Finite Group Databasebeta

← #35504326

Subgroups — #35504326

Order 21248.

734 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910174C211yesmaximal, characteristic
2#24908611C211yesmaximal, characteristic, complemented
3#24908606C211yesmaximal, characteristic
4#24908601C211yesmaximal, characteristic, complemented
5#24908596C211yesmaximal, characteristic
6#24908277C211yesmaximal, characteristic
7#24908272C211yesmaximal, characteristic
8#24907596C211yesmaximal, characteristic, complemented
9#24907590C211yesmaximal, characteristic, complemented
10#24907475C211yesmaximal, characteristic, complemented
11#24907470C211yesmaximal, characteristic
12#24907465C211yesmaximal, characteristic, complemented
13#24907460C211yesmaximal, characteristic
14#24907320C211yesmaximal, characteristic, complemented
15C83×#2660C211yesmaximal, characteristic, complemented
16C2×#14581780C2211yescharacteristic
17C2×C4×Dic166C2211yescharacteristic
18C2×C4×D332C2211yescharacteristic
19C2×C4⋊(C83⋊C8)C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×#14581715C2211yescharacteristic
22C2×#14581715C2211yescharacteristic
23C2×#14581715C2211yescharacteristic
24C2×#14581715C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×C4×C83⋊C8C2211yescharacteristic
31#19127607C2211yescharacteristic
32#19127603C2211yescharacteristic
33#19127593—21no—
34#19127593—21no—
35#19127575—21no—
36#19127575—21no—
37#19127571C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×C4⋊Q8×C83C2211yescharacteristic
49C2×#125×C83C2211yescharacteristic
50#398×C83—21no—
51#398×C83—21no—
52#395×C83—21no—
53#395×C83—21no—
54#384×C83C2211yescharacteristic
55#384×C83C2211yescharacteristic
56#382×C83C2211yescharacteristic
57#382×C83C2211yescharacteristic
58#376×C83C2211yescharacteristic
59C22×Dic166C2311yescharacteristic
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66#14581780—22no—
67#14581780Q814yescomplemented
68C4×Dic166—22no—
69C4×Dic166—22no—
70C4×D332—22no—
71C4×D332—22no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
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—
83C4⋊(C83⋊C8)—22no—
84C4⋊(C83⋊C8)—22no—
85#14581715—41no—
86#14581715—41no—
87#14581715—41no—
88#14581715—41no—
89#14581712—41no—
90#14581712—41no—
91#14581712—41no—
92#14581712—41no—
93#14581703—22no—
94C4×C83⋊C8—22no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C22×Q8×C83—21no—
98C4⋊Q8×C83—41no—
99C4⋊Q8×C83—41no—
100#125×C83—41no—
101#125×C83—41no—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C42×C83C2311yescharacteristic
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116#95×C83—21no—
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120#95×C83D411yescharacteristic
121#95×C83D411yescharacteristic
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124C23×D83—21no—
125C2×Dic166—21no—
126C2×Dic166—21no—
127C2×Dic166—21no—
128C2×Dic166—21no—
129C2×Dic166—21no—
130C2×Dic166C2×Q812yes—
131C2×D332—21no—
132C2×D332—21no—
133C2×D332—21no—
134C2×D332—21no—
135C2×D332—21no—
136C2×D332C2×Q812yes—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C2×C4×D83—42no—
141C2×C4×D83—81no—
142C2×C4×D83—81no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—81no—
160C4⋊Dic83—81no—
161Dic83⋊C4—42no—
162(C2×D83)⋊C4—24no—
163(C2×D83)⋊C4—24no—
164(C2×D83)⋊C4—42no—
165(C2×D83)⋊C4—81no—
166(C2×D83)⋊C4—81no—
167C4×Dic83—42no—
168C2×Q8×C83—42no—
169C2×Q8×C83—81no—
170C2×Q8×C83—81no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2411yescharacteristic
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83—42no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—81no—
