Finite Group Databasebeta

← #35462689

Subgroups — #35462689

Order 21248.

744 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#24912489C211yesmaximal, characteristic, complemented
2#24908851C211yesmaximal, characteristic
3#24908837C211yesmaximal, characteristic
4#24908599C211yesmaximal, characteristic
5#24908599C211yesmaximal, characteristic
6#24908415C211yesmaximal, characteristic, complemented
7#24908411C211yesmaximal, characteristic, complemented
8#24908229C211yesmaximal, characteristic, complemented
9#24908187C211yesmaximal, characteristic, complemented
10#24908055C211yesmaximal, characteristic, complemented
11#24908041C211yesmaximal, characteristic, complemented
12#24907596C211yesmaximal, characteristic, complemented
13#24907463C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15C83×#2357C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581791—21no—
19C2×#14581725C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29#19127622—21no—
30#19127622—21no—
31#19127600—21no—
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127571C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×C4×D4×C83C2211yescharacteristic
49#390×C83C2211yescharacteristic
50#390×C83C2211yescharacteristic
51#388×C83—21no—
52#388×C83—21no—
53#382×C83C2211yescharacteristic
54#381×C83—21no—
55C4×C4⋊C4×C83C2211yescharacteristic
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C22×Dic166—21no—
60C22×Dic166—21no—
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581792—22no—
66#14581792—22no—
67#14581792—41no—
68#14581792—41no—
69#14581791—44no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×C4×Dic83—21no—
81#14581725—22no—
82#14581723—22no—
83C4⋊(C83⋊C8)—22no—
84C4⋊(C83⋊C8)—41no—
85C4⋊(C83⋊C8)—41no—
86#14581713—22no—
87#14581713—22no—
88#14581713—41no—
89#14581713—41no—
90#14581703—22no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C22×D4×C83C2311yescharacteristic
95C23×C4×C83—21no—
96C4×D4×C83—22no—
97C4×D4×C83Q814yescomplemented
98C2×C4⋊C4×C83D411yescharacteristic
99C2×C4⋊C4×C83D411yescharacteristic
100C2×C4⋊C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83C2311yescharacteristic
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C22⋊C4×C83—21no—
104C2×C22⋊C4×C83—21no—
105C2×C22⋊C4×C83—21no—
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C42×C83—21no—
109C2×C42×C83—21no—
110C2×C42×C83C2311yescharacteristic
111C2×C42×C83C2311yescharacteristic
112C2×C42×C83C2311yescharacteristic
113#95×C83—21no—
114#95×C83—21no—
115#95×C83—21no—
116#95×C83—21no—
117#95×C83D411yescharacteristic
118#95×C83D411yescharacteristic
119C2×Dic166—42no—
120C2×Dic166—42no—
121C2×Dic166—42no—
122C2×Dic166—42no—
123C2×Dic166—81no—
124C2×Dic166—81no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—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—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—42no—
148C4⋊Dic83—42no—
149C4⋊Dic83—42no—
150C4⋊Dic83—81no—
151C4⋊Dic83#4112yes—
152C4⋊Dic83#4112yes—
153Dic83⋊C4—42no—
154Dic83⋊C4—42no—
155C4×Dic83—42no—
156C24×C83—21no—
157C2×D4×C83—21no—
158C2×D4×C83—21no—
159C2×D4×C83—21no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83C2×Q812yes—
163C22×C4×C83—21no—
164C22×C4×C83—24no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—81no—
168C22×C4×C83—81no—
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2411yescharacteristic
176C4⋊C4×C83—22no—
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×C83C2×Q812yes—
184C22⋊C4×C83—24no—
185C22⋊C4×C83—42no—
186C22⋊C4×C83—42no—
187C22⋊C4×C83—81no—
188C22⋊C4×C83—81no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—21no—
193C42×C83—81no—
