Finite Group Databasebeta

← #35466706

Subgroups — #35466706

Order 21248.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910420C211yesmaximal, characteristic, complemented
2#24909001C211yesmaximal, characteristic, complemented
3#24908984C211yesmaximal, characteristic
4#24908881C211yesmaximal, characteristic, complemented
5#24908866C211yesmaximal, characteristic
6#24908400C211yesmaximal, characteristic, complemented
7#24908396C211yesmaximal, characteristic
8#24908275C211yesmaximal, characteristic
9#24908231C211yesmaximal, characteristic, complemented
10#24908101C211yesmaximal, characteristic, complemented
11#24908094C211yesmaximal, characteristic, complemented
12#24908080C211yesmaximal, characteristic
13#24908080C211yesmaximal, characteristic
14C4×#14581725C211yesmaximal, characteristic, complemented
15C83×#2592C211yesmaximal, characteristic
16C2×#14581725C2211yescharacteristic
17C2×#14581725C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581723C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×#14581716C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×#14581713C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127608—21no—
31#19127587—21no—
32#19127587—21no—
33#19127571C2211yescharacteristic
34#19127567C2211yescharacteristic
35#19127567C2211yescharacteristic
36C4⋊C4×Dic83—21no—
37C4×C4⋊Dic83C2211yescharacteristic
38C4×C4⋊Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127456C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#124×C83C2211yescharacteristic
49C2×C4×Q8×C83C2211yescharacteristic
50#397×C83—21no—
51#393×C83—21no—
52#391×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#390×C83C2211yescharacteristic
55#390×C83C2211yescharacteristic
56#388×C83—21no—
57#388×C83—21no—
58C4×C4⋊C4×C83C2211yescharacteristic
59C22×C83⋊C8C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83C2311yescharacteristic
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4×Dic83—21no—
70C2×C4×Dic83—21no—
71C2×C4×Dic83—21no—
72#14581725—22no—
73#14581725—22no—
74#14581725—41no—
75#14581723—41no—
76C4⋊(C83⋊C8)—22no—
77#14581716—22no—
78#14581716—22no—
79#14581716—41no—
80#14581716—41no—
81#14581713—22no—
82#14581713—22no—
83#14581713—41no—
84#14581713—41no—
85C4×C83⋊C8—22no—
86C83⋊#95—21no—
87C83⋊#95—21no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C22×Q8×C83D411yescharacteristic
94C22×Q8×C83C2311yescharacteristic
95C22×D4×C83C2311yescharacteristic
96#124×C83—22no—
97#124×C83—22no—
98C4×Q8×C83—22no—
99C4×Q8×C83—22no—
100C2×C4⋊C4×C83D411yescharacteristic
101C2×C4⋊C4×C83D411yescharacteristic
102C2×C4⋊C4×C83D411yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C22⋊C4×C83—21no—
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83—21no—
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83—21no—
111C2×C42×C83—21no—
112C2×C42×C83C2311yescharacteristic
113C2×C42×C83C2311yescharacteristic
114C2×C42×C83C2311yescharacteristic
115#95×C83—21no—
116#95×C83—21no—
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C2×C83⋊C8—21no—
125C2×C83⋊C8—21no—
126C2×C83⋊C8—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8#4112yes—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—42no—
144C4⋊Dic83—81no—
145C4⋊Dic83#4112yes—
146C4×Dic83—42no—
147C4×Dic83—81no—
148C4×Dic83—81no—
149C24×C83—21no—
150C2×Q8×C83—22no—
151C2×Q8×C83—22no—
152C2×Q8×C83—21no—
153C2×Q8×C83—21no—
154C2×Q8×C83—21no—
155C2×Q8×C83—21no—
156C2×Q8×C83—21no—
157C2×Q8×C83—21no—
158C2×Q8×C83—41no—
