Finite Group Databasebeta

← #35408784

Subgroups — #35408784

Order 21248.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910114C211yesmaximal, characteristic, complemented
2#24910106C211yesmaximal, characteristic
3#24908346C211yesmaximal, characteristic
4#24908319C211yesmaximal, characteristic
5#24908202C211yesmaximal, characteristic, complemented
6#24908197C211yesmaximal, characteristic, complemented
7#24908162C211yesmaximal, characteristic
8#24908155C211yesmaximal, characteristic, complemented
9#24908148C211yesmaximal, characteristic
10#24908141C211yesmaximal, characteristic, complemented
11#24907360C211yesmaximal, characteristic, complemented
12C4×#14581716C211yesmaximal, characteristic, complemented
13C4×#14581713C211yesmaximal, characteristic
14#24907151C211yesmaximal, characteristic, complemented
15C2×#440×C83C211yesmaximal, characteristic
16C2×#14581726C2211yescharacteristic
17C2×#14581724C2211yescharacteristic
18C2×#14581716C2211yescharacteristic
19C2×#14581716C2211yescharacteristic
20C2×#14581713C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581702C2211yescharacteristic
23C2×#14581702C2211yescharacteristic
24C2×C4×C83⋊C8C2211yescharacteristic
25C2×C8×Dic83C2211yescharacteristic
26C2×C8×Dic83C2211yescharacteristic
27#19127608C2211yescharacteristic
28#19127608C2211yescharacteristic
29#19127604C2211yescharacteristic
30#19127604C2211yescharacteristic
31#19127600C2211yescharacteristic
32#19127600C2211yescharacteristic
33#19127600C2211yescharacteristic
34#19127600C2211yescharacteristic
35#19127594—21no—
36#19127594—21no—
37#19127587—21no—
38#19127587—21no—
39#19127569—21no—
40#19127567—21no—
41#19127564—21no—
42#19127561—21no—
43C4×C4⋊Dic83C2211yescharacteristic
44C4×C4⋊Dic83C2211yescharacteristic
45#19127547C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52C22×SD16×C83C2211yescharacteristic
53C2×C4×Q8×C83C2211yescharacteristic
54C2×C4×D4×C83C2211yescharacteristic
55#440×C83C418yescomplemented
56C2×#107×C83C2211yescharacteristic
57C2×#103×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#97×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C2×C4⋊Dic83—21no—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C22⋊Dic83—21no—
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83C2311yescharacteristic
77C2×C4×Dic83C2311yescharacteristic
78#14581726—22no—
79#14581724—22no—
80#14581716—22no—
81#14581716—22no—
82#14581713—22no—
83#14581713—22no—
84#14581702—22no—
85#14581702—22no—
86C4×C83⋊C8—22no—
87C8×Dic83—22no—
88C8×Dic83—22no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95D411yescharacteristic
100C83⋊#95D411yescharacteristic
101C83⋊#95D411yescharacteristic
102C83⋊#95D411yescharacteristic
103C22×Q8×C83C2311yescharacteristic
104C22×D4×C83C2311yescharacteristic
105C23×C4×C83—21no—
106C2×SD16×C83—22no—
107C2×SD16×C83—22no—
108C2×SD16×C83C2×C414yes—
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C4×Q8×C83—22no—
112C4×Q8×C83C2×C414yes—
113C4×D4×C83—22no—
114C4×D4×C83C2×C414yes—
115C2×C4⋊C4×C83—21no—
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C22⋊C4×C83—21no—
119C2×C42×C83—21no—
120C2×C42×C83C2311yescharacteristic
121#107×C83C2×C414yes—
122#103×C83C2×C414yes—
123#102×C83C2×C414yes—
124#97×C83C2×C414yes—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83C2×D411yescharacteristic
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C22×Dic83C2×D411yescharacteristic
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8#4112yes—
138C2×C83⋊C8#4112yes—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—81no—
148C4⋊Dic83—81no—
149C4⋊Dic83—81no—
150C4⋊Dic83—81no—
151C22⋊Dic83—81no—
152C22⋊Dic83—81no—
153C22⋊Dic83—81no—
154C22⋊Dic83—81no—
155C4×Dic83—21no—
156C4×Dic83—21no—
157C4×Dic83—21no—
158C4×Dic83—21no—
159C4×Dic83—81no—
160C4×Dic83—81no—
161C24×C83—21no—
162C2×Q8×C83—21no—
163C2×Q8×C83—21no—
164C2×Q8×C83—21no—
165C2×Q8×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83C22×C412yes—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C2×D4×C83—21no—
