Finite Group Databasebeta

← #35456062

Subgroups — #35456062

Order 21248.

754 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128100C211yesmaximal, characteristic, complemented
2C2×#19127998C211yesmaximal, characteristic
3#24912663C211yesmaximal, characteristic
4#24908944C211yesmaximal, characteristic, complemented
5#24908939C211yesmaximal, characteristic, complemented
6#24908848C211yesmaximal, characteristic
7#24908848C211yesmaximal, characteristic
8#24908420C211yesmaximal, characteristic, complemented
9#24908231C211yesmaximal, characteristic, complemented
10#24908221C211yesmaximal, characteristic, complemented
11#24908182C211yesmaximal, characteristic, complemented
12#24908090C211yesmaximal, characteristic
13#24908090C211yesmaximal, characteristic
14C4×#14581712C211yesmaximal, characteristic
15C83×#2370C211yesmaximal, characteristic, complemented
16C2×#14581784C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18#19128100—24no—
19#19127998—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581712C2211yescharacteristic
30C2×#14581712C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34#19127623—21no—
35#19127623—21no—
36#19127603C2211yescharacteristic
37#19127603C2211yescharacteristic
38#19127575—21no—
39#19127575—21no—
40#19127571C2211yescharacteristic
41#19127567C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×#125×C83C2211yescharacteristic
51#389×C83—21no—
52#389×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#387×C83—21no—
55#387×C83—21no—
56C4×C4⋊C4×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581784—22no—
68#14581784—22no—
69C4×D332—22no—
70C4×D332—22no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83#14581725—22no—
84#14581725—22no—
85#14581725—22no—
86#14581723—22no—
87#14581723—22no—
88#14581723—22no—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91C4⋊(C83⋊C8)—22no—
92#14581712—22no—
93#14581712—22no—
94#14581712—22no—
95#14581712Q814yes—
96C4×C83⋊C8—22no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99#125×C83—22no—
100#125×C83D414yes—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83D411yescharacteristic
106C2×C4⋊C4×C83D411yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C42×C83—21no—
112C2×C42×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124C23×D83—21no—
125C2×D332—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332C2×Q812yes—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C2×C4×D83—42no—
135C2×C4×D83—81no—
136C2×C4×D83—81no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8C2×Q812yes—
148C2×C83⋊C8#4112yes—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—81no—
156C4⋊Dic83—81no—
157Dic83⋊C4—42no—
158Dic83⋊C4—42no—
159(C2×D83)⋊C4—42no—
160(C2×D83)⋊C4—42no—
161(C2×D83)⋊C4—42no—
162(C2×D83)⋊C4—81no—
163(C2×D83)⋊C4—81no—
164C22×C4×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—24no—
176C4⋊C4×C83—24no—
177C4⋊C4×C83—22no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—41no—
184C4⋊C4×C83—42no—
185C4⋊C4×C83—81no—
186C4⋊C4×C83C2×D412yes—
187C4⋊C4×C83C2×D412yes—
188C4⋊C4×C83C2×Q812yes—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—21no—
193C42×C83—42no—
194C42×C83—81no—
195C42×C83—81no—
196C42×C83C2×D412yes—
197C42×C83#4112yes—
198C23×C83C22×D411yescharacteristic
199C22×D83—22no—
200C22×D83—41no—
201C22×D83—41no—
202C22×D83—41no—
203C22×D83—41no—
204C22×D83—41no—
205C22×D83—41no—
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—22no—
213C2×C4×C83—22no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—22no—
