Finite Group Databasebeta

← #35423836

Subgroups — #35423836

Order 21248.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127799C211yesmaximal, characteristic
2#24910410C211yesmaximal, characteristic
3#24908159C211yesmaximal, characteristic
4#24908145C211yesmaximal, characteristic
5#24908052C211yesmaximal, characteristic, complemented
6#24908031C211yesmaximal, characteristic, complemented
7#24907818C211yesmaximal, characteristic
8#24907818C211yesmaximal, characteristic
9#24907596C211yesmaximal, characteristic, complemented
10#24907470C211yesmaximal, characteristic
11#24907468C211yesmaximal, characteristic, complemented
12#24907338C211yesmaximal, characteristic, complemented
13#24907293C211yesmaximal, characteristic, complemented
14C4×#14581716C211yesmaximal, characteristic, complemented
15C83×#2351C211yesmaximal, characteristic, complemented
16C22×#13050577C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18C2×#14581750—21no—
19#19127799C418yescomplemented
20C2×#14581725C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×C4×C83⋊C8C2211yescharacteristic
31#19127603C2211yescharacteristic
32#19127603C2211yescharacteristic
33#19127593—21no—
34#19127593—21no—
35#19127571C2211yescharacteristic
36#19127558—21no—
37C4×C4⋊Dic83C2211yescharacteristic
38C4×C4⋊Dic83C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4×Q8×C83C2211yescharacteristic
50C2×C4×Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52#389×C83—21no—
53#389×C83—21no—
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#382×C83—21no—
57#382×C83C2211yescharacteristic
58C4×C4⋊C4×C83—21no—
59C4×C4⋊C4×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—
71#14581750—44no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×(C2×D83)⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84#14581725C2×C414yes—
85C4⋊(C83⋊C8)—22no—
86#14581716—22no—
87#14581716—22no—
88#14581716C2×C414yes—
89#14581712—41no—
90#14581712—41no—
91#14581712C2×C414yes—
92#14581703—41no—
93#14581703C2×C414yes—
94C4×C83⋊C8—41no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C22×Q8×C83C2311yescharacteristic
98C4×Q8×C83—22no—
99C4×Q8×C83—22no—
100C4×Q8×C83—22no—
101C4×Q8×C83—41no—
102C4×Q8×C83—41no—
103C4×Q8×C83C2×C414yes—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83D411yescharacteristic
110C2×C4⋊C4×C83D411yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C42×C83—21no—
115C2×C42×C83—21no—
116C2×C42×C83—21no—
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83D411yescharacteristic
125#95×C83D411yescharacteristic
126C23×D83—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332—21no—
132C2×D332C22×C412yes—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C2×C4×D83—24no—
137C2×C4×D83—42no—
138C2×C4×D83—42no—
139#13050577—28no—
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⋊C8—21no—
150C2×C83⋊C8C22×C412yes—
151C2×C83⋊C8C22×C412yes—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—42no—
158C4⋊Dic83—81no—
159C4⋊Dic83C22×C412yes—
160(C2×D83)⋊C4—24no—
161(C2×D83)⋊C4—42no—
162(C2×D83)⋊C4—42no—
163(C2×D83)⋊C4—81no—
164(C2×D83)⋊C4—81no—
165C4×Dic83—42no—
166C4×Dic83—42no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83C22×C412yes—
171C2×Q8×C83#4112yes—
172C2×Q8×C83#4112yes—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
184C4⋊C4×C83—24no—
185C4⋊C4×C83—22no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—41no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83—42no—
195C4⋊C4×C83—42no—
196C4⋊C4×C83C22×C412yes—
197C42×C83—24no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—42no—
203C42×C83—42no—
204C42×C83C22×C412yes—
205C42×C83#4112yes—
206C23×C83C22×D411yescharacteristic
207C22×D83—22no—
208C22×D83—41no—
209C22×D83—41no—
210C22×D83—41no—
