Finite Group Databasebeta

← #35335760

Subgroups — #35335760

Order 21248.

1770 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581787C211yesmaximal, characteristic
2C2×#19127568C211yesmaximal, characteristic, complemented
3C2×C42×Dic83C211yesmaximal, characteristic, complemented
4#24905701C211yesmaximal, characteristic
5#24905699C212yesmaximal
6#24905699C211yesmaximal, characteristic
7#24905699C212yesmaximal
8#24905697C212yesmaximal, complemented
9#24905695C211yesmaximal, characteristic, complemented
10#24905694C212yesmaximal, complemented
11C83×#1436C211yesmaximal, characteristic, complemented
12C2×#14581787—24no—
13C2×#14581787—24no—
14C2×#14581787C418yescomplemented
15C22×C4⋊Dic83C2211yescharacteristic
16C22×Dic83⋊C4C2211yescharacteristic
17C22×(C2×D83)⋊C4C2212yes—
18C22×(C2×D83)⋊C4C2212yes—
19C22×(C2×D83)⋊C4C2211yescharacteristic
20C22×(C2×D83)⋊C4C2211yescharacteristic
21C22×(C2×D83)⋊C4C2211yescharacteristic
22C22×C4×Dic83C2212yes—
23C22×C4×Dic83C2211yescharacteristic
24C22×C4×Dic83C2212yes—
25C22×C4×Dic83C2211yescharacteristic
26#19127568—24no—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2211yescharacteristic
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C42×Dic83—24no—
35C22×C4⋊C4×C83C2211yescharacteristic
36C22×C42×C83C2211yescharacteristic
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2212yes—
39C2×#95×C83C2212yes—
40C24×D83C2311yescharacteristic
41C23×Dic83C2312yes—
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2311yescharacteristic
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2312yes—
46#14581787—232no—
47C2×C4⋊Dic83—22no—
48C2×C4⋊Dic83—22no—
49C2×C4⋊Dic83C2×C414yes—
50C2×Dic83⋊C4—22no—
51C2×Dic83⋊C4—22no—
52C2×Dic83⋊C4C2×C414yes—
53C2×(C2×D83)⋊C4—22no—
54C2×(C2×D83)⋊C4—22no—
55C2×(C2×D83)⋊C4—22no—
56C2×(C2×D83)⋊C4—22no—
57C2×(C2×D83)⋊C4—24no—
58C2×(C2×D83)⋊C4—24no—
59C2×(C2×D83)⋊C4—24no—
60C2×(C2×D83)⋊C4—22no—
61C2×(C2×D83)⋊C4—22no—
62C2×(C2×D83)⋊C4—24no—
63C2×(C2×D83)⋊C4—24no—
64C2×(C2×D83)⋊C4—24no—
65C2×(C2×D83)⋊C4C2×C414yes—
66C2×(C2×D83)⋊C4C2×C414yes—
67C2×(C2×D83)⋊C4C2×C414yes—
68C2×C4×Dic83—24no—
69C2×C4×Dic83—24no—
70C2×C4×Dic83—24no—
71C2×C4×Dic83—24no—
72C2×C4×Dic83—24no—
73C2×C4×Dic83—22no—
74C2×C4×Dic83—24no—
75C2×C4×Dic83—24no—
76C2×C4×Dic83C2×C414yes—
77C2×C4×Dic83D414yes—
78C2×C4×Dic83D414yes—
79C83⋊#95—24no—
80C83⋊#95—24no—
81C83⋊#95—24no—
82C83⋊#95—24no—
83C83⋊#95—24no—
84C83⋊#95—22no—
85C83⋊#95—24no—
86C23×C4×C83C2312yes—
87C23×C4×C83C2311yescharacteristic
88C23×C4×C83C2311yescharacteristic
89C23×C4×C83C2311yescharacteristic
90C23×C4×C83C2312yes—
91C2×C4⋊C4×C83—22no—
92C2×C4⋊C4×C83—22no—
93C2×C4⋊C4×C83C2×C414yes—
94C2×C42×C83—22no—
95C2×C42×C83—24no—
96#95×C83—22no—
97#95×C83—24no—
98#95×C83—24no—
99C23×D83—21no—
100C23×D83—21no—
101C23×D83—22no—
102C23×D83—22no—
103C23×D83—22no—
104C23×D83—22no—
105C23×D83—22no—
106C23×D83—22no—
107C23×D83C22×C412yes—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—21no—
117C22×Dic83—21no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—22no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83C22×C412yes—
128C22×Dic83C22×C412yes—
129C22×Dic83C22×C412yes—
130C22×Dic83C2×D412yes—