194C42×C83—81no—
195C42×C83C2×Q812yes—
196C42×C83#4112yes—
197C23×C83—22no—
198C23×C83—41no—
199C23×C83—41no—
200C23×C83—41no—
201C23×C83—41no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83C22×D411yescharacteristic
205D4×C83—22no—
206D4×C83—22no—
207D4×C83—41no—
208D4×C83—41no—
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—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—84no—
233C2×C4×C83C22⋊Q812yes—
234C2×C4×C83C22⋊Q812yes—
235C2×C4×C83AGL(1,ℤ/8)12yes—
236C2×C4×C83#13712yes—
237C2×C4×C83C22×Q811yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83H(5,𝔽2)11yescharacteristic
242C2×C4×C832−1+411yescharacteristic
243C2×Dic83—22no—
244C2×Dic83—22no—
245C2×Dic83—41no—
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—
255Dic166—84no—
256Dic166—84no—
257C83⋊C8—41no—
258C83⋊C8—41no—
259C83⋊C8—41no—
260C83⋊C8—41no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83—42no—
264C22×C83—81no—
265C22×C83—81no—
266C22×C83—81no—
267C22×C83—81no—
268C22×C83C2×C22⋊Q811yescharacteristic
269C22×C83#54611yescharacteristic
270C22×C83#54811yescharacteristic
271C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C83C2×#13711yescharacteristic
273C22×C83#57611yescharacteristic
274C22×C83#57711yescharacteristic
275C4×C83—22no—
276C4×C83—22no—
277C4×C83—22no—
278C4×C83—22no—
279C4×C83—42no—
280C4×C83—41no—
281C4×C83—41no—
282C4×C83—81no—
283C4×C83—81no—
284C4×C83#53311yescharacteristic
285C4×C83#54411yescharacteristic
286C4×C83#55511yescharacteristic
287C4×C83#55611yescharacteristic
288Dic83—42no—
289Dic83—42no—
290Dic83—81no—
291Dic83—81no—
292#26036—831nomaximal, Sylow
293C2×C83—82no—
294C2×C83#243911yescharacteristic
295C2×C83#307911yescharacteristic
296C2×C83#308111yescharacteristic
297C2×C83#331811yescharacteristic
298C2×C83#332011yescharacteristic
299C2×C83#337311yescharacteristic
300C2×C83#337511yescharacteristic
301#2551—831no—
302#2357D8311yescharacteristic
303#2067—831no—
304#2065—831no—
305#2023—831no—
306#2023—831no—
307#1988—831no—
308#1987—831no—
309#1949—831no—
310#1940—831no—
311#1921—831no—
312#1919—831no—
313#1836—831no—
314#1802—831no—
315#1762—831no—
316C83#2603611yescharacteristic, complemented, Sylow
317C2×C4⋊Q8—831no—
318C2×C4⋊Q8—831no—
319C2×C4⋊Q8—1661no—
320C2×C4×D4C2×D8311yescharacteristic
321C2×#107—831no—
322C2×#106—831no—
323C2×C4⋊C8—831no—
324C2×C4⋊C8—831no—
325C2×C4⋊C8—831no—
326C2×#102—831no—
327C2×#102—831no—
328C2×#102—831no—
329C2×#102—831no—
330C2×#97—831no—
331#395—1661no—
332#395—1661no—
333#390C2×D8311yescharacteristic
334#390C2×D8311yescharacteristic
335#389—831no—
336#389—831no—
337#389—1661no—
338#388—21no—
339#388—21no—
340#384—831no—
341#382C2×D8311yescharacteristic
342#381—21no—
343C4×C4⋊C4—831no—
344C4×C4⋊C4C2×D8311yescharacteristic
345C4×C4⋊C4C2×D8311yescharacteristic
346C4×C4⋊C4C2×D8311yescharacteristic
347C4×C22⋊C4—21no—
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#340—831no—
357#340—831no—
358#336—831no—
359#336—831no—
360C22×Q8—1661no—
361C22×Q8—1661no—
362C22×D4C22×D8311yescharacteristic
363C23×C4—21no—
364C22×C8—831no—
365C22×C8—831no—
366C22×C8—831no—
367C22×C8—831no—
368C4⋊Q8—3321no—
369C4⋊Q8—3321no—
370C4⋊Q8—3324no—
371C4⋊Q8—1662no—
372C4⋊Q8—1662no—
373C4×D4—22no—
374C4×D4Dic16614yescomplemented
375C2×C4⋊C4—831no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4C22⋊D8311yescharacteristic
386C2×C4⋊C4C22⋊D8311yescharacteristic
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C4⋊C4C22×D8311yescharacteristic
389C2×C4⋊C4C22×D8311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1661no—
398C2×C42C22×D8311yescharacteristic
399C2×C42C22×D8311yescharacteristic
