Finite Group Databasebeta

← #35477507

Subgroups — #35477507

Order 21248.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910488C211yesmaximal, characteristic
2#24908872C211yesmaximal, characteristic, complemented
3#24908855C211yesmaximal, characteristic, complemented
4#24908659C211yesmaximal, characteristic
5#24908659C211yesmaximal, characteristic
6#24908224C211yesmaximal, characteristic, complemented
7#24908203C211yesmaximal, characteristic, complemented
8#24908078C211yesmaximal, characteristic
9#24908048C211yesmaximal, characteristic, complemented
10#24908045C211yesmaximal, characteristic, complemented
11#24907730C211yesmaximal, characteristic, complemented
12#24907660C211yesmaximal, characteristic
13#24907555C211yesmaximal, characteristic
14#24907296C211yesmaximal, characteristic, complemented
15C83×#2497C211yesmaximal, characteristic
16C2×#14581771C2211yescharacteristic
17C2×C4×Dic166C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×#14581715C2211yescharacteristic
25C2×#14581715C2211yescharacteristic
26C2×#14581715C2211yescharacteristic
27C2×#14581715C2211yescharacteristic
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×C22⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31#19127607C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127595—21no—
34#19127589—21no—
35#19127583—21no—
36C4×C4⋊Dic83C2211yescharacteristic
37#19127552—21no—
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48#400×C83—21no—
49#396×C83—21no—
50#389×C83C2211yescharacteristic
51#389×C83C2211yescharacteristic
52#388×C83—21no—
53#387×C83—21no—
54#384×C83C2211yescharacteristic
55#380×C83C2211yescharacteristic
56#380×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59C22×Dic166C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581771—22no—
65#14581771—22no—
66C4×Dic166—22no—
67C4×Dic166—41no—
68#14581751—22no—
69#14581751—41no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×C22⋊Dic83—21no—
77C2×C22⋊Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83—21no—
80C2×C4×Dic83—21no—
81#14581725—41no—
82#14581723—22no—
83#14581723—22no—
84#14581723—22no—
85C4⋊(C83⋊C8)—22no—
86#14581715—22no—
87#14581715—22no—
88#14581715—22no—
89#14581715—41no—
90C22⋊(C83⋊C8)—22no—
91C22⋊(C83⋊C8)—41no—
92#14581703—41no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C23×C4×C83C2311yescharacteristic
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83D411yescharacteristic
109C2×C22⋊C4×C83D411yescharacteristic
110C2×C42×C83—21no—
111C2×C42×C83C2311yescharacteristic
112C2×C42×C83C2311yescharacteristic
113#95×C83—21no—
114#95×C83—21no—
115#95×C83—21no—
116#95×C83—21no—
117#95×C83—21no—
118#95×C83D411yescharacteristic
119#95×C83D411yescharacteristic
120C2×Dic166—21no—
121C2×Dic166—21no—
122C2×Dic166—21no—
123C2×Dic166—21no—
124C2×Dic166—21no—
125C2×Dic166#4112yes—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8#4112yes—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83#4112yes—
147C4⋊Dic83#4112yes—
148Dic83⋊C4—24no—
149Dic83⋊C4—42no—
150Dic83⋊C4—42no—
151C22⋊Dic83—24no—
152C22⋊Dic83—42no—
153C4×Dic83—42no—
154C4×Dic83—42no—
155C4×Dic83—42no—
156C24×C83C2×D411yescharacteristic
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2411yescharacteristic
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—42no—
182C4⋊C4×C83—81no—
183C4⋊C4×C83—81no—
184C4⋊C4×C83—81no—
185C22⋊C4×C83—22no—
186C22⋊C4×C83—41no—
187C22⋊C4×C83—81no—
188C22⋊C4×C83—81no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—21no—
193C42×C83—81no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83—21no—
199C23×C83—21no—
200C23×C83—21no—
201C23×C83C22×D411yescharacteristic
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
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#13712yes—
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83H(5,𝔽2)11yescharacteristic
238C2×C4×C832−1+411yescharacteristic
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251Dic166—22no—
252Dic166—22no—
253Dic166—41no—
254Dic166—41no—
255C83⋊C8—22no—
256C83⋊C8—41no—
257C83⋊C8—41no—
258C83⋊C8—41no—
259C22×C83—41no—
260C22×C83—41no—
261C22×C83—41no—
262C22×C83—41no—
263C22×C83—41no—
264C22×C83—41no—
265C22×C83—41no—
266C22×C83#52511yescharacteristic
267C22×C83#54611yescharacteristic
268C22×C83#54711yescharacteristic
269C22×C83C2×#13711yescharacteristic
270C22×C83#57511yescharacteristic
271C22×C83#57711yescharacteristic
272C22×C83#57711yescharacteristic
273C4×C83—22no—
274C4×C83—41no—
275C4×C83—41no—
276C4×C83—41no—
277C4×C83—41no—
278C4×C83—81no—
279C4×C83—81no—
280C4×C83—81no—
281C4×C83—81no—
282C4×C83#52911yescharacteristic
283C4×C83#53911yescharacteristic
284C4×C83#55311yescharacteristic
285C4×C83#55611yescharacteristic
286Dic83—42no—
287Dic83—42no—
288Dic83—42no—
289Dic83—81no—
290#27060—831nomaximal, Sylow
291C2×C83—81no—
292C2×C83#246611yescharacteristic
293C2×C83#311611yescharacteristic
294C2×C83#312611yescharacteristic
295C2×C83#332111yescharacteristic
296C2×C83#333111yescharacteristic
297C2×C83#334411yescharacteristic