171C2×D4×C83—21no—
172C2×D4×C83—21no—
173C2×D4×C83C22×C412yes—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—24no—
178C22×C4×C83—81no—
179C22×C4×C83—81no—
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2411yescharacteristic
183SD16×C83—28no—
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83C22×C412yes—
187C2×C8×C83C22×C412yes—
188C2×C8×C83#4112yes—
189C2×C8×C83#4112yes—
190C4⋊C4×C83—24no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83C22×C412yes—
194C4⋊C4×C83C22×C412yes—
195C22⋊C4×C83—24no—
196C42×C83—24no—
197C42×C83C22×C412yes—
198C42×C83#4112yes—
199C23×C83—22no—
200C23×C83—41no—
201C23×C83—41no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83C22×D411yescharacteristic
207Q8×C83—22no—
208Q8×C83—22no—
209Q8×C83—41no—
210Q8×C83—41no—
211D4×C83—22no—
212D4×C83—22no—
213D4×C83—41no—
214D4×C83—41no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—22no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
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—84no—
229C2×C4×C83C4×D412yes—
230C2×C4×C83C4×D412yes—
231C2×C4×C83C23×C411yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C8×C83—22no—
262C8×C83#11714yes—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C22×C83—42no—
266C22×C83—42no—
267C22×C83—42no—
268C22×C83—81no—
269C22×C83—81no—
270C22×C83—81no—
271C22×C83—81no—
272C22×C83C2×C4×D411yescharacteristic
273C22×C83#53211yescharacteristic
274C22×C83#53211yescharacteristic
275C22×C83#57611yescharacteristic
276C22×C83#57711yescharacteristic
277C22×C83#57711yescharacteristic
278C22×C83#57811yescharacteristic
279C4×C83—22no—
280C4×C83—22no—
281C4×C83—42no—
282C4×C83—42no—
283C4×C83—42no—
284C4×C83C2×#11711yescharacteristic
285C4×C83C4×#4111yescharacteristic
286C4×C83#52911yescharacteristic
287C4×C83#52911yescharacteristic
288Dic83—41no—
289Dic83—41no—
290Dic83—41no—
291Dic83—41no—
292Dic83—81no—
293Dic83—81no—
294Dic83—81no—
295Dic83—81no—
296#21291—831nomaximal, Sylow
297C2×C83—82no—
298C2×C83#229311yescharacteristic
299C2×C83#294611yescharacteristic
300C2×C83#294711yescharacteristic
301C2×C83#317411yescharacteristic
302C2×C83#317511yescharacteristic
303C2×C83#317811yescharacteristic
304C2×C83#317911yescharacteristic
305C2×#440D8311yescharacteristic
306#2320—831no—
307#2319—831no—
308#1973—831no—
309#1968—831no—
310#1943—831no—
311#1942—831no—
312#1936—831no—
313#1935—831no—
314#1934—831no—
315#1933—831no—
316#1776—831no—
317C4×#103—831no—
318C4×#102—831no—
319#1726—831no—
320C83#2129111yescharacteristic, complemented, Sylow
321C22×SD16C2×D8311yescharacteristic
322C2×C4×Q8C2×D8311yescharacteristic
323C2×C4×D4C2×D8311yescharacteristic
324#440Dic8318yescomplemented
325C2×#107—831no—
326C2×#107C2×D8311yescharacteristic
327C2×#106—831no—
328C2×#103—831no—
329C2×#103—831no—
330C2×#103C2×D8311yescharacteristic
331C2×#102—831no—
332C2×#102—831no—
333C2×#102C2×D8311yescharacteristic
334C2×#97—831no—
335C2×#97—831no—
336C2×#97C2×D8311yescharacteristic
337C2×C4×C8—831no—
338C2×C4×C8—831no—
339C2×C4×C8—831no—
340#391—831no—
341#391—831no—
342#390—831no—
343#390—831no—
344#389—831no—
345#389—831no—
346#389—831no—
347#389—831no—
348#388—1661no—
349#388—1661no—
350#387—1661no—
351#387—1661no—
352#383—1661no—
353#382—1661no—
354#381—1661no—
355#380—1661no—
356C4×C4⋊C4—831no—
357C4×C4⋊C4—831no—
358#376—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365C22×Q8C22×D8311yescharacteristic
366C22×D4C22×D8311yescharacteristic
367C23×C4—21no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C2×Dic8314yes—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8C22×D8311yescharacteristic
374C22×C8C22×D8311yescharacteristic
375C4×Q8—22no—
376C4×Q8C2×Dic8314yes—
377C4×D4—22no—
378C4×D4C2×Dic8314yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—831no—
382C2×C4⋊C4—831no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4—1661no—
386C2×C4⋊C4—1661no—
