Finite Group Databasebeta

← #35426372

Subgroups — #35426372

Order 21248.

878 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910523C211yesmaximal, characteristic, complemented
2#24910373C211yesmaximal, characteristic
3#24908437C211yesmaximal, characteristic, complemented
4#24908233C211yesmaximal, characteristic, complemented
5#24908207C211yesmaximal, characteristic, complemented
6#24908192C211yesmaximal, characteristic, complemented
7#24908050C211yesmaximal, characteristic, complemented
8#24908043C211yesmaximal, characteristic, complemented
9#24907726C211yesmaximal, characteristic
10#24907712C211yesmaximal, characteristic
11#24907340C211yesmaximal, characteristic, complemented
12#24907290C211yesmaximal, characteristic
13#24907288C211yesmaximal, characteristic
14#102×Dic83C211yesmaximal, characteristic
15C2×#476×C83C211yesmaximal, characteristic
16C2×Q8×Dic83—21no—
17C2×D4×Dic83C2211yescharacteristic
18C2×D4×Dic83C2211yescharacteristic
19C2×#14581726C2211yescharacteristic
20C2×#14581724C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581703C2211yescharacteristic
24C2×#14581702C2211yescharacteristic
25C2×C8×Dic83C2211yescharacteristic
26#19127608—21no—
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127600C2211yescharacteristic
31#19127587—21no—
32#19127587—21no—
33#19127576—21no—
34#19127576—21no—
35#19127567C2211yescharacteristic
36#19127561—21no—
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41C2×C83⋊#95—21no—
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C2×C4⋊Q8×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#476×C83C418yescomplemented
55C2×#106×C83C2211yescharacteristic
56C2×#106×C83C2211yescharacteristic
57C2×C4⋊C8×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#102×C83C2211yescharacteristic
60C23×Dic83—21no—
61C23×Dic83—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64Q8×Dic83—44no—
65D4×Dic83—22no—
66D4×Dic83—22no—
67D4×Dic83—41no—
68D4×Dic83—41no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83D411yescharacteristic
74C2×C4⋊Dic83D411yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83C2311yescharacteristic
85C2×C4×Dic83C2311yescharacteristic
86#14581726—41no—
87#14581724—41no—
88#14581713—22no—
89#14581713Q814yescomplemented
90#14581703—22no—
91#14581702—41no—
92C8×Dic83—41no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95D411yescharacteristic
103C83⋊#95D411yescharacteristic
104C22×Q8×C83—21no—
105C22×D4×C83C2311yescharacteristic
106C23×C4×C83—21no—
107C22×C8×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C4⋊Q8×C83—22no—
110C4⋊Q8×C83C2×C414yescomplemented
111C4×D4×C83—22no—
112C4×D4×C83C2×C414yes—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C42×C83C2311yescharacteristic
119#106×C83C2×C414yescomplemented
120#106×C83C2×C414yescomplemented
121C4⋊C8×C83C2×C414yescomplemented
122#102×C83C2×C414yes—
123#102×C83C2×C414yes—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—42no—
129C22×Dic83—42no—
130C22×Dic83—42no—
131C22×Dic83—42no—
132C22×Dic83—81no—
133C22×Dic83—81no—
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C22×Dic83C2×D411yescharacteristic
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8C2×Q812yes—
144C4⋊Dic83—21no—
145C4⋊Dic83—22no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—41no—
153C4⋊Dic83—42no—
154C4⋊Dic83—42no—
155C4⋊Dic83—81no—
156C4⋊Dic83—81no—
157C4⋊Dic83C2×Q812yes—
158C22⋊Dic83—42no—
159C22⋊Dic83—42no—
160C4×Dic83—21no—
161C4×Dic83—21no—
162C4×Dic83—21no—
163C4×Dic83—21no—
164C4×Dic83—42no—
165C4×Dic83—42no—
166C4×Dic83—42no—
167C24×C83—21no—
168C2×Q8×C83—24no—
169C2×Q8×C83—42no—
170C2×Q8×C83—42no—
171C2×D4×C83—21no—
172C2×D4×C83—21no—
173C2×D4×C83—21no—
174C2×D4×C83C22×C412yes—
175C2×D4×C83C2×Q812yes—
176C2×D4×C83#4112yes—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—24no—
181C22×C4×C83—81no—
182C22×C4×C83—81no—
