Finite Group Databasebeta

← #35335534

Subgroups — #35335534

Order 21248.

1486 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127563C211yesmaximal, characteristic
2C2×#19127560C211yesmaximal, characteristic
3C2×#19127558C211yesmaximal, characteristic
4#24905726C211yesmaximal, characteristic, complemented
5#24905726C212yesmaximal, complemented
6#24905720C211yesmaximal, characteristic, complemented
7#24905719C211yesmaximal, characteristic, complemented
8#24905718C212yesmaximal, complemented
9#24905707C212yesmaximal
10#24905705C212yesmaximal
11C83×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4C2211yescharacteristic
15C22×Dic83⋊C4C2211yescharacteristic
16C22×Dic83⋊C4C2212yes—
17C22×(C2×D83)⋊C4C2212yes—
18C22×(C2×D83)⋊C4C2211yescharacteristic
19C22×(C2×D83)⋊C4C2211yescharacteristic
20C22×(C2×D83)⋊C4C2212yes—
21#19127563C418yescomplemented
22#19127560—24no—
23#19127558—24no—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2211yescharacteristic
33C22×C4⋊C4×C83C2211yescharacteristic
34C22×C4⋊C4×C83C2212yes—
35C2×#95×C83C2211yescharacteristic
36C2×#95×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C24×D83C2311yescharacteristic
39C23×Dic83C2311yescharacteristic
40C23×Dic83C2311yescharacteristic
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C23×Dic83C2311yescharacteristic
44C22×C4×D83—42no—
45C22×C4×D83—42no—
46C22×C4×D83C2×C414yes—
47C22×C4×D83D414yes—
48C22×C4×D83Q814yes—
49C2×C4⋊Dic83—22no—
50C2×C4⋊Dic83—42no—
51C2×Dic83⋊C4—22no—
52C2×Dic83⋊C4—22no—
53C2×Dic83⋊C4—42no—
54C2×Dic83⋊C4—42no—
55C2×Dic83⋊C4—42no—
56C2×Dic83⋊C4—42no—
57C2×Dic83⋊C4—42no—
58C2×(C2×D83)⋊C4—24no—
59C2×(C2×D83)⋊C4—22no—
60C2×(C2×D83)⋊C4—22no—
61C2×(C2×D83)⋊C4—42no—
62C2×(C2×D83)⋊C4—42no—
63C2×(C2×D83)⋊C4—42no—
64C2×(C2×D83)⋊C4—42no—
65C2×(C2×D83)⋊C4—42no—
66C2×(C2×D83)⋊C4—42no—
67C2×(C2×D83)⋊C4C2×C418yes—
68C83⋊#95—24no—
69C83⋊#95—22no—
70C83⋊#95—42no—
71C83⋊#95—42no—
72C83⋊#95—42no—
73C83⋊#95—42no—
74C83⋊#95—42no—
75C83⋊#95C2×C418yes—
76C83⋊#95C2×C414yes—
77C23×C4×C83C2311yescharacteristic
78C23×C4×C83C2312yes—
79C23×C4×C83C2311yescharacteristic
80C23×C4×C83C2312yes—
81C23×C4×C83C2311yescharacteristic
82C2×C4⋊C4×C83—22no—
83C2×C4⋊C4×C83—42no—
84C2×C4⋊C4×C83—42no—
85C2×C4⋊C4×C83—42no—
86C2×C4⋊C4×C83—42no—
87#95×C83—22no—
88#95×C83—42no—
89#95×C83C2×C414yes—
90C23×D83—22no—
91C23×D83—22no—
92C23×D83—22no—
93C23×D83—21no—
94C23×D83—21no—
95C23×D83—21no—
96C23×D83—21no—
97C23×D83—22no—
98C23×D83C22×C412yes—
99C23×D83C2×D412yes—
100C23×D83C2×Q812yes—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—21no—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83C22×C414yes—
127C22×Dic83C22×C412yes—
128C22×Dic83C2×D412yes—
129C22×Dic83C2×Q812yes—
130C2×C4×D83—48no—
131C2×C4×D83—44no—
132C2×C4×D83—44no—
133C2×C4×D83—48no—
134C2×C4×D83C4⋊C4116yescomplemented
135C4⋊Dic83—44no—
136Dic83⋊C4—44no—
137Dic83⋊C4—44no—
138Dic83⋊C4—48no—
139(C2×D83)⋊C4—48no—
140(C2×D83)⋊C4—48no—
141(C2×D83)⋊C4—44no—
142(C2×D83)⋊C4—44no—
143C24×C83C2411yescharacteristic