131C22×Dic83C2×D412yes—
132C22×Dic83C2×D412yes—
133C22×Dic83C2×D412yes—
134C22×Dic83#4114yes—
135C22×Dic83#4114yes—
136C22×Dic83#4114yes—
137C4⋊Dic83—28no—
138Dic83⋊C4—28no—
139(C2×D83)⋊C4—28no—
140(C2×D83)⋊C4—28no—
141(C2×D83)⋊C4—28no—
142(C2×D83)⋊C4—48no—
143(C2×D83)⋊C4—48no—
144C4×Dic83—216no—
145C4×Dic83—216no—
146C4×Dic83—28no—
147C4×Dic83C22⋊C4116yescomplemented
148C24×C83C2411yescharacteristic
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—21no—
158C22×C4×C83—21no—
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—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83C22×C412yes—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C22×C412yes—
173C22×C4×C83C2×D412yes—
174C22×C4×C83C2×D412yes—
175C22×C4×C83#4114yes—
176C22×C4×C83#4114yes—
177C22×C4×C83#4114yes—
178C4⋊C4×C83—28no—
179C42×C83—28no—
180C23×C83C23×C411yescharacteristic
181C23×C83C22×D411yescharacteristic
182C23×C83C22×D411yescharacteristic
183C23×C83C2×#4112yes—
184C23×C83C2×#4112yes—
185C23×C83C2×#4112yes—
186C23×C83C2×#4112yes—
187C23×C83C2×#4112yes—
188C23×C83C2×#4112yes—
189C22×D83—22no—
190C22×D83—24no—
191C22×D83—22no—
192C22×D83—22no—
193C22×D83—24no—
194C22×D83—41no—
195C22×D83—41no—
196C22×D83—42no—
197C22×D83—42no—
198C22×D83—42no—
199C22×D83—42no—
200C22×D83—42no—
201C22×D83—42no—
202C22×D83—42no—
203C22×D83—42no—
204C22×D83—42no—
205C22×D83—42no—
206C22×D83—42no—
207C22×D83—42no—
208C22×D83—41no—
209C22×D83—41no—
210C2×C4×C83—24no—
211C2×C4×C83—22no—
212C2×C4×C83—24no—
213C2×C4×C83—22no—
214C2×C4×C83—24no—
215C2×C4×C83—24no—
216C2×C4×C83—24no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—24no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
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×C83—42no—
234C2×C4×C83—42no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83C2×C22⋊C414yes—
238C2×C4×C83#12618yescomplemented
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—24no—
245C2×Dic83—24no—
246C2×Dic83—24no—
247C2×Dic83—24no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—22no—
252C2×Dic83—22no—
253C2×Dic83—22no—
254C2×Dic83—24no—
255C2×Dic83—42no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260C2×Dic83—42no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83C2×C22⋊C414yes—
266C2×Dic83C2×C22⋊C414yes—
267C2×Dic83#12618yescomplemented
268C22×C83C22×C22⋊C411yescharacteristic
269C22×C83C4×#4112yes—
270C22×C83C4×#4111yescharacteristic
271C22×C83C4×#4111yescharacteristic
272C22×C83C4×#4112yes—
273C22×C83#52511yescharacteristic
274C22×C83#52511yescharacteristic
275C22×C83#52512yes—
276C22×C83#52512yes—
277C22×C83C2×#12612yes—
278C22×C83C2×#12612yes—
279C22×C83#52912yes—
280C22×C83#52912yes—
281C22×C83#52912yes—
282C22×C83#52912yes—
283C22×C83#52912yes—
284C22×C83#52912yes—
285C22×C83#53212yes—
286C22×C83#53212yes—
287C22×C83#53211yescharacteristic
288C22×C83#53211yescharacteristic
289C2×D83—44no—
290C2×D83—44no—
291C2×D83—42no—
292C2×D83—42no—
293C2×D83—42no—
294C2×D83—82no—
295C2×D83—82no—
296C2×D83—82no—
297C2×D83—82no—
298C4×C83—28no—
299C4×C83—24no—
300C4×C83—42no—
301C4×C83—42no—
302C4×C83—82no—
303Dic83—28no—
304Dic83—24no—
305Dic83—24no—
306Dic83—42no—
307Dic83—82no—