400C2×C42C22×D8311yescharacteristic
401#107—1662no—
402#106—1662no—
403C4⋊C8—3321no—
404C4⋊C8—3321no—
405C4⋊C8—1662no—
406#102—3321no—
407#102—3321no—
408#102—1662no—
409#102—1662no—
410#97—1662no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1661no—
416#95—1661no—
417#95—1661no—
418#95C22⋊D8311yescharacteristic
419#95C22⋊D8311yescharacteristic
420C24—21no—
421C2×Q8—3322no—
422C2×Q8—3322no—
423C2×Q8—3322no—
424C2×Q8—3322no—
425C2×Q8—6641no—
426C2×Q8—6641no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C2×Dic16612yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—1661no—
440C22×C4—1661no—
441C22×C4—1661no—
442C22×C4—1661no—
443C22×C4C2×C22⋊D8311yescharacteristic
444C22×C4C2×C22⋊D8311yescharacteristic
445C22×C4C2×C22⋊D8311yescharacteristic
446C22×C4C2×C22⋊D8311yescharacteristic
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C23×D8311yescharacteristic
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—
460C2×C8—1661no—
461C2×C8—1661no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—3322no—
470C4⋊C4—3322no—
471C4⋊C4—3322no—
472C4⋊C4—3322no—
473C4⋊C4—3322no—
474C4⋊C4—832no—
475C4⋊C4—832no—
476C4⋊C4—6641no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4C2×Dic16612yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—81no—
494C42—81no—
495C42—3322no—
496C42C2×Dic16612yes—
497C42#1305059012yes—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C22×C22⋊D8311yescharacteristic
506Q8—6644no—
507Q8—6644no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
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—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
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—84no—
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—3321no—
545C2×C4—3321no—
546C2×C4—1662no—
547C2×C4—1662no—
548C2×C4#1458177112yes—
549C2×C4#1458177112yes—
550C2×C4#1458181712yes—
551C2×C4#1458182212yes—
552C2×C4C22×Dic16611yescharacteristic
553C2×C4#41×D8311yescharacteristic
554C2×C4#41×D8311yescharacteristic
555C2×C4C2×#1305059011yescharacteristic
556C2×C4#1458183811yescharacteristic
557C2×C4#1458183911yescharacteristic
558C8—3321no—
559C8—3321no—
560C8—3321no—
561C8—3321no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1458177111yescharacteristic
570C22#1912833911yescharacteristic
571C22#1912835811yescharacteristic
572C22C2×#1458181711yescharacteristic
573C22C2×#1458182211yescharacteristic
574C22#1912850411yescharacteristic
575C22#1912851211yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—3322no—
586C4—3322no—
587C4—6641no—
588C4—6641no—
589C4#1912825411yescharacteristic
590C4#1912832911yescharacteristic
591C4#1912841411yescharacteristic
592C4#1912841911yescharacteristic
593C2—82no—
594C2#2491114811yescharacteristic
595C2#2491727811yescharacteristic
596C2#2491729411yescharacteristic
597C2#2491959811yescharacteristic
598C2#2491962811yescharacteristic
599C2#2492025111yescharacteristic
600C2#2492027311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2357C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2357D83
hypocenterC83#26036
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #24912489C2semidirect
6. #24908415C2semidirect
7. #24908411C2semidirect
8. #24908229C2semidirect
9. #24908187C2semidirect
10. #24908055C2semidirect
11. #24908041C2semidirect
12. #24907596C2semidirect
97. C4×D4×C83Q8semidirect
316. C83#26036semidirect
374. C4×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
#249124892
#249088512
#249088372
#249085992
#249084152
#249084112
#249082292
#249081872
#249080552
#249080412
#249075962
#249074632
#249072882
C83×#23572
#2603683