2×C8×C83C22×C411yescharacteristic
242C2×C8×C83C2×D411yescharacteristic
243C2×C8×C83C2×D411yescharacteristic
244C42×C83—21no—
245C42×C83—21no—
246C23×C83—21no—
247C23×C83—21no—
248C23×C83C2×#4111yescharacteristic
249Q8×C83—21no—
250Q8×C83—21no—
251Q8×C83—21no—
252Q8×C83#11712yes—
253D4×C83—22no—
254D4×C83—22no—
255D4×C83—21no—
256D4×C83—21no—
257D4×C83—21no—
258D4×C83—41no—
259D4×C83—41no—
260D4×C83#11712yes—
261C2×C4×C83—21no—
262C2×C4×C83—22no—
263C2×C4×C83—22no—
264C2×C4×C83—22no—
265C2×C4×C83—21no—
266C2×C4×C83—41no—
267C2×C4×C83—41no—
268C2×C4×C83—41no—
269C2×C4×C83—41no—
270C2×C4×C83C23×C411yescharacteristic
271C2×C4×C83C22×D411yescharacteristic
272C2×C4×C83C2×#4111yescharacteristic
273C2×C4×C83C2×#4111yescharacteristic
274C2×C4×C83H(5,𝔽2)11yescharacteristic
275C2×C4×C832−1+411yescharacteristic
276C2×Dic83—21no—
277C2×Dic83—21no—
278C2×Dic83—21no—
279C2×Dic83—21no—
280C2×Dic83—41no—
281C2×Dic83—41no—
282C2×Dic83—42no—
283C2×Dic83—42no—
284C2×Dic83—41no—
285C2×Dic83—41no—
286C2×Dic83—41no—
287C2×Dic83—81no—
288C2×Dic83—81no—
289C2×Dic83C4×D411yescharacteristic
290C2×Dic83C4×D411yescharacteristic
291C2×Dic83C4×D411yescharacteristic
292C2×Dic83C4×D411yescharacteristic
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×C83—41no—
311C22×C83—41no—
312C22×C83—41no—
313C22×C83C2×C4×D411yescharacteristic
314C22×C83#55211yescharacteristic
315C22×C83#55311yescharacteristic
316C4×C83—22no—
317C4×C83—42no—
318C4×C83—41no—
319C4×C83C2×#11711yescharacteristic
320C4×C83#52011yescharacteristic
321C4×C83#54611yescharacteristic
322C4×C83#54811yescharacteristic
323Dic83—21no—
324Dic83—21no—
325Dic83—21no—
326Dic83—21no—
327Dic83—41no—
328Dic83—41no—
329Dic83—41no—
330Dic83—41no—
331Dic83—81no—
332#22784—831nomaximal, Sylow
333C2×C83—21no—
334C2×C83—21no—
335C2×C83—42no—
336C2×C83—81no—
337C2×C83#235211yescharacteristic
338C2×C83#295511yescharacteristic
339C2×C83#295611yescharacteristic
340#2943—831no—
341#2925—831no—
342#2906—831no—
343#1938—831no—
344#1937—831no—
345#1922—831no—
346#1922—831no—
347#1876—831no—
348#1876D8311yescharacteristic
349#1838—831no—
350#1806—831no—
351#1806—831no—
352#1771—831no—
353#1765—831no—
354#1764—831no—
355C83#2278411yescharacteristic, complemented, Sylow
356C2×#135C2×D8311yescharacteristic
357C2×#131—831no—
358C2×#131C2×D8311yescharacteristic
359C4×#41—831no—
360C4×#41—831no—
361#442—831no—
362#442—831no—
363#441—831no—
364#441—831no—
365#440—831no—
366#440—831no—
367#440—831no—
368#440—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×#104C2×D8311yescharacteristic
386C2×#104C2×D8311yescharacteristic
387#416—831no—
388#416—831no—
389#408—831no—
390#408C2×D8311yescharacteristic
391#405—831no—
392#405—831no—
393C2×#97—831no—
394C2×#97—831no—
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#339—831no—
405#336—831no—
406#336C2×D8311yescharacteristic
407C2×#41C22×D8311yescharacteristic
408C2×#41C22×D8311yescharacteristic
409#135—22no—
410#135—22no—
411C2×Q16C4×D8311yescharacteristic
412C2×SD16C4×D8311yescharacteristic
413C2×SD16C4×D8311yescharacteristic
414C2×D8C4×D8311yescharacteristic
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—831no—
445#117—1661no—
446#117—1661no—
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—41no—
471#104—3321no—
472#104—1662no—
473#103—831no—
474#103—831no—
475#103—831no—
476#103—831no—
477#102—831no—
478#102—831no—
479#102—831no—
480#102—831no—
481C22⋊C8—21no—
482C22⋊C8—21no—
483C22⋊C8—1661no—