308#14711—831nomaximal, Sylow
309C2×C83#229711yescharacteristic
310C2×C83#229711yescharacteristic
311C2×C83#229711yescharacteristic
312C2×C83C4×#12612yes—
313C2×C83C4×#12612yes—
314C2×C83#285212yes—
315C2×C83#285212yes—
316C2×C83#286012yes—
317C2×C83#286012yes—
318D83—82no—
319C22×#126—831no—
320C2×#383—831no—
321C2×C43—831no—
322#1437—831no—
323#1437—831no—
324#1437—832no—
325#1437—832no—
326#1436—832no—
327#1436—832no—
328#1436—831no—
329#1436D8311yescharacteristic, complemented
330C83#1471111yescharacteristic, complemented, Sylow
331C2×#126—838no—
332C2×#126—1664no—
333C2×#126—1664no—
334C22×C4⋊C4—831no—
335C22×C4⋊C4—831no—
336C22×C4⋊C4C2×D8311yescharacteristic
337C22×C22⋊C4—831no—
338C22×C22⋊C4—831no—
339C22×C22⋊C4—831no—
340C22×C22⋊C4—832no—
341C22×C22⋊C4—832no—
342C22×C42—831no—
343C22×C42—832no—
344C22×C42—832no—
345C22×C42—831no—
346C22×C42C2×D8311yescharacteristic
347#383—1664no—
348C2×#95—832no—
349C2×#95—832no—
350C2×#95—832no—
351C2×#95—832no—
352C2×#95—832no—
353C2×#95—832no—
354C2×#95—831no—
355C2×#95C2×D8312yes—
356C2×#95C2×D8312yes—
357C2×#95C2×D8311yescharacteristic
358C43—1664no—
359C25—831no—
360C23×C4—831no—
361C23×C4—831no—
362C23×C4—832no—
363C23×C4—831no—
364C23×C4—832no—
365C23×C4C22×D8312yes—
366C23×C4C22×D8311yescharacteristic
367C23×C4C22×D8311yescharacteristic
368C23×C4C22×D8311yescharacteristic
369C23×C4C22×D8312yes—
370#126—16632no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—834no—
374C2×C4⋊C4—834no—
375C2×C4⋊C4—1662no—
376C2×C4⋊C4—1662no—
377C2×C4⋊C4—1662no—
378C2×C4⋊C4—1662no—
379C2×C4⋊C4C4×D8314yes—
380C2×C22⋊C4—834no—
381C2×C22⋊C4—834no—
382C2×C22⋊C4—834no—
383C2×C22⋊C4—1664no—
384C2×C22⋊C4—1664no—
385C2×C22⋊C4—1664no—
386C2×C22⋊C4—1664no—
387C2×C22⋊C4—1664no—
388C2×C22⋊C4—1664no—
389C2×C22⋊C4—1662no—
390C2×C22⋊C4—1662no—
391C2×C22⋊C4—1662no—
392C2×C22⋊C4—1662no—
393C2×C22⋊C4—1662no—
394C2×C22⋊C4—1662no—
395C2×C42—22no—
396C2×C42—24no—
397C2×C42—834no—
398C2×C42—834no—
399C2×C42—834no—
400C2×C42—1664no—
401C2×C42—1664no—
402C2×C42—1664no—
403C2×C42—1664no—
404C2×C42—1664no—
405C2×C42—1662no—
406C2×C42—1664no—
407C2×C42—1664no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—1664no—
412#95—1664no—
413#95—1664no—
414#95—1664no—
415#95—1664no—
416#95—1662no—
417#95—1664no—
418C24—832no—
419C24—1661no—
420C24—1661no—
421C24—1662no—
422C24—1662no—
423C24—1662no—
424C24—1662no—
425C24—1662no—
426C24—1662no—
427C24C23×D8311yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—832no—
450C22×C4—832no—
451C22×C4—832no—
452C22×C4—832no—
453C22×C4—832no—
454C22×C4—832no—
455C22×C4—834no—
456C22×C4—834no—
457C22×C4—834no—
458C22×C4—832no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1662no—
466C22×C4—1661no—
467C22×C4—1661no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1662no—
473C22×C4—1662no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4C2×C4×D8312yes—
479C22×C4C2×C4×D8312yes—
480C22×C4C2×C4×D8312yes—
481C22×C4D4×D8312yes—
482C22×C4D4×D8312yes—
483C22×C4#1305058914yes—
484C22×C4#1305059014yes—
485C22×C4#1305059114yes—
486C4⋊C4—28no—
487C4⋊C4—1668no—
488C4⋊C4—1668no—
489C22⋊C4—3328no—
490C22⋊C4—3328no—
491C22⋊C4—1668no—
492C22⋊C4—1668no—
