Finite Group Databasebeta

← #35326704

Subgroups — #35326704

Order 21248.

1586 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127568C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic83C211yesmaximal, characteristic
3#24905718C212yesmaximal, complemented
4#24905706C212yesmaximal
5#24905700C212yesmaximal
6Dic83⋊#95C211yesmaximal, characteristic, complemented
7#24905689C212yesmaximal, complemented
8#24905687C211yesmaximal, characteristic, complemented
9#24905686C211yesmaximal, characteristic
10#95×Dic83C211yesmaximal, characteristic, complemented
11C2×#381×C83C211yesmaximal, characteristic
12C22×Dic83⋊C4C2212yes—
13C22×C22⋊Dic83C2212yes—
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C22⋊Dic83C2211yescharacteristic
16C22×C4×Dic83C2212yes—
17C22×C4×Dic83C2211yescharacteristic
18C22×C4×Dic83C2211yescharacteristic
19#19127568—24no—
20C4×C22⋊Dic83C418yescomplemented
21C2×C83⋊#95C2211yescharacteristic
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2211yescharacteristic
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2211yescharacteristic
31C2×C83⋊#95C2211yescharacteristic
32C24×C4×C83C2211yescharacteristic
33C22×C22⋊C4×C83C2212yes—
34#381×C83—24no—
35C2×#95×C83C2211yescharacteristic
36C2×#95×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C23×Dic83C2312yes—
39C23×Dic83C2311yescharacteristic
40C23×Dic83C2311yescharacteristic
41C23×Dic83C2312yes—
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2311yescharacteristic
44C2×Dic83⋊C4—24no—
45C2×Dic83⋊C4—42no—
46C2×Dic83⋊C4—42no—
47C2×C22⋊Dic83—22no—
48C2×C22⋊Dic83—24no—
49C2×C22⋊Dic83—22no—
50C2×C22⋊Dic83—24no—
51C2×C22⋊Dic83—24no—
52C2×C22⋊Dic83—42no—
53C2×C22⋊Dic83C2×C414yes—
54C2×C22⋊Dic83C2×C414yes—
55C2×C4×Dic83—24no—
56C2×C4×Dic83—24no—
57C2×C4×Dic83—22no—
58C2×C4×Dic83—22no—
59C2×C4×Dic83—42no—
60C2×C4×Dic83—42no—
61C2×C4×Dic83C2×C414yes—
62C2×C4×Dic83C2×C414yes—
63C83⋊#95—24no—
64C83⋊#95—24no—
65C83⋊#95—24no—
66C83⋊#95—24no—
67C83⋊#95—24no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—42no—
71C83⋊#95—42no—
72C83⋊#95C2×C414yes—
73C83⋊#95C2×C414yes—
74C25×C83C2311yescharacteristic
75C23×C4×C83—24no—
76C23×C4×C83—22no—
77C23×C4×C83—22no—
78C23×C4×C83—42no—
79C23×C4×C83C2×C414yes—
80C23×C4×C83C2312yes—
81C23×C4×C83C2311yescharacteristic
82C23×C4×C83C2312yes—
83C23×C4×C83C2311yescharacteristic
84C2×C22⋊C4×C83—24no—
85C2×C22⋊C4×C83—24no—
86C2×C22⋊C4×C83—42no—
87#95×C83—24no—
88#95×C83—22no—
89#95×C83D414yes—
90C22×Dic83—22no—
91C22×Dic83—22no—
92C22×Dic83—22no—
93C22×Dic83—21no—
94C22×Dic83—21no—
95C22×Dic83—21no—
96C22×Dic83—21no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—21no—
102C22×Dic83—21no—
103C22×Dic83—22no—
104C22×Dic83—21no—
105C22×Dic83—21no—
106C22×Dic83—22no—
107C22×Dic83—21no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83C22×C412yes—
119C22×Dic83C22×C412yes—
120C22×Dic83C22×C412yes—
121C22×Dic83C22×C412yes—
122C22×Dic83#4114yes—
123C22×Dic83#4112yes—
124C22×Dic83#4114yes—
125C22×Dic83#4112yes—
126Dic83⋊C4—48no—
127C22⋊Dic83—28no—
128C22⋊Dic83—28no—
129C22⋊Dic83—48no—
130C4×Dic83—28no—