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—21no—
153C22×C4×C83—21no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
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—21no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83C22×C414yes—
170C22×C4×C83C22×C412yes—
171C22×C4×C83C2×D412yes—
172C22×C4×C83C2×Q812yes—
173C4⋊C4×C83—44no—
174C4⋊C4×C83—48no—
175C23×C83C23×C411yescharacteristic
176C23×C83C22×D411yescharacteristic
177C23×C83C22×Q811yescharacteristic
178C23×C83H(5,𝔽2)12yes—
179C23×C83H(5,𝔽2)11yescharacteristic
180C23×C83H(5,𝔽2)12yes—
181C23×C83H(5,𝔽2)12yes—
182C23×C83H(5,𝔽2)11yescharacteristic
183C23×C832−1+411yescharacteristic
184C23×C832−1+412yes—
185C23×C832−1+411yescharacteristic
186C22×D83—24no—
187C22×D83—22no—
188C22×D83—24no—
189C22×D83—22no—
190C22×D83—22no—
191C22×D83—22no—
192C22×D83—22no—
193C22×D83—22no—
194C22×D83—22no—
195C22×D83—22no—
196C22×D83—24no—
197C22×D83—22no—
198C22×D83—24no—
199C22×D83—22no—
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×D83C2×C4⋊C414yes—
209C2×C4×C83—24no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—24no—
213C2×C4×C83—24no—
214C2×C4×C83—24no—
215C2×C4×C83—24no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—42no—
219C2×C4×C83—42no—
220C2×C4×C83—42no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—42no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—42no—
228C2×C4×C83—41no—
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×C4⋊C414yes—
238C2×Dic83—24no—
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—24no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—42no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—42no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—42no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260C2×Dic83—42no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83C2×C4⋊C414yes—
267C4×D83—416no—
268C22×C83C22×C4⋊C411yescharacteristic
269C22×C83#51812yes—
270C22×C83#52011yescharacteristic
271C22×C83#52012yes—
272C22×C83#52011yescharacteristic
273C22×C83#53411yescharacteristic
274C22×C83#53411yescharacteristic
275C22×C83#53511yescharacteristic
276C22×C83#53511yescharacteristic
277C22×C83#53611yescharacteristic
278C22×C83#53611yescharacteristic
279C22×C83#54312yes—
280C22×C83#54412yes—
281C22×C83#54411yescharacteristic
282C22×C83#54411yescharacteristic
283C22×C83#56012yes—
284C22×C83#56012yes—
285C22×C83#56212yes—
286C22×C83#56212yes—
287C22×C83#56212yes—
288C22×C83#56212yes—
289C22×C83#56312yes—
290C22×C83#56312yes—
291C2×D83—24no—
292C2×D83—24no—
293C2×D83—24no—
294C2×D83—42no—
295C2×D83—42no—
296C2×D83—42no—
297C2×D83—44no—
298C2×D83—44no—
299C2×D83—44no—
300C2×D83—44no—
301C2×D83—42no—
302C4×C83—24no—
303C4×C83—44no—
304C4×C83—81no—
305C4×C83—82no—
306C4×C83—81no—
307Dic83—24no—
308Dic83—44no—
309Dic83—81no—
310Dic83—82no—
311Dic83—81no—
312#14674—831nomaximal, Sylow
313C2×C83#231511yescharacteristic
314C2×C83#231711yescharacteristic
315C2×C83#231711yescharacteristic
316C2×C83#237612yes—
317C2×C83#237612yes—
318C2×C83#281511yescharacteristic