211C22×D83—41no—
212C22×D83—41no—
213C22×D83—41no—
214Q8×C83—22no—
215Q8×C83—22no—
216Q8×C83—22no—
217Q8×C83#11714yescomplemented
218C2×C4×C83—22no—
219C2×C4×C83—21no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—22no—
228C2×C4×C83—21no—
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×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83C4×D412yes—
246C2×C4×C83C4×D412yes—
247C2×C4×C83AGL(1,ℤ/8)12yes—
248C2×C4×C83C23×C411yescharacteristic
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83H(5,𝔽2)11yescharacteristic
253C2×C4×C832−1+411yescharacteristic
254C2×Dic83—22no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263D332—22no—
264D332—22no—
265D332—41no—
266D332—41no—
267C4×D83—48no—
268C83⋊C8—22no—
269C83⋊C8—22no—
270C83⋊C8—41no—
271C83⋊C8—41no—
272C22×C83C2×C4×D411yescharacteristic
273C22×C83#54611yescharacteristic
274C22×C83#54811yescharacteristic
275C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C83#57511yescharacteristic
277C22×C83#57711yescharacteristic
278C22×C83#57811yescharacteristic
279C2×D83—42no—
280C2×D83—42no—
281C2×D83—42no—
282C2×D83—81no—
283C2×D83—81no—
284C2×D83—81no—
285C2×D83—81no—
286C4×C83—24no—
287C4×C83—22no—
288C4×C83—22no—
289C4×C83—22no—
290C4×C83—42no—
291C4×C83—41no—
292C4×C83—41no—
293C4×C83—41no—
294C4×C83—81no—
295C4×C83—81no—
296C4×C83C2×#11711yescharacteristic
297C4×C83#52011yescharacteristic
298C4×C83#55211yescharacteristic
299C4×C83#55311yescharacteristic
300Dic83—42no—
301Dic83—81no—
302Dic83—81no—
303#22781—831nomaximal, Sylow
304C2×C83#235211yescharacteristic
305C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C83#294711yescharacteristic
307C2×C83#330811yescharacteristic
308C2×C83#332111yescharacteristic
309C2×C83#336811yescharacteristic
310C2×C83#337711yescharacteristic
311D83—82no—
312C2×#440—831no—
313#2354—831no—
314#2351D8311yescharacteristic, complemented
315#1936—831no—
316#1934—831no—
317#1921—831no—
318#1918—831no—
319#1880—831no—
320#1880—831no—
321#1836—831no—
322#1804—831no—
323#1803—831no—
324#1771—831no—
325#1763—831no—
326C4×#103—831no—
327C83#2278111yescharacteristic, complemented, Sylow
328C22×SD16—831no—
329C2×C4×Q8C2×D8311yescharacteristic
330C2×C4×Q8C2×D8311yescharacteristic
331C2×C4×Q8C2×D8311yescharacteristic
332C2×C4×D4—831no—
333C2×#117—1661no—
334#440—838no—
335C2×#107—831no—
336C2×C4⋊C8—831no—
337C2×#103—831no—
338C2×#103—831no—
339C2×#103—831no—
340C2×#102—831no—
341C2×#102—831no—
342C2×#102—831no—
343C2×#97—831no—
344C2×#97—831no—
345C2×C4×C8—831no—
346#390—831no—
347#390—831no—
348#389—21no—
349#389—21no—
350#389C2×D8311yescharacteristic
351#389C2×D8311yescharacteristic
352#388—1661no—
353#388—1661no—
354#384—831no—
355#382—21no—
356#382C2×D8311yescharacteristic
357#380—1661no—
358C4×C4⋊C4—21no—
359C4×C4⋊C4—831no—
360C4×C4⋊C4—831no—
361C4×C4⋊C4C2×D8311yescharacteristic
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#336—831no—
371#336—831no—
372C22×Q8C22×D8311yescharacteristic
373C22×D4—831no—
374C23×C4—1661no—
375C2×SD16—834no—
376C2×SD16—1662no—
377C2×SD16—1662no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—831no—
381C22×C8—831no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—41no—
386C4×Q8—41no—
387C4×Q8C4×D8314yes—
388C4×D4—834no—
389C4×D4—1662no—
390#117—3324no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4C22⋊D8311yescharacteristic
402C2×C4⋊C4C22⋊D8311yescharacteristic
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C4⋊C4C22×D8311yescharacteristic
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C22⋊C4—1661no—
407C2×C22⋊C4—1661no—
408C2×C22⋊C4—1661no—
409C2×C22⋊C4—1661no—
410C2×C22⋊C4—1661no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—1661no—
415C2×C42—1661no—