131C4×Dic83—44no—
132C4×Dic83—48no—
133C24×C83—21no—
134C24×C83—21no—
135C24×C83—22no—
136C24×C83—21no—
137C24×C83—21no—
138C24×C83—22no—
139C24×C83—22no—
140C24×C83—21no—
141C24×C83—21no—
142C24×C83C22×C412yes—
143C24×C83#4114yes—
144C24×C83C2411yescharacteristic
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—28no—
148C22×C4×C83—21no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—21no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—28no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—44no—
167C22×C4×C83—48no—
168C22×C4×C83—48no—
169C22×C4×C83C22×C412yes—
170C22×C4×C83C22×C412yes—
171C22×C4×C83C2×D414yes—
172C22×C4×C83C2×D412yes—
173C22×C4×C83#4112yes—
174C22⋊C4×C83—48no—
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—22no—
178C23×C83—22no—
179C23×C83—22no—
180C23×C83—22no—
181C23×C83—24no—
182C23×C83—24no—
183C23×C83—24no—
184C23×C83—24no—
185C23×C83—24no—
186C23×C83—24no—
187C23×C83—42no—
188C23×C83—42no—
189C23×C83—42no—
190C23×C83—42no—
191C23×C83—41no—
192C23×C83—41no—
193C23×C83—42no—
194C23×C83—42no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83#11714yes—
198C23×C83C23×C411yescharacteristic
199C23×C83C22×D411yescharacteristic
200C23×C83C2×#4112yes—
201C23×C83C2×#4111yescharacteristic
202C23×C83C2×#4111yescharacteristic
203C23×C83C2×#4112yes—
204C23×C83C2×#4112yes—
205C23×C83C2×#4111yescharacteristic
206C23×C83H(5,𝔽2)12yes—
207C23×C83H(5,𝔽2)11yescharacteristic
208C23×C832−1+411yescharacteristic
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—24no—
212C2×C4×C83—24no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—24no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—416no—
220C2×C4×C83—42no—
221C2×C4×C83—42no—
222C2×C4×C83—42no—
223C2×C4×C83—42no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83—42no—
228C2×C4×C83—42no—
229C2×C4×C83—42no—
230C2×C4×C83—42no—
231C2×C4×C83—42no—
232C2×C4×C83—42no—
233C2×C4×C83C4×D414yes—
234C2×Dic83—22no—
235C2×Dic83—22no—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—24no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83—42no—
254C2×Dic83—42no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—42no—
258C2×Dic83—42no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83#11714yes—
269C2×Dic83#11714yes—
270C22×C83—24no—
271C22×C83—24no—
272C22×C83—24no—
273C22×C83—44no—
274C22×C83—44no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83—44no—
280C22×C83—44no—
281C22×C83C2×#11711yescharacteristic
282C22×C83C2×#11711yescharacteristic
283C22×C83C2×#11711yescharacteristic
284C22×C83C2×C4×D411yescharacteristic
285C22×C83C4×#4111yescharacteristic
286C22×C83#51811yescharacteristic
287C22×C83#52011yescharacteristic
288C22×C83#52912yes—
289C22×C83#52912yes—
290C22×C83#52912yes—
291C22×C83#53212yes—
292C22×C83#53812yes—
293C22×C83#53912yes—
294C22×C83#54612yes—
295C22×C83#54711yescharacteristic
296C22×C83#55011yescharacteristic
297C22×C83#55112yes—
298C22×C83#55112yes—
299C22×C83#55112yes—
300C22×C83#55212yes—
301C22×C83#55312yes—
302C22×C83#55511yescharacteristic
303C22×C83#55611yescharacteristic
