Finite Group Databasebeta

← #35407525

Subgroups — #35407525

Order 21248.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127799C211yesmaximal, characteristic
2#24910107C211yesmaximal, characteristic
3#24908090C211yesmaximal, characteristic
4#24908083C211yesmaximal, characteristic
5#24908055C211yesmaximal, characteristic, complemented
6#24908034C211yesmaximal, characteristic, complemented
7#24907818C211yesmaximal, characteristic
8#24907818C211yesmaximal, characteristic
9#24907731C211yesmaximal, characteristic, complemented
10#24907717C211yesmaximal, characteristic, complemented
11#24907361C211yesmaximal, characteristic, complemented
12C4×#14581716C211yesmaximal, characteristic, complemented
13C4×#14581712C211yesmaximal, characteristic
14#24907215C211yesmaximal, characteristic, complemented
15C83×#2276C211yesmaximal, characteristic, complemented
16C22×#13050577C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18#19127799C418yescomplemented
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127603C2211yescharacteristic
31#19127603C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127590—21no—
35#19127590—21no—
36#19127563—21no—
37#19127560—21no—
38C4×C4⋊Dic83C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4×Q8×C83C2211yescharacteristic
50C2×C4×Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C4×C4⋊C4×C83—21no—
53C4×C4⋊C4×C83—21no—
54C4×C4⋊C4×C83—21no—
55C4×C4⋊C4×C83C2211yescharacteristic
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58#376×C83—21no—
59#376×C83C2211yescharacteristic
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62C2×#13050577—22no—
63C2×#13050577—22no—
64C2×#13050577C2×C414yes—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C4×D332—22no—
70C4×D332C2×C414yes—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×(C2×D83)⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×C4×Dic83—21no—
82#14581725—22no—
83#14581725C2×C414yes—
84#14581723—22no—
85#14581716—22no—
86#14581716—22no—
87#14581716C2×C414yes—
88#14581712—22no—
89#14581712—22no—
90#14581712C2×C414yes—
91#14581703C2×C414yes—
92C4×C83⋊C8—22no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C22×Q8×C83C2311yescharacteristic
97C4×Q8×C83—22no—
98C4×Q8×C83—22no—
99C4×Q8×C83—22no—
100C4×Q8×C83—22no—
101C4×Q8×C83—22no—
102C4×Q8×C83C2×C414yes—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C42×C83—21no—
110C2×C42×C83—21no—
111C2×C42×C83—21no—
112C2×C42×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83—21no—
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
121#95×C83D411yescharacteristic
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124#95×C83D411yescharacteristic
125C23×D83—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332C22×C412yes—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C2×C4×D83—24no—
136C2×C4×D83—81no—
137C2×C4×D83—81no—
138#13050577—28no—
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⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8C22×C412yes—
150C2×C83⋊C8C22×C412yes—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83C22×C412yes—
157Dic83⋊C4—81no—
158Dic83⋊C4—81no—
159(C2×D83)⋊C4—24no—
160(C2×D83)⋊C4—81no—
161(C2×D83)⋊C4—81no—
162(C2×D83)⋊C4—81no—
163(C2×D83)⋊C4—81no—
164C4×Dic83—81no—
165C2×Q8×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83C22×C412yes—
169C2×Q8×C83#4112yes—
170C2×Q8×C83#4112yes—
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—24no—
183C4⋊C4×C83—24no—
184C4⋊C4×C83—24no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83C22×C412yes—
189C4⋊C4×C83#4112yes—
190C4⋊C4×C83#4112yes—
191C42×C83—24no—
192C42×C83—24no—