159C2×D4×C83—21no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2411yescharacteristic
175C4⋊C4×C83—22no—
176C4⋊C4×C83—22no—
177C4⋊C4×C83—22no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C22⋊C4×C83—42no—
183C22⋊C4×C83—42no—
184C22⋊C4×C83—42no—
185C22⋊C4×C83—42no—
186C22⋊C4×C83—81no—
187C22⋊C4×C83—81no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—81no—
193C42×C83#4112yes—
194C42×C83#4112yes—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83—41no—
199C23×C83—41no—
200C23×C83—41no—
201C23×C83—41no—
202C23×C83C22×D411yescharacteristic
203Q8×C83—44no—
204Q8×C83—41no—
205Q8×C83—41no—
206Q8×C83—41no—
207Q8×C83—41no—
208D4×C83—41no—
209D4×C83—41no—
210D4×C83—41no—
211D4×C83—41no—
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—21no—
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#12312yes—
237C2×C4×C83#12312yes—
238C2×C4×C83#13512yes—
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83H(5,𝔽2)11yescharacteristic
244C2×C4×C832−1+411yescharacteristic
245C2×Dic83—22no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C83⋊C8—22no—
258C83⋊C8—41no—
259C83⋊C8—41no—
260C83⋊C8—41no—
261C22×C83—81no—
262C22×C83—81no—
263C22×C83—81no—
264C22×C83—81no—
265C22×C83—81no—
266C22×C83—81no—
267C22×C83—81no—
268C22×C83C2×#12311yescharacteristic
269C22×C83#54811yescharacteristic
270C22×C83#55011yescharacteristic
271C22×C83C2×#13511yescharacteristic
272C22×C83#57511yescharacteristic
273C22×C83#57611yescharacteristic
274C22×C83#57811yescharacteristic
275C4×C83—22no—
276C4×C83—22no—
277C4×C83—22no—
278C4×C83—41no—
279C4×C83—41no—
280C4×C83—41no—
281C4×C83—81no—
282C4×C83—81no—
283C4×C83—81no—
284C4×C83#52911yescharacteristic
285C4×C83#54211yescharacteristic
286C4×C83#55611yescharacteristic
287C4×C83#55911yescharacteristic
288Dic83—42no—
289Dic83—81no—
290Dic83—81no—
291Dic83—81no—
292#26325—831nomaximal, Sylow
293C2×C83—161no—
294C2×C83#244811yescharacteristic
295C2×C83#309111yescharacteristic
296C2×C83#309811yescharacteristic
297C2×C83#336111yescharacteristic
298C2×C83#337711yescharacteristic
299C2×C83#339311yescharacteristic
300C2×C83#340311yescharacteristic
301#2592D8311yescharacteristic
302#2355—831no—
303#2095—831no—
304#2092—831no—
305#2074—831no—
306#2071—831no—
307#1984—831no—
308#1983—831no—
309#1958—831no—
310#1949—831no—
311#1928—831no—
312#1927—831no—
313#1925—831no—
314#1925—831no—
315C4×#107—831no—
316C83#2632511yescharacteristic, complemented, Sylow
317C2×#124C2×D8311yescharacteristic
318C2×C4×Q8C2×D8311yescharacteristic
319C2×#107—831no—
320C2×#107—831no—
321C2×#107—831no—
322C2×#106—831no—
323C2×C4⋊C8—831no—
324C2×#103—831no—
325C2×#103—831no—
326C2×#103—831no—
327C2×#103—831no—
328C2×#102—831no—
329C2×#102—831no—
330C2×#102—831no—
331C2×#102—831no—
332C2×C4×C8—831no—
333#397—21no—
334#393—21no—
335#391—1661no—
336#391C2×D8311yescharacteristic
337#391C2×D8311yescharacteristic
338#390C2×D8311yescharacteristic
339#390C2×D8311yescharacteristic
340#388—21no—
341#388—21no—
342#387—1661no—
343#387—1661no—
344#384—831no—
345#382—831no—
346#382—831no—
347C4×C4⋊C4—831no—
348C4×C4⋊C4—831no—
349C4×C4⋊C4—831no—
350C4×C4⋊C4—1661no—
351C4×C4⋊C4C2×D8311yescharacteristic
352#340—831no—
353#340—831no—
354#340—831no—
355#340—831no—
356#336—831no—
357#336—831no—
358#336—831no—
359#336—831no—
360C22×Q8C22⋊D8311yescharacteristic
361C22×Q8C22×D8311yescharacteristic
362C22×D4C22×D8311yescharacteristic
363C22×C8—831no—
364C22×C8—831no—
365C22×C8—831no—
366C22×C8—831no—