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2411yescharacteristic
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83C22×C412yes—
191C2×C8×C83C22×C412yes—
192C4⋊C4×C83—24no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83C22×C412yes—
197C4⋊C4×C83C22×C412yes—
198C4⋊C4×C83C22×C412yes—
199C22⋊C4×C83—24no—
200C42×C83C22×C412yes—
201C42×C83#4112yes—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83C22×D411yescharacteristic
210Q8×C83—48no—
211D4×C83—22no—
212D4×C83—22no—
213D4×C83—22no—
214D4×C83C4×Q814yescomplemented
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×C83C22×Q811yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83H(5,𝔽2)11yescharacteristic
236C2×C4×C832−1+411yescharacteristic
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—22no—
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—84no—
261C2×Dic83—84no—
262C2×Dic83AGL(1,ℤ/8)12yes—
263C8×C83—22no—
264C8×C83—22no—
265C83⋊C8—22no—
266C83⋊C8—41no—
267C22×C83—24no—
268C22×C83—42no—
269C22×C83—42no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83C2×C4×D411yescharacteristic
275C22×C83#54611yescharacteristic
276C22×C83D4×Q811yescharacteristic
277C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C83#57511yescharacteristic
279C22×C83#57711yescharacteristic
280C22×C83#57711yescharacteristic
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—42no—
284C4×C83—42no—
285C4×C83—42no—
286C4×C83C2×C4×Q811yescharacteristic
287C4×C83#52011yescharacteristic
288C4×C83#55211yescharacteristic
289C4×C83#55411yescharacteristic
290Dic83—22no—
291Dic83—42no—
292Dic83—41no—
293Dic83—41no—
294Dic83—41no—
295Dic83—81no—
296Dic83—81no—
297Dic83—81no—
298#22975—831nomaximal, Sylow
299C2×C83—44no—
300C2×C83#236011yescharacteristic
301C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C83#294911yescharacteristic
303C2×C83#330611yescharacteristic
304C2×C83#331111yescharacteristic
305C2×C83#338411yescharacteristic
306C2×C83#338811yescharacteristic
307C2×#476D8311yescharacteristic
308#2365—831no—
309#2349—831no—
310#1993—831no—
311#1949—831no—
312#1944—831no—
313#1941—831no—
314#1920—831no—
315#1919—831no—
316#1864—831no—
317#1862—831no—
318#1771—831no—
319#1762—831no—
320#1762—831no—
321C4×#102—831no—
322C83#2297511yescharacteristic, complemented, Sylow
323C2×C4⋊Q8C2×D8311yescharacteristic
324C2×C4×Q8—1661no—
325C2×C4×D4—831no—
326C2×C4×D4—831no—
327C2×C4×D4C2×D8311yescharacteristic
328#476Dic8318yescomplemented
329C2×#107—831no—
330C2×#106—831no—
331C2×#106C2×D8311yescharacteristic
332C2×#106C2×D8311yescharacteristic
333C2×C4⋊C8C2×D8311yescharacteristic
334C2×#102—831no—
335C2×#102—831no—
336C2×#102C2×D8311yescharacteristic
337C2×#102C2×D8311yescharacteristic
338C2×#97—831no—
339C2×#97—831no—
340C2×C4×C8—831no—
341#391—1661no—
342#389—831no—
343#389—831no—
344#389—831no—
345#389—831no—
346#387—1661no—
347#387—1661no—
348#385—1661no—
349#385—1661no—
350#382—831no—
351#380—1661no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—831no—
356C2×#95—1661no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#336—831no—
366#336—831no—
367C22×Q8—21no—
368C22×D4C22×D8311yescharacteristic
369C23×C4—21no—
370C23×C4—1661no—
371C23×C4—1661no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8C22×D8311yescharacteristic
375C22×C8C22×D8311yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8C2×Dic8314yescomplemented
378C4×Q8—3324no—
379C4×D4—22no—
380C4×D4—3321no—
381C4×D4—3321no—
382C4×D4—1662no—
383C4×D4—1662no—
384C4×D4C2×Dic8314yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C42—831no—
403C2×C42—831no—
404C2×C42—1661no—
405C2×C42—1661no—
406C2×C42—1661no—
407C2×C42C22×D8311yescharacteristic
408#107—3321no—
409#106—3321no—
410#106C2×Dic8314yescomplemented
411#106C2×Dic8314yescomplemented
412C4⋊C8C2×Dic8314yescomplemented
413#102—834no—
414#102—1662no—
415#102C2×Dic8314yes—
416#102C2×Dic8314yes—