304C4×C83—24no—
305C4×C83—42no—
306C4×C83—44no—
307C4×C83—82no—
308Dic83—24no—
309Dic83—24no—
310Dic83—44no—
311Dic83—42no—
312Dic83—42no—
313Dic83—82no—
314#13913—831nomaximal, Sylow
315C2×C83—44no—
316C2×C83#228311yescharacteristic
317C2×C83#229311yescharacteristic
318C2×C83#231811yescharacteristic
319C2×C83#234211yescharacteristic
320C2×C83#234311yescharacteristic
321C2×C83#235311yescharacteristic
322C2×C83#235411yescharacteristic
323C2×C83#255312yes—
324C2×C83#255412yes—
325C2×C83#257212yes—
326C2×C83#257312yes—
327C2×#383—831no—
328C2×#381D8311yescharacteristic
329C2×C4×C22⋊C4—831no—
330#1442—832no—
331#1439—832no—
332#1437—832no—
333C4⋊#95—831no—
334#1434—831no—
335#1434—832no—
336C22⋊#95—831no—
337C4×#95—831no—
338C83#1391311yescharacteristic, complemented, Sylow
339C24×C4C2×D8311yescharacteristic
340C22×C4⋊C4—832no—
341C22×C22⋊C4—832no—
342C22×C22⋊C4—831no—
343C22×C22⋊C4—831no—
344C22×C22⋊C4C2×D8312yes—
345C22×C42—831no—
346C22×C42—832no—
347C22×C42—831no—
348#383—1664no—
349#381—24no—
350C4×C22⋊C4—838no—
351C2×#95—832no—
352C2×#95—832no—
353C2×#95—831no—
354C2×#95—831no—
355C2×#95—831no—
356C2×#95—832no—
357C2×#95—831no—
358C2×#95—832no—
359C2×#95—832no—
360C2×#95—832no—
361C2×#95—832no—
362C2×#95C2×D8311yescharacteristic
363C2×#95C2×D8312yes—
364C2×#95C2×D8311yescharacteristic
365C25C22×D8311yescharacteristic
366C23×C4—24no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—831no—
371C23×C4—831no—
372C23×C4—831no—
373C23×C4—831no—
374C23×C4—832no—
375C23×C4—832no—
376C23×C4C4×D8314yes—
377C23×C4C22×D8312yes—
378C23×C4C22×D8311yescharacteristic
379C23×C4C22×D8312yes—
380C23×C4C22×D8311yescharacteristic
381C2×C4⋊C4—3322no—
382C2×C4⋊C4—3322no—
383C2×C4⋊C4—1664no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—3322no—
388C2×C22⋊C4—834no—
389C2×C22⋊C4—834no—
390C2×C22⋊C4—1664no—
391C2×C22⋊C4—1664no—
392C2×C22⋊C4—1662no—
393C2×C22⋊C4—1664no—
394C2×C22⋊C4—1662no—
395C2×C42—3322no—
396C2×C42—3322no—
397C2×C42—834no—
398C2×C42—834no—
399C2×C42—1664no—
400C2×C42—1664no—
401C2×C42—1662no—
402C2×C42—1662no—
403#95—24no—
404#95—22no—
405#95—3322no—
406#95—3322no—
407#95—834no—
408#95—834no—
409#95—1664no—
410#95—1664no—
411#95—1664no—
412#95—1662no—
413#95—1664no—
414#95—1664no—
415#95—1662no—
416#95C22⋊D8314yes—
417C24—21no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—21no—
423C24—22no—
424C24—21no—
425C24—21no—
426C24C2×C4×D8312yes—
427C24#1305058914yes—
428C24C23×D8311yescharacteristic
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—832no—
454C22×C4—832no—
455C22×C4—832no—
456C22×C4—834no—
457C22×C4—832no—
458C22×C4—832no—
459C22×C4—832no—
460C22×C4—834no—
461C22×C4—1662no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1662no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1661no—
479C22×C4—1661no—
480C22×C4—1662no—
481C22×C4—1661no—
482C22×C4—1662no—
483C22×C4—1661no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1661no—
488C22×C4—1661no—
489C22×C4C2×C4×D8312yes—
490C22×C4C2×C4×D8312yes—
491C22×C4C2×C22⋊D8314yes—
492C22×C4C2×C22⋊D8312yes—