416C2×C42C22×D8311yescharacteristic
417C2×C42C22×D8311yescharacteristic
418C2×C42C22×D8311yescharacteristic
419#107—834no—
420C4⋊C8—1662no—
421#103—834no—
422#103—1662no—
423#103—1662no—
424#102—3321no—
425#102—3321no—
426#102—834no—
427#97—3321no—
428#97—834no—
429C4×C8—3321no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1661no—
435#95—1661no—
436#95C22⋊D8311yescharacteristic
437#95C22⋊D8311yescharacteristic
438C24—1661no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D8312yes—
443C2×Q8#1305058912yes—
444C2×Q8#1305059112yes—
445C2×D4—832no—
446C2×D4—1661no—
447C2×D4—1661no—
448C2×D4—1661no—
449C2×D4—1661no—
450C2×D4—1661no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—3322no—
456C22×C4—3322no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1664no—
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4C23×D8311yescharacteristic
468SD16—1668no—
469C2×C8—832no—
470C2×C8—832no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—3322no—
494C4⋊C4—832no—
495C4⋊C4—6641no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4C2×C4×D8312yes—
502C22⋊C4—3322no—
503C22⋊C4—3322no—
504C22⋊C4—6641no—
505C22⋊C4—6641no—
506C22⋊C4—1664no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—24no—
512C42—42no—
513C42—42no—
514C42—3322no—
515C42—3322no—
516C42C2×C4×D8312yes—
517C42#1305059012yes—
518C23—3321no—
519C23—3321no—
520C23—3321no—
521C23—3321no—
522C23—3321no—
523C23—3321no—
524C23—1662no—
525C23C22×C22⋊D8311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8#1458175014yescomplemented
530D4—3321no—
531D4—3321no—
532D4—1662no—
533D4—1662no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—3328no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—1662no—
571C2×C4C4×C22⋊D8312yes—
572C2×C4C4×C22⋊D8312yes—
573C2×C4#1458181712yes—
574C2×C4C22×C4×D8311yescharacteristic
575C2×C4C2×#1305058911yescharacteristic
576C2×C4C2×#1305059011yescharacteristic
577C2×C4C2×#1305059111yescharacteristic
578C2×C4#1458183711yescharacteristic
579C2×C4#1458184011yescharacteristic
580C8—3321no—
581C8—3321no—
582C8—1662no—
583C8—1662no—
584C22—3322no—
585C22—3322no—
586C22—3322no—
587C22—6641no—
588C22—6641no—
589C22—6641no—
590C22—6641no—
591C22C2×C4×C22⋊D8311yescharacteristic
592C22#1912834511yescharacteristic
593C22#1912836211yescharacteristic
594C22C2×#1458181711yescharacteristic
595C22#1912849911yescharacteristic
596C22#1912850911yescharacteristic
597C22#1912851611yescharacteristic
598C4—22no—
599C4—24no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—3322no—
609C4—6641no—
610C4—6641no—
611C4C2×#1458175011yescharacteristic
612C4#1912818211yescharacteristic
613C4#1912838611yescharacteristic
614C4#1912839911yescharacteristic
615C2—6642no—
616C2#2491038811yescharacteristic
617C2C4×#1458181711yescharacteristic
618C2#2491630611yescharacteristic
619C2#2491945411yescharacteristic
620C2#2491964911yescharacteristic
621C2#2492019211yescharacteristic
622C2#2492030511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2351C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#2351D83
hypocenterC83#22781
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
5. #24908052C2semidirect
6. #24908031C2semidirect
9. #24907596C2semidirect
11. #24907468C2semidirect
12. #24907338C2semidirect
13. #24907293C2semidirect
14. C4×#14581716C2semidirect
15. C83×#2351C2semidirect
19. #19127799C4semidirect
217. Q8×C83#117semidirect
314. #2351D83semidirect
327. C83#22781semidirect
529. Q8#14581750semidirect

Maximal subgroups

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

subgroupindex
C2×#191277992
#249104102
#249081592
#249081452
#249080522
#249080312
#249078182
#249075962
#249074702
#249074682
#249073382
#249072932
C4×#145817162
C83×#23512
#2278183