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—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—41no—
233C2×C4×C83#12112yes—
234C2×C4×C83#12112yes—
235C2×C4×C83#13512yes—
236C2×C4×C83C22×D411yescharacteristic
237C2×C4×C83C22×Q811yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C832−1+411yescharacteristic
241C2×C4×C832−1+411yescharacteristic
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—
251D332—22no—
252D332—22no—
253D332—41no—
254D332—41no—
255C4×D83—84no—
256C83⋊C8—22no—
257C83⋊C8—22no—
258C83⋊C8—41no—
259C83⋊C8—41no—
260C22×C83C2×#12111yescharacteristic
261C22×C83#54711yescharacteristic
262C22×C83D4×Q811yescharacteristic
263C22×C83C2×#13511yescharacteristic
264C22×C83#57511yescharacteristic
265C22×C83#57611yescharacteristic
266C22×C83#57711yescharacteristic
267C2×D83—42no—
268C2×D83—42no—
269C2×D83—42no—
270C2×D83—81no—
271C2×D83—81no—
272C2×D83—81no—
273C2×D83—81no—
274C4×C83—22no—
275C4×C83—22no—
276C4×C83—22no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83—41no—
280C4×C83—42no—
281C4×C83—42no—
282C4×C83—42no—
283C4×C83—81no—
284C4×C83#53311yescharacteristic
285C4×C83#54111yescharacteristic
286C4×C83#54811yescharacteristic
287C4×C83D4×Q811yescharacteristic
288Dic83—81no—
289Dic83—81no—
290Dic83—81no—
291#25457—831nomaximal, Sylow
292C2×C83#242911yescharacteristic
293C2×C83#303511yescharacteristic
294C2×C83#304511yescharacteristic
295C2×C83#333911yescharacteristic
296C2×C83#335111yescharacteristic
297C2×C83#338111yescharacteristic
298C2×C83#338911yescharacteristic
299D83—82no—
300C2×#499—831no—
301C2×#478—831no—
302#2566—831no—
303#2370D8311yescharacteristic, complemented
304#2085—831no—
305#2084—831no—
306#2067—831no—
307#2067—831no—
308#1989—831no—
309#1949—831no—
310#1947—831no—
311#1939—831no—
312#1926—831no—
313#1926—831no—
314C4×#102—831no—
315C83#2545711yescharacteristic, complemented, Sylow
316C2×#125—831no—
317C2×#125C2×D8311yescharacteristic
318C2×C4×D4—831no—
319#499—1664no—
320#478—1664no—
321C2×#107—831no—
322C2×#107—831no—
323C2×#107—831no—
324C2×#106—831no—
325C2×#106—831no—
326C2×#106—831no—
327C2×C4⋊C8—831no—
328C2×C4⋊C8—831no—
329C2×C4⋊C8—831no—
330C2×#102—831no—
331C2×#102—831no—
332C2×#102—831no—
333C2×#102—831no—
334C2×C4×C8—831no—
335#396—1661no—
336#396—1661no—
337#390—831no—
338#390—831no—
339#389—21no—
340#389C2×D8311yescharacteristic
341#389C2×D8311yescharacteristic
342#387—21no—
343#387—21no—
344#385—1661no—
345#385—1661no—
346#384—831no—
347#382—831no—
348C4×C4⋊C4—21no—
349C4×C4⋊C4C2×D8311yescharacteristic
350C4×C4⋊C4C2×D8311yescharacteristic
351C4×C4⋊C4C2×D8311yescharacteristic
352C4×C4⋊C4C2×D8311yescharacteristic
353#340—831no—
354#340—831no—
355#340—831no—
356#340—831no—
357#340—831no—
358#340—831no—
359#336—831no—
360#336—831no—
361C22×D4—831no—
362C23×C4—1661no—
363C22×C8—831no—
364C22×C8—831no—
365C22×C8—831no—
366C22×C8—831no—
367#125—22no—
368#125—1662no—
369#125—1662no—
370#125D33214yes—
371C4×D4—1662no—
372C4×D4—1662no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4C22⋊D8311yescharacteristic
385C2×C4⋊C4C22⋊D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C4⋊C4C22×D8311yescharacteristic
389C2×C4⋊C4C22×D8311yescharacteristic
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C22⋊C4—1661no—
393C2×C22⋊C4—1661no—
394C2×C22⋊C4—1661no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C22×D8311yescharacteristic
399C2×C42C22×D8311yescharacteristic
400C2×C42C22×D8311yescharacteristic
401#107—1662no—
402#107—1662no—
403#107—1662no—