193C42×C83—24no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—81no—
199C42×C83—81no—
200C42×C83—81no—
201C42×C83C22×C412yes—
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—22no—
212Q8×C83—22no—
213Q8×C83—22no—
214Q8×C83#11714yes—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—22no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83C4×D412yes—
244C2×C4×C83C4×D412yes—
245C2×C4×C83C23×C411yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C2×C4×C83C2×#4111yescharacteristic
248C2×C4×C83C2×#4111yescharacteristic
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×Dic83—22no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260D332—22no—
261D332—22no—
262D332—41no—
263D332—41no—
264C4×D83—84no—
265C83⋊C8—22no—
266C83⋊C8—22no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C22×C83C2×C4×D411yescharacteristic
270C22×C83#52511yescharacteristic
271C22×C83#52511yescharacteristic
272C22×C83#57611yescharacteristic
273C22×C83#57711yescharacteristic
274C22×C83#57711yescharacteristic
275C22×C83#57811yescharacteristic
276C2×D83—42no—
277C2×D83—42no—
278C2×D83—42no—
279C2×D83—81no—
280C2×D83—81no—
281C2×D83—81no—
282C2×D83—81no—
283C4×C83—22no—
284C4×C83—22no—
285C4×C83—24no—
286C4×C83—41no—
287C4×C83—42no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—42no—
291C4×C83—41no—
292C4×C83—42no—
293C4×C83C2×#11711yescharacteristic
294C4×C83C4×#4111yescharacteristic
295C4×C83#52911yescharacteristic
296C4×C83#52911yescharacteristic
297Dic83—42no—
298Dic83—81no—
299Dic83—81no—
300#21128—831nomaximal, Sylow
301C2×C83#229011yescharacteristic
302C2×C83#294611yescharacteristic
303C2×C83#294711yescharacteristic
304C2×C83#311911yescharacteristic
305C2×C83#312111yescharacteristic
306C2×C83#312411yescharacteristic
307C2×C83#312611yescharacteristic
308D83—82no—
309C2×#440—831no—
310#2319—831no—
311#2276D8311yescharacteristic, complemented
312#1926—831no—
313#1925—831no—
314#1921—831no—
315#1918—831no—
316#1880—831no—
317#1880—831no—
318#1865—831no—
319#1863—831no—
320#1776—831no—
321C4×#103—831no—
322C4×#102—831no—
323#1746—831no—
324C83#2112811yescharacteristic, complemented, Sylow
325C22×SD16—831no—
326C2×C4×Q8C2×D8311yescharacteristic
327C2×C4×Q8C2×D8311yescharacteristic
328C2×C4×Q8C2×D8311yescharacteristic
329C2×C4×D4—831no—
330#440—838no—
331C2×#107—831no—
332C2×#107—831no—
333C2×#106—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×#103—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#102—831no—
340C2×#97—831no—
341C2×C4×C8—831no—
342#390—831no—
343#390—831no—
344#389—831no—
345#389—831no—
346#388—1661no—
347#388—1661no—
348#381—1661no—
349#380—1661no—
350C4×C4⋊C4—21no—
351C4×C4⋊C4—21no—
352C4×C4⋊C4—21no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4C2×D8311yescharacteristic
355C4×C4⋊C4C2×D8311yescharacteristic
356C4×C4⋊C4C2×D8311yescharacteristic
357#376—21no—
358#376C2×D8311yescharacteristic
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#336—831no—
366#336—831no—
367#336—831no—
368#336—831no—
369C22×Q8C22×D8311yescharacteristic
370C22×D4—831no—
371C23×C4—1661no—
372C2×SD16—834no—
373C2×SD16—1662no—
374C2×SD16—1662no—
375C22×C8—831no—
376C22×C8—831no—
377C22×C8—831no—
378C22×C8—831no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8C4×D8314yes—
385C4×D4—834no—
386C4×D4—1662no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—1661no—
410C2×C42C22×D8311yescharacteristic
411C2×C42C22×D8311yescharacteristic
412C2×C42C22×D8311yescharacteristic
413#107—834no—
414#107—1662no—
415#106—1662no—
416#103—834no—
417#103—1662no—
418#103—1662no—
419#102—834no—
420#102—1662no—
421#102—1662no—
422#97—834no—