319C2×C83#281611yescharacteristic
320C2×C83#281611yescharacteristic
321C2×C83#281711yescharacteristic
322C2×C83#284112yes—
323C2×C83#284112yes—
324D83—44no—
325C2×#381—831no—
326C2×#380—831no—
327C2×#380—831no—
328#1444—831no—
329#1444—832no—
330#1444D8311yescharacteristic, complemented
331#1442—832no—
332#1442—831no—
333#1442—831no—
334#1439—832no—
335#1439—832no—
336C83#1467411yescharacteristic, complemented, Sylow
337C24×C4—831no—
338C22×C4⋊C4—832no—
339C22×C4⋊C4—831no—
340C22×C4⋊C4—831no—
341C22×C4⋊C4—831no—
342C22×C4⋊C4C2×D8311yescharacteristic
343C22×C4⋊C4C2×D8312yes—
344C22×C22⋊C4—832no—
345C22×C22⋊C4—831no—
346C22×C22⋊C4—832no—
347C22×C22⋊C4—831no—
348#381—838no—
349#380—1664no—
350#380—1664no—
351C2×#95—832no—
352C2×#95—832no—
353C2×#95—832no—
354C2×#95—832no—
355C2×#95—832no—
356C2×#95—832no—
357C2×#95—831no—
358C2×#95—832no—
359C2×#95—831no—
360C2×#95C2×D8311yescharacteristic
361C2×#95C2×D8312yes—
362C2×#95C2×D8311yescharacteristic
363C25—831no—
364C23×C4—3322no—
365C23×C4—3322no—
366C23×C4—834no—
367C23×C4—834no—
368C23×C4—834no—
369C23×C4—831no—
370C23×C4—832no—
371C23×C4—831no—
372C23×C4—831no—
373C23×C4—832no—
374C23×C4C22×D8311yescharacteristic
375C23×C4C22×D8312yes—
376C23×C4C22×D8311yescharacteristic
377C23×C4C22×D8312yes—
378C23×C4C22×D8311yescharacteristic
379C2×C4⋊C4—22no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—42no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—3322no—
385C2×C4⋊C4—3322no—
386C2×C4⋊C4—3322no—
387C2×C4⋊C4—3322no—
388C2×C4⋊C4—3322no—
389C2×C4⋊C4—3322no—
390C2×C4⋊C4—1662no—
391C2×C4⋊C4—1662no—
392C2×C4⋊C4—1662no—
393C2×C22⋊C4—3322no—
394C2×C22⋊C4—3322no—
395C2×C22⋊C4—3322no—
396C2×C22⋊C4—3322no—
397C2×C22⋊C4—3322no—
398C2×C22⋊C4—3322no—
399C2×C22⋊C4—838no—
400C2×C22⋊C4—1662no—
401C2×C22⋊C4—1662no—
402C2×C22⋊C4—1664no—
403#95—22no—
404#95—42no—
405#95—3322no—
406#95—3322no—
407#95—3322no—
408#95—3322no—
409#95—3322no—
410#95—838no—
411#95—834no—
412#95—1664no—
413#95—1662no—
414#95C4×D8314yes—
415C24—832no—
416C24—832no—
417C24—832no—
418C24—1662no—
419C24—1661no—
420C24—1662no—
421C24—1661no—
422C24—1662no—
423C24—1661no—
424C24—1661no—
425C24—1662no—
426C24C23×D8311yescharacteristic
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—3328no—
453C22×C4—3328no—
454C22×C4—3324no—
455C22×C4—3324no—
456C22×C4—8316no—
457C22×C4—832no—
458C22×C4—832no—
459C22×C4—832no—
460C22×C4—834no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1662no—
464C22×C4—1661no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1662no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4C2×C4×D8314yes—
487C22×C4C2×C4×D8312yes—
488C22×C4D4×D8312yes—
489C22×C4Q8×D8312yes—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—3324no—
493C4⋊C4—3328no—
494C4⋊C4—3324no—
495C4⋊C4—3324no—
496C22⋊C4—3328no—
497C22⋊C4—3324no—
498C22⋊C4—3324no—
499C22⋊C4—3328no—
500C23—3322no—
501C23—3322no—
502C23—3322no—
503C23—3322no—
504C23—3322no—
505C23—3322no—
506C23—3322no—
507C23—3322no—
508C23—834no—
509C23—1664no—
510C23—1662no—