493C22×C4#1305059012yes—
494C4⋊C4—3328no—
495C22⋊C4—48no—
496C22⋊C4—3328no—
497C22⋊C4—1668no—
498C22⋊C4—1668no—
499C42—3324no—
500C42—3328no—
501C42—1668no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#1458174814yes—
525C23C22×C4×D8311yescharacteristic
526C23C22×C22⋊D8311yescharacteristic
527C23#41×D8311yescharacteristic
528C23#41×D8311yescharacteristic
529C23#41×D8312yes—
530C23#41×D8312yes—
531C23C2×#1305058912yes—
532C23C2×#1305059011yescharacteristic
533C23#1458183712yes—
534C23#1458183811yescharacteristic
535C23#1458183911yescharacteristic
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—416no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3322no—
563C2×C4—3322no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3322no—
567C2×C4—3322no—
568C2×C4—3322no—
569C2×C4—3322no—
570C2×C4—3322no—
571C2×C4—3322no—
572C2×C4—3322no—
573C2×C4—3322no—
574C2×C4—3322no—
575C2×C4—3322no—
576C2×C4—3322no—
577C2×C4—834no—
578C2×C4—834no—
579C2×C4—1664no—
580C2×C4—1664no—
581C2×C4—1662no—
582C2×C4—1662no—
583C2×C4—1662no—
584C2×C4—1662no—
585C2×C4—1662no—
586C2×C4—1662no—
587C2×C4—1664no—
588C2×C4—1664no—
589C2×C4—1664no—
590C2×C4—1662no—
591C2×C4—1662no—
592C2×C4—1662no—
593C2×C4—1662no—
594C2×C4—1664no—
595C2×C4—1664no—
596C2×C4C4×C22⋊D8314yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22#117×D8311yescharacteristic
609C22#117×D8311yescharacteristic
610C22C2×#1458174811yescharacteristic
611C22C2×C4×C22⋊D8311yescharacteristic
612C22#1912816911yescharacteristic
613C22#1912817411yescharacteristic
614C22#1912818111yescharacteristic
615C22#1912822812yes—
616C22#1912822812yes—
617C22#1912822912yes—
618C22#1912824712yes—
619C22#1912828712yes—
620C22#1912829412yes—
621C22#1912834512yes—
622C22#1912835011yescharacteristic
623C22#1912837011yescharacteristic
624C22#1912837412yes—
625C22#1912837712yes—
626C22#1912837912yes—
627C22#1912838812yes—
628C22#1912839612yes—
629C22#1912841411yescharacteristic
630C22#1912841911yescharacteristic
631C4—24no—
632C4—44no—
633C4—42no—
634C4—82no—
635C4—3324no—
636C4—3322no—
637C4—3322no—
638C4—6642no—
639C4—1664no—
640C4—1664no—
641C2—44no—
642C2#2490985411yescharacteristic
643C2#2490992211yescharacteristic
644C2#2491009211yescharacteristic
645C2#2491030811yescharacteristic
646C2#2491032511yescharacteristic
647C2#2491040011yescharacteristic
648C2#2491041311yescharacteristic
649C2#2491250412yes—
650C2#2491251912yes—
651C2#2491270912yes—
652C2#2491271812yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#381×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#381D83
hypocenterC83#13913
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
1. C2×#19127568C2semidirect
3. #24905718C2semidirect
6. Dic83⋊#95C2semidirect
7. #24905689C2semidirect
8. #24905687C2semidirect
10. #95×Dic83C2semidirect
20. C4×C22⋊Dic83C4semidirect
338. C83#13913semidirect

Maximal subgroups

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

subgroupindex
C2×#191275682
C2×C4×C22⋊Dic832
#249057182
#249057062
#249057002
Dic83⋊#952
#249056892
#249056872
#249056862
#95×Dic832
C2×#381×C832
#1391383