367#124—22no—
368#124—22no—
369C4×Q8—22no—
370C4×Q8—22no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4C22⋊D8311yescharacteristic
378C2×C4⋊C4C22⋊D8311yescharacteristic
379C2×C4⋊C4C22⋊D8311yescharacteristic
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C4⋊C4C22×D8311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C42—21no—
389C2×C42—1661no—
390C2×C42—1661no—
391C2×C42—1661no—
392C2×C42C22×D8311yescharacteristic
393C2×C42C22×D8311yescharacteristic
394C2×C42C22×D8311yescharacteristic
395#107—3321no—
396#107—1662no—
397#107—1662no—
398#106—3321no—
399C4⋊C8—1662no—
400#103—3321no—
401#103—3321no—
402#103—1662no—
403#103—1662no—
404#102—3321no—
405#102—3321no—
406#102—1662no—
407#102—1662no—
408C4×C8—1662no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—1661no—
415#95—1661no—
416#95—1661no—
417#95—1661no—
418#95—1661no—
419#95—1661no—
420#95—1661no—
421C24—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—41no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—1661no—
441C22×C4—1661no—
442C22×C4—1661no—
443C22×C4—1661no—
444C22×C4C2×C22⋊D8311yescharacteristic
445C22×C4C2×C22⋊D8311yescharacteristic
446C22×C4C2×C22⋊D8311yescharacteristic
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C23×D8311yescharacteristic
451C2×C8—832no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—3322no—
471C4⋊C4—832no—
472C4⋊C4—6641no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—81no—
491C42—3322no—
492C42—6641no—
493C42—6641no—
494C42#1305058912yes—
495C42#1305058912yes—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22×C22⋊D8311yescharacteristic
504Q8—44no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—1662no—
549C2×C4#1458177312yes—
550C2×C4#1458177312yes—
551C2×C4#1458181212yes—
552C2×C4#41×D8311yescharacteristic
553C2×C4#41×D8311yescharacteristic
554C2×C4C2×#1305058911yescharacteristic
555C2×C4C2×#1305058911yescharacteristic
556C2×C4#1458183811yescharacteristic
557C2×C4#1458183911yescharacteristic
558C8—3321no—
559C8—3321no—
560C8—3321no—
561C8—1662no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1458177311yescharacteristic
570C22#1912835811yescharacteristic
571C22#1912837011yescharacteristic
572C22C2×#1458181211yescharacteristic
573C22#1912849911yescharacteristic
574C22#1912850411yescharacteristic
575C22#1912851611yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—3322no—
586C4—6641no—
587C4—6641no—
588C4—6641no—
589C4#1912823011yescharacteristic
590C4#1912832011yescharacteristic
591C4#1912842011yescharacteristic
592C4#1912843311yescharacteristic
593C2—161no—
594C2#2491125711yescharacteristic
595C2#2491737211yescharacteristic
596C2#2491743711yescharacteristic
597C2#2492011911yescharacteristic
598C2#2492029911yescharacteristic
599C2#2492044711yescharacteristic
600C2#2492054711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2592C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2592D83
hypocenterC83#26325
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
1. #24910420C2semidirect
2. #24909001C2semidirect
4. #24908881C2semidirect
6. #24908400C2semidirect
9. #24908231C2semidirect
10. #24908101C2semidirect
11. #24908094C2semidirect
14. C4×#14581725C2semidirect
316. C83#26325semidirect

Maximal subgroups

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

subgroupindex
#249104202
#249090012
#249089842
#249088812
#249088662
#249084002
#249083962
#249082752
#249082312
#249081012
#249080942
#249080802
C4×#145817252
C83×#25922
#2632583