404#106—1662no—
405#106—1662no—
406#106—1662no—
407C4⋊C8—1662no—
408C4⋊C8—1662no—
409C4⋊C8—1662no—
410#102—834no—
411#102—1662no—
412#102—1662no—
413#102—1662no—
414C4×C8—1662no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—1661no—
421#95—1661no—
422#95C22⋊D8311yescharacteristic
423#95C22⋊D8311yescharacteristic
424C24—1661no—
425C2×D4—832no—
426C2×D4—1661no—
427C2×D4—1661no—
428C2×D4—1661no—
429C2×D4—1661no—
430C2×D4—1661no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—3322no—
436C22×C4—6641no—
437C22×C4—6641no—
438C22×C4—1661no—
439C22×C4—1661no—
440C22×C4—1661no—
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C2×C22⋊D8311yescharacteristic
444C22×C4C2×C22⋊D8311yescharacteristic
445C22×C4C2×C22⋊D8311yescharacteristic
446C22×C4C2×C22⋊D8311yescharacteristic
447C22×C4C23×D8311yescharacteristic
448C2×C8—832no—
449C2×C8—832no—
450C2×C8—1661no—
451C2×C8—1661no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—3322no—
472C4⋊C4—3322no—
473C4⋊C4—6641no—
474C4⋊C4—6641no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4C2×D33212yes—
482C4⋊C4C2×D33212yes—
483C4⋊C4Q8×D8312yes—
484C22⋊C4—3322no—
485C22⋊C4—3322no—
486C22⋊C4—3322no—
487C22⋊C4—6641no—
488C22⋊C4—6641no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—81no—
495C42—81no—
496C42C2×D33212yes—
497C42#1305059012yes—
498C23—3321no—
499C23—3321no—
500C23—3321no—
501C23—3321no—
502C23—3321no—
503C23—3321no—
504C23—1662no—
505C23C22×C22⋊D8311yescharacteristic
506D4—3321no—
507D4—3321no—
508D4—1662no—
509D4—1662no—
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—22no—
520C2×C4—22no—
521C2×C4—22no—
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—6644no—
547C2×C4#1458176612yes—
548C2×C4#1458176612yes—
549C2×C4#1458181212yes—
550C2×C4C22×D33211yescharacteristic
551C2×C4C2×Q8×D8311yescharacteristic
552C2×C4#41×D8311yescharacteristic
553C2×C4C2×#1305059011yescharacteristic
554C2×C4#1458183911yescharacteristic
555C2×C4#1458184011yescharacteristic
556C8—3321no—
557C8—3321no—
558C8—1662no—
559C8—1662no—
560C22—3322no—
561C22—3322no—
562C22—3322no—
563C22—6641no—
564C22—6641no—
565C22—6641no—
566C22—6641no—
567C22C2×#1458176611yescharacteristic
568C22#1912835011yescharacteristic
569C22Q8×C22⋊D8311yescharacteristic
570C22C2×#1458181211yescharacteristic
571C22#1912849911yescharacteristic
572C22#1912850411yescharacteristic
573C22#1912850911yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—6641no—
585C4—6641no—
586C4—6641no—
587C4#1912825511yescharacteristic
588C4#1912830911yescharacteristic
589C4#1912836311yescharacteristic
590C4Q8×D33211yescharacteristic
591C2—6642no—
592C2#2491105611yescharacteristic
593C2#2491691811yescharacteristic
594C2#2491700211yescharacteristic
595C2#2491985911yescharacteristic
596C2#2492002311yescharacteristic
597C2#2492033411yescharacteristic
598C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2370C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2370D83
hypocenterC83#25457
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C2×#19128100C2semidirect
4. #24908944C2semidirect
5. #24908939C2semidirect
8. #24908420C2semidirect
9. #24908231C2semidirect
10. #24908221C2semidirect
11. #24908182C2semidirect
15. C83×#2370C2semidirect
303. #2370D83semidirect
315. C83#25457semidirect

Maximal subgroups

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

subgroupindex
C2×#191281002
C2×#191279982
#249126632
#249089442
#249089392
#249088482
#249084202
#249082312
#249082212
#249081822
#249080902
C4×#145817122
C83×#23702
#2545783