493C22⋊C4—1668no—
494C42—28no—
495C42—8316no—
496C42—1668no—
497C42—16616no—
498C42—16616no—
499C23—3321no—
500C23—3321no—
501C23—3322no—
502C23—3322no—
503C23—3322no—
504C23—3322no—
505C23—3322no—
506C23—3322no—
507C23—3322no—
508C23—3322no—
509C23—3322no—
510C23—3322no—
511C23—3322no—
512C23—3322no—
513C23—3321no—
514C23—3321no—
515C23—1662no—
516C23—1664no—
517C23—1662no—
518C23—1662no—
519C23—1664no—
520C23C22×C4×D8311yescharacteristic
521C23C2×D4×D8311yescharacteristic
522C23C2×D4×D8311yescharacteristic
523C23#41×D8312yes—
524C23#41×D8312yes—
525C23#41×D8312yes—
526C23C2×#1305058912yes—
527C23C2×#1305059012yes—
528C23C2×#1305059112yes—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
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—3322no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3322no—
560C2×C4—3322no—
561C2×C4—3322no—
562C2×C4—3322no—
563C2×C4—3322no—
564C2×C4—3322no—
565C2×C4—3322no—
566C2×C4—834no—
567C2×C4—834no—
568C2×C4—838no—
569C2×C4—1662no—
570C2×C4—1664no—
571C2×C4—1662no—
572C2×C4—1662no—
573C2×C4—1664no—
574C2×C4—1664no—
575C2×C4—1664no—
576C2×C4—1664no—
577C2×C4—1664no—
578C2×C4—1664no—
579C2×C4—1664no—
580C2×C4—1664no—
581C2×C4—1664no—
582C2×C4—1664no—
583C2×C4—1662no—
584C2×C4—1662no—
585C2×C4C22⋊C4×D8314yes—
586C2×C4#1458178718yescomplemented
587C22—3322no—
588C22—3322no—
589C22—3322no—
590C22—3324no—
591C22—3324no—
592C22—6642no—
593C22—6642no—
594C22—6642no—
595C22—6642no—
596C22C2×C22⋊C4×D8311yescharacteristic
597C22C4×#1305058912yes—
598C22#1912816911yescharacteristic
599C22C4×#1305059011yescharacteristic
600C22C4×#1305059112yes—
601C22#1912820711yescharacteristic
602C22#1912820911yescharacteristic
603C22#1912821012yes—
604C22#1912821412yes—
605C22#126×D8312yes—
606C22C2×#1458178712yes—
607C22#1912822812yes—
608C22#1912822912yes—
609C22#1912823112yes—
610C22#1912823612yes—
611C22#1912823712yes—
612C22#1912823812yes—
613C22#1912824612yes—
614C22#1912824812yes—
615C22#1912825011yescharacteristic
616C22#1912825211yescharacteristic
617C4—28no—
618C4—24no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—3322no—
623C4—6642no—
624C4—1668no—
625C4—1664no—
626C4—1664no—
627C2—6642no—
628C2#2490994811yescharacteristic
629C2#2490994911yescharacteristic
630C2#2490995211yescharacteristic
631C2#2491001312yes—
632C2C4×#1458178712yes—
633C2#2491578012yes—
634C2#2491578112yes—
635C2#2491582912yes—
636C2#2491583112yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1436C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenter#1436D83
hypocenterC83#14711
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
2. C2×#19127568C2semidirect
3. C2×C42×Dic83C2semidirect
8. #24905697C2semidirect
9. #24905695C2semidirect
10. #24905694C2semidirect
11. C83×#1436C2semidirect
14. C2×#14581787C4semidirect
147. C4×Dic83C22⋊C4semidirect
238. C2×C4×C83#126semidirect
267. C2×Dic83#126semidirect
329. #1436D83semidirect
330. C83#14711semidirect
586. C2×C4#14581787semidirect

Maximal subgroups

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

subgroupindex
C22×#145817872
C2×#191275682
C2×C42×Dic832
#249057012
#249056992
#249056972
#249056952
#249056942
C83×#14362
#1471183