298C2×C83#335011yescharacteristic
299#2497D8311yescharacteristic
300#2362—831no—
301#2072—831no—
302#2068—831no—
303#2035—831no—
304#2035—831no—
305#1948—831no—
306#1944—831no—
307#1924—831no—
308#1920—831no—
309#1920—831no—
310#1865—831no—
311#1854—831no—
312#1825—831no—
313#1763—831no—
314C83#2706011yescharacteristic, complemented, Sylow
315C2×C22⋊Q8—831no—
316C2×C4×Q8—831no—
317C2×#117—831no—
318C2×#107—831no—
319C2×#106—831no—
320C2×#106—831no—
321C2×#106—831no—
322C2×C4⋊C8—831no—
323C2×#103—831no—
324C2×#103—831no—
325C2×#103—831no—
326C2×#103—831no—
327C2×C22⋊C8—831no—
328C2×C22⋊C8—831no—
329C2×#97—831no—
330#400—21no—
331#396—21no—
332#391—831no—
333#389—831no—
334#389C2×D8311yescharacteristic
335#389C2×D8311yescharacteristic
336#388—21no—
337#388—1661no—
338#388—1661no—
339#387—21no—
340#387—1661no—
341#384C2×D8311yescharacteristic
342#380C2×D8311yescharacteristic
343#380C2×D8311yescharacteristic
344C4×C4⋊C4—831no—
345C4×C4⋊C4—1661no—
346C4×C4⋊C4C2×D8311yescharacteristic
347C4×C22⋊C4C2×D8311yescharacteristic
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#340—831no—
355#340—831no—
356#336—831no—
357#336—831no—
358C22×Q8—831no—
359C23×C4C22×D8311yescharacteristic
360C22×C8—831no—
361C22×C8—831no—
362C22×C8—831no—
363C22×C8—831no—
364C22⋊Q8—1662no—
365C22⋊Q8—1662no—
366C4×Q8—3321no—
367C4×Q8—1662no—
368#117—3321no—
369#117—1662no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C4⋊C4C22×D8311yescharacteristic
382C2×C4⋊C4C22×D8311yescharacteristic
383C2×C4⋊C4C22×D8311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1661no—
387C2×C22⋊C4—1661no—
388C2×C22⋊C4C22⋊D8311yescharacteristic
389C2×C22⋊C4C22⋊D8311yescharacteristic
390C2×C42—21no—
391C2×C42—1661no—
392C2×C42—1661no—
393C2×C42—1661no—
394C2×C42C22×D8311yescharacteristic
395C2×C42C22×D8311yescharacteristic
396#107—3321no—
397#106—1662no—
398#106—1662no—
399#106—1662no—
400C4⋊C8—1662no—
401#103—3321no—
402#103—1662no—
403#103—1662no—
404#103—1662no—
405C22⋊C8—3321no—
406C22⋊C8—1662no—
407#97—3321no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416#95—1661no—
417#95C22⋊D8311yescharacteristic
418#95C22⋊D8311yescharacteristic
419C24C2×C22⋊D8311yescharacteristic
420C2×Q8—832no—
421C2×Q8—1661no—
422C2×Q8—1661no—
423C2×Q8—1661no—
424C2×Q8—1661no—
425C2×Q8—1661no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1661no—
437C22×C4—1661no—
438C22×C4—1661no—
439C22×C4—1661no—
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C2×C22⋊D8311yescharacteristic
444C22×C4C2×C22⋊D8311yescharacteristic
445C22×C4C23×D8311yescharacteristic
446C2×C8—832no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8—1661no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—3322no—
471C4⋊C4—3322no—
472C4⋊C4—832no—
473C4⋊C4—832no—
474C4⋊C4—1664no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—3322no—
484C22⋊C4—1664no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—81no—
490C42—3322no—
491C42—3322no—
492C42—3322no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×C22⋊D8311yescharacteristic
501Q8—3321no—
502Q8—3321no—
503Q8—1662no—
504Q8—1662no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—1662no—
545C2×C4—1662no—
546C2×C4—1662no—
547C2×C4#1458182312yes—
548C2×C4#41×D8311yescharacteristic
549C2×C4#41×D8311yescharacteristic
550C2×C4#41×D8311yescharacteristic
551C2×C4#41×D8311yescharacteristic
552C2×C4#1458183811yescharacteristic
553C2×C4#1458183911yescharacteristic
554C8—3321no—
555C8—3321no—
556C8—3321no—
557C8—1662no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1912820811yescharacteristic
566C22#1912833911yescharacteristic
567C22#1912835011yescharacteristic
568C22C2×#1458182311yescharacteristic
569C22#1912849911yescharacteristic
570C22#1912851211yescharacteristic
571C22#1912851211yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—3322no—
582C4—3322no—
583C4—3322no—
584C4—6641no—
585C4#1912823411yescharacteristic
586C4#1912829211yescharacteristic
587C4#1912840211yescharacteristic
588C4#1912841811yescharacteristic
589C2—81no—
590C2#2491149511yescharacteristic
591C2#2491761911yescharacteristic
592C2#2491774911yescharacteristic
593C2#2491964311yescharacteristic
594C2#2491977311yescharacteristic
595C2#2491991211yescharacteristic
596C2#2492000211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2497C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2497D83
hypocenterC83#27060
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2497

Complements

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

NH ≅ G/Ntype
2. #24908872C2semidirect
3. #24908855C2semidirect
6. #24908224C2semidirect
7. #24908203C2semidirect
9. #24908048C2semidirect
10. #24908045C2semidirect
11. #24907730C2semidirect
14. #24907296C2semidirect
314. C83#27060semidirect

Maximal subgroups

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

subgroupindex
#249104882
#249088722
#249088552
#249086592
#249082242
#249082032
#249080782
#249080482
#249080452
#249077302
#249076602
#249075552
#249072962
C83×#24972
#2706083