194C4⋊C4×C83—81no—
195C4⋊C4×C83—81no—
196C4⋊C4×C83—81no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83C2×Q812yes—
200C42×C83#4112yes—
201C42×C83#4112yes—
202C42×C83#4112yes—
203C23×C83C22×D411yescharacteristic
204C22×D83—22no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209C22×D83—41no—
210C22×D83—41no—
211Q8×C83—84no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83C22⋊Q812yes—
239C2×C4×C83C22⋊Q812yes—
240C2×C4×C83#12412yes—
241C2×C4×C83#12412yes—
242C2×C4×C83C22×Q811yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C2×C4×C83H(5,𝔽2)11yescharacteristic
247C2×C4×C83H(5,𝔽2)11yescharacteristic
248C2×Dic83—22no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257D332—22no—
258D332—22no—
259D332—41no—
260D332—41no—
261C4×D83—84no—
262Dic166—22no—
263Dic166—22no—
264Dic166—41no—
265Dic166—41no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C22×C83C2×C22⋊Q811yescharacteristic
271C22×C83C2×#12411yescharacteristic
272C22×C83#53511yescharacteristic
273C22×C83#57611yescharacteristic
274C22×C83#57711yescharacteristic
275C22×C83#57711yescharacteristic
276C22×C83#57811yescharacteristic
277C2×D83—42no—
278C2×D83—42no—
279C2×D83—42no—
280C2×D83—81no—
281C2×D83—81no—
282C2×D83—81no—
283C2×D83—81no—
284C4×C83—22no—
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—81no—
291C4×C83—81no—
292C4×C83—81no—
293C4×C83—81no—
294C4×C83#55411yescharacteristic
295C4×C83#55411yescharacteristic
296C4×C83#55511yescharacteristic
297C4×C83#55511yescharacteristic
298Dic83—42no—
299Dic83—81no—
300Dic83—81no—
301#29151—831nomaximal, Sylow
302C2×C83#256311yescharacteristic
303C2×C83#308311yescharacteristic
304C2×C83#308411yescharacteristic
305C2×C83#313611yescharacteristic
306C2×C83#313711yescharacteristic
307C2×C83#322411yescharacteristic
308C2×C83#322511yescharacteristic
309D83—82no—
310#2660D8311yescharacteristic, complemented
311#2326—831no—
312#2026—831no—
313#2025—831no—
314#2024—831no—
315#2023—831no—
316#1959—831no—
317#1958—831no—
318#1836—831no—
319#1834—831no—
320#1805—831no—
321#1804—831no—
322#1803—831no—
323#1802—831no—
324#1768—831no—
325C83#2915111yescharacteristic, complemented, Sylow
326C2×C4⋊Q8C2×D8311yescharacteristic
327C2×#125C2×D8311yescharacteristic
328C2×#124—831no—
329C2×C4×Q8—831no—
330C2×C4×D4—831no—
331C2×C4⋊C8—831no—
332C2×C4⋊C8—831no—
333C2×#103—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×#103—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#102—831no—
340C2×#102—831no—
341C2×#97—831no—
342C2×C4×C8—831no—
343#398—21no—
344#398—21no—
345#395—21no—
346#395—21no—
347#391—831no—
348#390—831no—
349#388—1661no—
350#388—1661no—
351#385—1661no—
352#385—1661no—
353#384—831no—
354#384C2×D8311yescharacteristic
355#384C2×D8311yescharacteristic
356#382C2×D8311yescharacteristic
357#382C2×D8311yescharacteristic
358#376C2×D8311yescharacteristic
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#336—831no—
368#336—831no—
369C22×Q8—21no—
370C22×Q8—831no—
371C22×D4—831no—
372C23×C4—1661no—
373C22×C8—831no—
374C22×C8—831no—
375C22×C8—831no—
376C22×C8—831no—
377C4⋊Q8—41no—
378C4⋊Q8—41no—
379#125—41no—
380#125—41no—
381#124—834no—
382#124—1662no—
383C4×Q8—1662no—
384C4×Q8—1662no—
385C4×D4—1662no—
386C4×D4—1662no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C4⋊C4C22×D8311yescharacteristic
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C22⋊C4—1661no—
404C2×C22⋊C4—1661no—
405C2×C22⋊C4—1661no—
406C2×C22⋊C4—1661no—
407C2×C22⋊C4—1661no—
408C2×C42—1661no—
409C2×C42C22×D8311yescharacteristic
410C2×C42C22×D8311yescharacteristic
411C2×C42C22×D8311yescharacteristic