484C22⋊C8—1661no—
485#97—3321no—
486#97—831no—
487#97—831no—
488#97—831no—
489#97—831no—
490#97—1661no—
491#97—1661no—
492C4×C8—3321no—
493C4×C8—1661no—
494C4×C8—1661no—
495#95—1661no—
496#41—21no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—21no—
501#41—21no—
502#41#1305058912yes—
503#41#1305058912yes—
504C2×Q8C2×C4×D8311yescharacteristic
505C2×Q8C2×C4×D8311yescharacteristic
506C2×D4—21no—
507C2×D4—21no—
508C2×D4C2×C4×D8311yescharacteristic
509C2×D4C2×C4×D8311yescharacteristic
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—831no—
514C22×C4—831no—
515C22×C4—1661no—
516C22×C4—1661no—
517C22×C4—1661no—
518C22×C4—1661no—
519C22×C4—1661no—
520C22×C4C23×D8311yescharacteristic
521Q16—22no—
522SD16—22no—
523SD16—22no—
524D8—22no—
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—831no—
561C4⋊C4—831no—
562C4⋊C4—1661no—
563C4⋊C4—1661no—
564C4⋊C4—1661no—
565C4⋊C4—1661no—
566C22⋊C4—1661no—
567C22⋊C4—1661no—
568C22⋊C4—1661no—
569C22⋊C4—1661no—
570C22⋊C4—1661no—
571C22⋊C4—1661no—
572C42—21no—
573C42—21no—
574C42—831no—
575C42—831no—
576C42—831no—
577C42—831no—
578C42—1661no—
579C42—1661no—
580C42—1661no—
581C42—1661no—
582C42—1661no—
583C42—1661no—
584C42—1661no—
585C42—1661no—
586C23—21no—
587C23—21no—
588C23#41×D8311yescharacteristic
589Q8—21no—
590Q8—21no—
591Q8—21no—
592Q8#1458174812yes—
593D4—22no—
594D4—22no—
595D4—21no—
596D4—21no—
597D4—21no—
598D4—41no—
599D4—41no—
600D4#1458174812yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—3322no—
611C2×C4—3322no—
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—1661no—
627C2×C4C22×C4×D8311yescharacteristic
628C2×C4C2×D4×D8311yescharacteristic
629C2×C4C2×#1305058911yescharacteristic
630C2×C4C2×#1305058911yescharacteristic
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—41no—
651C22—41no—
652C22—41no—
653C22C4×D4×D8311yescharacteristic
654C22#1912838811yescharacteristic
655C22#1912839411yescharacteristic
656C4—22no—
657C4—42no—
658C4—41no—
659C4—3321no—
660C4—3321no—
661C4—3321no—
662C4—3321no—
663C4—6641no—
664C4—1661no—
665C4—1661no—
666C4—1661no—
667C4—1661no—
668C4C2×#1458174811yescharacteristic
669C4#1912818011yescharacteristic
670C4#1912833811yescharacteristic
671C4#1912835711yescharacteristic
672C2—21no—
673C2—21no—
674C2—42no—
675C2—81no—
676C2#2491039011yescharacteristic
677C2#2491636311yescharacteristic
678C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1876C2
socleC22×C83C2×C4×D4
prime radicalC2×#135×C83C22
cosocleC22×C4×C83C24
hypercenter#1876D83
hypocenterC83#22784
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1876

Complements

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

NH ≅ G/Ntype
2. #24916125C2semidirect
3. #24916029C2semidirect
4. #24908177C2semidirect
5. #24908164C2semidirect
6. #24908063C2semidirect
7. #24908061C2semidirect
9. #24907602C2semidirect
10. #24907481C2semidirect
11. #24907481C2semidirect
12. #24907334C2semidirect
13. #24907305C2semidirect
14. #24907301C2semidirect
19. #19127813C4semidirect
20. #19127813C4semidirect
21. #19127808C4semidirect
22. #19127808C4semidirect
23. #19127802C4semidirect
24. #19127802C4semidirect
25. #19127802C4semidirect
26. #19127802C4semidirect
355. C83#22784semidirect

Maximal subgroups

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

subgroupindex
#249162642
#249161252
#249160292
#249081772
#249081642
#249080632
#249080612
#249077922
#249076022
#249074812
#249073342
#249073052
#249073012
C83×#18762
#2278483