511C23—1664no—
512C23—1662no—
513C23—1664no—
514C23—1662no—
515C23—1664no—
516C23—1662no—
517C23—1662no—
518C23—1662no—
519C23—1662no—
520C23—1662no—
521C23—1662no—
522C23—1662no—
523C23C22×C4×D8311yescharacteristic
524C23C2×D4×D8311yescharacteristic
525C23C2×Q8×D8311yescharacteristic
526C23#1458183712yes—
527C23#1458183711yescharacteristic
528C23#1458183712yes—
529C23#1458183812yes—
530C23#1458183811yescharacteristic
531C23#1458183911yescharacteristic
532C23#1458184012yes—
533C23#1458184011yescharacteristic
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
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—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—33216no—
563C2×C4—3322no—
564C2×C4—3322no—
565C2×C4—3322no—
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—3321no—
576C2×C4—3321no—
577C2×C4—3322no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3322no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3322no—
584C2×C4—834no—
585C2×C4—1664no—
586C2×C4—1664no—
587C2×C4—1664no—
588C2×C4—1664no—
589C2×C4—1664no—
590C2×C4—1662no—
591C2×C4—1662no—
592C2×C4C4⋊C4×D8314yes—
593C22—3324no—
594C22—3324no—
595C22—3324no—
596C22—3324no—
597C22—3322no—
598C22—3322no—
599C22—3322no—
600C22—3322no—
601C22—1664no—
602C22—1664no—
603C22—1664no—
604C22C2×C4⋊C4×D8311yescharacteristic
605C22#1912817312yes—
606C22#1912818011yescharacteristic
607C22#1912818212yes—
608C22#1912818311yescharacteristic
609C22#1912826111yescharacteristic
610C22#1912826211yescharacteristic
611C22#1912826411yescharacteristic
612C22#1912826711yescharacteristic
613C22#1912827411yescharacteristic
614C22#1912827611yescharacteristic
615C22#1912832612yes—
616C22#1912832812yes—
617C22#1912832811yescharacteristic
618C22#1912833111yescharacteristic
619C22#1912843912yes—
620C22#1912844012yes—
621C22#1912844712yes—
622C22#1912844712yes—
623C22#1912844812yes—
624C22#1912844912yes—
625C22#1912845012yes—
626C22#1912845212yes—
627C4—24no—
628C4—44no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4—3324no—
633C4—6641no—
634C4—6641no—
635C4—6642no—
636C4—1664no—
637C2—3324no—
638C2#2491007411yescharacteristic
639C2#2491008411yescharacteristic
640C2#2491008511yescharacteristic
641C2#2491060812yes—
642C2#2491061212yes—
643C2#2491548411yescharacteristic
644C2#2491549211yescharacteristic
645C2#2491549611yescharacteristic
646C2#2491549711yescharacteristic
647C2#2491571712yes—
648C2#2491571912yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1444C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenter#1444D83
hypocenterC83#14674
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
4. #24905726C2semidirect
5. #24905726C2semidirect
6. #24905720C2semidirect
7. #24905719C2semidirect
8. #24905718C2semidirect
11. C83×#1444C2semidirect
21. #19127563C4semidirect
134. C2×C4×D83C4⋊C4semidirect
330. #1444D83semidirect
336. C83#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#191275632
C2×#191275602
C2×#191275582
#249057262
#249057202
#249057192
#249057182
#249057072
#249057052
C83×#14442
#1467483