417#97—3321no—
418#97—1662no—
419C4×C8—3321no—
420#95—831no—
421#95—831no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431C24—21no—
432C2×Q8—24no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C22×Dic8312yes—
439C2×D4Q8×D8312yes—
440C2×D4#1305059112yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—3322no—
448C22×C4—3322no—
449C22×C4—3322no—
450C22×C4—3322no—
451C22×C4—831no—
452C22×C4—831no—
453C22×C4—831no—
454C22×C4—831no—
455C22×C4—6641no—
456C22×C4—6641no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4D4×D8311yescharacteristic
462C22×C4D4×D8311yescharacteristic
463C22×C4C23×D8311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—832no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8C22×Dic8312yes—
475C2×C8C22×Dic8312yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—3321no—
481C4⋊C4—3322no—
482C4⋊C4—3322no—
483C4⋊C4—832no—
484C4⋊C4—6641no—
485C4⋊C4—6641no—
486C4⋊C4—1662no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4C22×Dic8312yes—
495C4⋊C4C22×Dic8312yes—
496C4⋊C4C22×Dic8312yes—
497C22⋊C4—24no—
498C22⋊C4—3322no—
499C22⋊C4—3322no—
500C42—3322no—
501C42—3322no—
502C42—3322no—
503C42—1661no—
504C42—1661no—
505C42—1661no—
506C42—1661no—
507C42C22×Dic8312yes—
508C42#1305058912yes—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×D4×D8311yescharacteristic
517Q8—48no—
518D4—22no—
519D4—22no—
520D4—22no—
521D4Q8×Dic8314yescomplemented
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
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—3321no—
547C2×C4—832no—
548C2×C4—6644no—
549C2×C4—6644no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1662no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4—1661no—
561C2×C4—1661no—
562C2×C4D4×Dic8312yes—
563C2×C4D4×Dic8312yes—
564C2×C4C23×Dic8311yescharacteristic
565C2×C4C2×Q8×D8311yescharacteristic
566C2×C4C2×#1305058911yescharacteristic
567C2×C4C2×#1305059111yescharacteristic
568C2×C4#1458183811yescharacteristic
569C2×C4#1458183911yescharacteristic
570C8—22no—
571C8—22no—
572C8—3321no—
573C8—1662no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22C2×D4×Dic8311yescharacteristic
582C22#1912834211yescharacteristic
583C22#1912836411yescharacteristic
584C22AGL(1,ℤ/8)×D8311yescharacteristic
585C22#1912849711yescharacteristic
586C22#1912850711yescharacteristic
587C22#1912850711yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—3322no—
594C4—3321no—
595C4—3321no—
596C4—3321no—
597C4—6641no—
598C4—6641no—
599C4—6641no—
600C4—1662no—
601C4C2×Q8×Dic8311yescharacteristic
602C4#1912818411yescharacteristic
603C4#1912839311yescharacteristic
604C4#1912840911yescharacteristic
605C2—44no—
606C2#2491047211yescharacteristic
607C2AGL(1,ℤ/8)×Dic8311yescharacteristic
608C2#2491632311yescharacteristic
609C2#2491942111yescharacteristic
610C2#2491949611yescharacteristic
611C2#2492035711yescharacteristic
612C2#2492040111yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#476×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#476D83
hypocenterC83#22975
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #24910523C2semidirect
3. #24908437C2semidirect
4. #24908233C2semidirect
5. #24908207C2semidirect
6. #24908192C2semidirect
7. #24908050C2semidirect
8. #24908043C2semidirect
11. #24907340C2semidirect
54. #476×C83C4semidirect
89. #14581713Q8semidirect
110. C4⋊Q8×C83C2×C4semidirect
119. #106×C83C2×C4semidirect
120. #106×C83C2×C4semidirect
121. C4⋊C8×C83C2×C4semidirect
214. D4×C83C4×Q8semidirect
322. C83#22975semidirect
328. #476Dic83semidirect
377. C4⋊Q8C2×Dic83semidirect
410. #106C2×Dic83semidirect
411. #106C2×Dic83semidirect
412. C4⋊C8C2×Dic83semidirect
521. D4Q8×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249105232
#249103732
#249084372
#249082332
#249082072
#249081922
#249080502
#249080432
#249077262
#249077122
#249073402
#249072902
#249072882
#102×Dic832
C2×#476×C832
#2297583