387C2×C4⋊C4—1661no—
388C2×C4⋊C4C22×D8311yescharacteristic
389C2×C4⋊C4C22×D8311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1661no—
392C2×C22⋊C4—1661no—
393C2×C22⋊C4—1661no—
394C2×C22⋊C4—1661no—
395C2×C42—21no—
396C2×C42—831no—
397C2×C42—831no—
398C2×C42—1661no—
399C2×C42—1661no—
400C2×C42C22×D8311yescharacteristic
401#107—1662no—
402#107C2×Dic8314yes—
403#106—1662no—
404#103—1662no—
405#103—1662no—
406#103C2×Dic8314yes—
407#102—1662no—
408#102—1662no—
409#102C2×Dic8314yes—
410#97—1662no—
411#97—1662no—
412#97C2×Dic8314yes—
413C4×C8—1662no—
414C4×C8—1662no—
415C4×C8—1662no—
416#95—831no—
417#95—831no—
418#95—831no—
419#95—831no—
420#95—1661no—
421#95—1661no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C22×Dic8312yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C22×Dic8312yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—831no—
450C22×C4—831no—
451C22×C4—831no—
452C22×C4—831no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4D4×D8311yescharacteristic
458C22×C4D4×D8311yescharacteristic
459C22×C4C23×D8311yescharacteristic
460SD16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—832no—
464C2×C8—832no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8C22×Dic8312yes—
470C2×C8C22×Dic8312yes—
471C2×C8#1305059012yes—
472C2×C8#1305059012yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—6641no—
477C4⋊C4—6641no—
478C4⋊C4—6641no—
479C4⋊C4—6641no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4C22×Dic8312yes—
489C4⋊C4C22×Dic8312yes—
490C22⋊C4—24no—
491C22⋊C4—6641no—
492C22⋊C4—6641no—
493C22⋊C4—6641no—
494C22⋊C4—6641no—
495C42—24no—
496C42—6641no—
497C42—6641no—
498C42—1661no—
499C42—1661no—
500C42—1661no—
501C42—1661no—
502C42C22×Dic8312yes—
503C42#1305058912yes—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D8311yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
534C2×C4—3321no—
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—3321no—
545C2×C4—3321no—
546C2×C4—1661no—
547C2×C4—1661no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4D4×Dic8312yes—
559C2×C4D4×Dic8312yes—
560C2×C4C23×Dic8311yescharacteristic
561C2×C4#41×D8311yescharacteristic
562C2×C4#41×D8311yescharacteristic
563C2×C4C2×#1305058911yescharacteristic
564C2×C4C2×#1305059011yescharacteristic
565C2×C4C2×#1305059011yescharacteristic
566C8—22no—
567C8—1662no—
568C8—1662no—
569C8#1458175114yes—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×D4×Dic8311yescharacteristic
578C22#1912824811yescharacteristic
579C22#1912824811yescharacteristic
580C22#1912850511yescharacteristic
581C22#1912851011yescharacteristic
582C22#1912851111yescharacteristic
583C22#1912851711yescharacteristic
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—3321no—
590C4—3321no—
591C4—3321no—
592C4—3321no—
593C4—6641no—
594C4—6641no—
595C4—6641no—
596C4—6641no—
597C4C2×#1458175111yescharacteristic
598C4#41×Dic8311yescharacteristic
599C4#1912823311yescharacteristic
600C4#1912823311yescharacteristic
601C2—82no—
602C2#2490992811yescharacteristic
603C2#2491629611yescharacteristic
604C2#2491630711yescharacteristic
605C2#2491808911yescharacteristic
606C2#2491810011yescharacteristic
607C2#2491812911yescharacteristic
608C2#2491814411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#440×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#440D83
hypocenterC83#21291
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #24910114C2semidirect
5. #24908202C2semidirect
6. #24908197C2semidirect
8. #24908155C2semidirect
10. #24908141C2semidirect
11. #24907360C2semidirect
12. C4×#14581716C2semidirect
14. #24907151C2semidirect
55. #440×C83C4semidirect
320. C83#21291semidirect
324. #440Dic83semidirect

Maximal subgroups

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

subgroupindex
#249101142
#249101062
#249083462
#249083192
#249082022
#249081972
#249081622
#249081552
#249081482
#249081412
#249073602
C4×#145817162
C4×#145817132
#249071512
C2×#440×C832
#2129183