412C4⋊C8—1662no—
413C4⋊C8—1662no—
414#103—3321no—
415#103—3321no—
416#103—3321no—
417#103—3321no—
418#102—3321no—
419#102—3321no—
420#102—3321no—
421#102—3321no—
422#97—1662no—
423C4×C8—1662no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1661no—
429#95—1661no—
430#95C22⋊D8311yescharacteristic
431#95C22⋊D8311yescharacteristic
432#95C22⋊D8311yescharacteristic
433#95C22⋊D8311yescharacteristic
434C24—1661no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—81no—
438C2×Q8—832no—
439C2×Q8—1661no—
440C2×Q8—1661no—
441C2×Q8—1661no—
442C2×Q8—1661no—
443C2×Q8—1661no—
444C2×D4—832no—
445C2×D4—1661no—
446C2×D4—1661no—
447C2×D4—1661no—
448C2×D4—1661no—
449C2×D4—1661no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—3322no—
455C22×C4—6641no—
456C22×C4—6641no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C23×D8311yescharacteristic
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—3322no—
495C4⋊C4—6641no—
496C4⋊C4—6641no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C22⋊C4—3322no—
502C22⋊C4—6641no—
503C22⋊C4—6641no—
504C22⋊C4—1664no—
505C22⋊C4—1664no—
506C42—21no—
507C42—21no—
508C42—3322no—
509C42Q8×D8312yes—
510C42#1305059112yes—
511C42#1305059112yes—
512C42#1305059112yes—
513C23—3321no—
514C23—3321no—
515C23—3321no—
516C23—3321no—
517C23—3321no—
518C23—3321no—
519C23—1662no—
520C23C22×C22⋊D8311yescharacteristic
521Q8—84no—
522Q8—3321no—
523Q8—3321no—
524Q8—1662no—
525Q8—1662no—
526D4—3321no—
527D4—3321no—
528D4—1662no—
529D4—1662no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—6644no—
565C2×C4—1662no—
566C2×C4#1458176812yes—
567C2×C4#1458176812yes—
568C2×C4#1458178112yes—
569C2×C4#1458178112yes—
570C2×C4C2×Q8×D8311yescharacteristic
571C2×C4C2×#1305059111yescharacteristic
572C2×C4C2×#1305059111yescharacteristic
573C2×C4C2×#1305059111yescharacteristic
574C2×C4#1458183711yescharacteristic
575C2×C4#1458183711yescharacteristic
576C8—3321no—
577C8—3321no—
578C8—3321no—
579C8—3321no—
580C22—3322no—
581C22—3322no—
582C22—3322no—
583C22—6641no—
584C22—6641no—
585C22—6641no—
586C22—6641no—
587C22C2×#1458176811yescharacteristic
588C22C2×#1458178111yescharacteristic
589C22#1912826811yescharacteristic
590C22#1912850411yescharacteristic
591C22#1912850911yescharacteristic
592C22#1912851211yescharacteristic
593C22#1912851611yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—3322no—
605C4—6641no—
606C4—6641no—
607C4#1912841211yescharacteristic
608C4#1912841211yescharacteristic
609C4#1912841711yescharacteristic
610C4#1912841711yescharacteristic
611C2—6642no—
612C2#2491263711yescharacteristic
613C2#2491731011yescharacteristic
614C2#2491732111yescharacteristic
615C2#2491782211yescharacteristic
616C2#2491783311yescharacteristic
617C2#2491855711yescharacteristic
618C2#2491857211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2660C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2660D83
hypocenterC83#29151
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2660

Complements

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

NH ≅ G/Ntype
2. #24908611C2semidirect
4. #24908601C2semidirect
8. #24907596C2semidirect
9. #24907590C2semidirect
10. #24907475C2semidirect
12. #24907465C2semidirect
14. #24907320C2semidirect
15. C83×#2660C2semidirect
67. #14581780Q8semidirect
310. #2660D83semidirect
325. C83#29151semidirect

Maximal subgroups

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

subgroupindex
#249101742
#249086112
#249086062
#249086012
#249085962
#249082772
#249082722
#249075962
#249075902
#249074752
#249074702
#249074652
#249074602
#249073202
C83×#26602
#2915183