423C4×C8—1662no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95C22⋊D8311yescharacteristic
431#95C22⋊D8311yescharacteristic
432#95C22⋊D8311yescharacteristic
433#95C22⋊D8311yescharacteristic
434C24—1661no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D8312yes—
439C2×Q8#1305059012yes—
440C2×Q8#1305059012yes—
441C2×D4—832no—
442C2×D4—1661no—
443C2×D4—1661no—
444C2×D4—1661no—
445C2×D4—1661no—
446C2×D4—1661no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—6641no—
452C22×C4—6641no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1664no—
457C22×C4C2×C22⋊D8311yescharacteristic
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4C2×C22⋊D8311yescharacteristic
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C23×D8311yescharacteristic
464SD16—1668no—
465C2×C8—832no—
466C2×C8—832no—
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—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—832no—
484C4⋊C4—6641no—
485C4⋊C4—6641no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4C2×C4×D8312yes—
492C4⋊C4#1305059012yes—
493C4⋊C4#1305059012yes—
494C22⋊C4—6641no—
495C22⋊C4—6641no—
496C22⋊C4—6641no—
497C22⋊C4—6641no—
498C22⋊C4—1664no—
499C42—24no—
500C42—24no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—24no—
506C42—81no—
507C42—81no—
508C42—81no—
509C42—6641no—
510C42C2×C4×D8312yes—
511C42#1305059012yes—
512C23—3321no—
513C23—3321no—
514C23—3321no—
515C23—3321no—
516C23—3321no—
517C23—3321no—
518C23—1662no—
519C23C22×C22⋊D8311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#1458174914yes—
524D4—3321no—
525D4—3321no—
526D4—1662no—
527D4—1662no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
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×C4C4×C22⋊D8312yes—
567C2×C4C4×C22⋊D8312yes—
568C2×C4C22×C4×D8311yescharacteristic
569C2×C4C2×#1305059011yescharacteristic
570C2×C4C2×#1305059011yescharacteristic
571C2×C4C2×#1305059011yescharacteristic
572C2×C4C2×#1305059011yescharacteristic
573C2×C4C2×#1305059011yescharacteristic
574C8—3321no—
575C8—3321no—
576C8—1662no—
577C8—1662no—
578C22—3322no—
579C22—3322no—
580C22—3322no—
581C22—6641no—
582C22—6641no—
583C22—6641no—
584C22—6641no—
585C22C2×C4×C22⋊D8311yescharacteristic
586C22#1912821211yescharacteristic
587C22#1912821211yescharacteristic
588C22#1912850411yescharacteristic
589C22#1912850911yescharacteristic
590C22#1912850911yescharacteristic
591C22#1912851611yescharacteristic
592C4—22no—
593C4—24no—
594C4—22no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—3322no—
603C4—6641no—
604C4—6641no—
605C4C2×#1458174911yescharacteristic
606C4C4×#1305059011yescharacteristic
607C4#1912823511yescharacteristic
608C4#1912823511yescharacteristic
609C2—6642no—
610C2#2490990611yescharacteristic
611C2#2491629111yescharacteristic
612C2#2491630611yescharacteristic
613C2#2491766211yescharacteristic
614C2#2491769211yescharacteristic
615C2#2491772511yescharacteristic
616C2#2491775411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2276C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#2276D83
hypocenterC83#21128
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2276

Complements

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

NH ≅ G/Ntype
5. #24908055C2semidirect
6. #24908034C2semidirect
9. #24907731C2semidirect
10. #24907717C2semidirect
11. #24907361C2semidirect
12. C4×#14581716C2semidirect
14. #24907215C2semidirect
15. C83×#2276C2semidirect
18. #19127799C4semidirect
311. #2276D83semidirect
324. C83#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#191277992
#249101072
#249080902
#249080832
#249080552
#249080342
#249078182
#249077312
#249077172
#249073612
C4×#145817162
C4×#145817122
#249072152
C83×#22762
#2112883