Finite Group Databasebeta

← #35331127

Subgroups — #35331127

Order 21248.

1566 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×#19127563C211yesmaximal, characteristic
2C2×#19127558C211yesmaximal, characteristic
3C2×#19127548C211yesmaximal, characteristic
4#24905728C211yesmaximal, characteristic, complemented
5#24905726C211yesmaximal, characteristic, complemented
6#24905719C211yesmaximal, characteristic, complemented
7#24905709C212yesmaximal
8#24905707C212yesmaximal
9#24905697C212yesmaximal, complemented
10#24905694C212yesmaximal, complemented
11C83×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×C4⋊Dic83C2212yes—
15C22×Dic83⋊C4C2212yes—
16C22×(C2×D83)⋊C4C2211yescharacteristic
17C22×(C2×D83)⋊C4C2211yescharacteristic
18C22×(C2×D83)⋊C4C2212yes—
19C22×(C2×D83)⋊C4C2212yes—
20C22×C4×Dic83C2211yescharacteristic
21C22×C4×Dic83C2211yescharacteristic
22#19127563C418yescomplemented
23#19127558—24no—
24#19127548—24no—
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
34C2×#95×C83C2211yescharacteristic
35C2×#95×C83C2212yes—
36C2×#95×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C24×D83C2311yescharacteristic
39C23×Dic83C2312yes—
40C23×Dic83C2312yes—
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2311yescharacteristic
44C22×C4×D83—42no—
45C22×C4×D83—42no—
46C22×C4×D83C2×C414yes—
47C22×C4×D83D414yes—
48C22×C4×D83Q814yes—
49C2×C4⋊Dic83—24no—
50C2×C4⋊Dic83—22no—
51C2×C4⋊Dic83—42no—
52C2×C4⋊Dic83—42no—
53C2×C4⋊Dic83—42no—
54C2×Dic83⋊C4—24no—
55C2×Dic83⋊C4—42no—
56C2×Dic83⋊C4—42no—
57C2×(C2×D83)⋊C4—22no—
58C2×(C2×D83)⋊C4—22no—
59C2×(C2×D83)⋊C4—22no—
60C2×(C2×D83)⋊C4—22no—
61C2×(C2×D83)⋊C4—24no—
62C2×(C2×D83)⋊C4—22no—
63C2×(C2×D83)⋊C4—22no—
64C2×(C2×D83)⋊C4—42no—
65C2×(C2×D83)⋊C4—42no—
66C2×(C2×D83)⋊C4—42no—
67C2×(C2×D83)⋊C4—42no—
68C2×(C2×D83)⋊C4C2×C418yes—
69C2×C4×Dic83—22no—
70C2×C4×Dic83—22no—
71C2×C4×Dic83—42no—
72C2×C4×Dic83—42no—
73C83⋊#95—24no—
74C83⋊#95—22no—
75C83⋊#95—24no—
76C83⋊#95—42no—
77C83⋊#95—42no—
78C83⋊#95—42no—
79C83⋊#95C2×C418yes—
80C83⋊#95C2×C414yes—
81C23×C4×C83C2312yes—
82C23×C4×C83C2312yes—
83C23×C4×C83C2311yescharacteristic
84C23×C4×C83C2311yescharacteristic
85C23×C4×C83C2311yescharacteristic
86C2×C4⋊C4×C83—22no—
87C2×C4⋊C4×C83—42no—
88#95×C83—22no—
89#95×C83—24no—
90#95×C83—42no—
91#95×C83C2×C414yes—
92C23×D83—21no—
93C23×D83—21no—
94C23×D83—21no—
95C23×D83—21no—
96C23×D83—22no—
97C23×D83—22no—
98C23×D83—22no—
99C23×D83—22no—
100C23×D83C22×C412yes—
101C23×D83C2×D412yes—
102C23×D83C2×Q812yes—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—21no—
106C22×Dic83—21no—
107C22×Dic83—22no—
108C22×Dic83—21no—
109C22×Dic83—21no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
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—22no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83C22×C414yes—
127C22×Dic83C22×C412yes—
128C22×Dic83C2×D412yes—
129C22×Dic83C2×Q812yes—
130C22×Dic83#4112yes—
131C22×Dic83#4112yes—
132C2×C4×D83—48no—
133C2×C4×D83—44no—
134C2×C4×D83—44no—
135C2×C4×D83—48no—
136C2×C4×D83C4⋊C4116yescomplemented
137C4⋊Dic83—48no—
138C4⋊Dic83—44no—
139Dic83⋊C4—48no—
140(C2×D83)⋊C4—28no—
141(C2×D83)⋊C4—28no—
142(C2×D83)⋊C4—48no—
143(C2×D83)⋊C4—48no—
144C4×Dic83—44no—
145C4×Dic83—44no—
146C24×C83C2411yescharacteristic
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—21no—
150C22×C4×C83—21no—
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—22no—
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×C414yes—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C2×D412yes—
173C22×C4×C83C2×Q812yes—
174C22×C4×C83#4112yes—
175C22×C4×C83#4112yes—
176C4⋊C4×C83—44no—
177C23×C83C23×C411yescharacteristic
178C23×C83C22×D411yescharacteristic
179C23×C83C22×Q811yescharacteristic
180C23×C83C2×#4111yescharacteristic
181C23×C83C2×#4111yescharacteristic
182C23×C83C2×#4111yescharacteristic
183C23×C83C2×#4111yescharacteristic
184C23×C83H(5,𝔽2)12yes—
185C23×C83H(5,𝔽2)12yes—
186C23×C83H(5,𝔽2)12yes—
187C23×C832−1+412yes—
188C22×D83—22no—
189C22×D83—22no—
190C22×D83—24no—
191C22×D83—24no—
192C22×D83—22no—
193C22×D83—22no—
194C22×D83—22no—
195C22×D83—22no—
196C22×D83—22no—
197C22×D83—22no—
198C22×D83—22no—
199C22×D83—22no—
200C22×D83—24no—
201C22×D83—24no—
202C22×D83—42no—
203C22×D83—42no—
204C22×D83—42no—
205C22×D83—42no—
206C22×D83—42no—
207C22×D83—42no—
208C22×D83—42no—
209C22×D83—42no—
210C22×D83C2×C4⋊C414yes—
211C2×C4×C83—24no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—24no—
215C2×C4×C83—24no—
216C2×C4×C83—24no—
217C2×C4×C83—24no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—24no—
221C2×C4×C83—24no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
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×C83—42no—
234C2×C4×C83—42no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83—42no—
238C2×C4×C83—42no—
239C2×C4×C83C2×C4⋊C414yes—
240C2×Dic83—24no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—24no—
244C2×Dic83—24no—
245C2×Dic83—24no—
246C2×Dic83—24no—
247C2×Dic83—22no—
248C2×Dic83—22no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—42no—
256C2×Dic83—42no—
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×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83C2×C4⋊C414yes—
269C4×D83—416no—
270C22×C83C22×C4⋊C411yescharacteristic
271C22×C83C4×#4111yescharacteristic
272C22×C83C4×#4111yescharacteristic
273C22×C83#51812yes—
274C22×C83#52012yes—
275C22×C83#52511yescharacteristic
276C22×C83#52511yescharacteristic
277C22×C83#53311yescharacteristic
278C22×C83#53311yescharacteristic
279C22×C83#53411yescharacteristic
280C22×C83#53411yescharacteristic
281C22×C83#53511yescharacteristic
282C22×C83#53611yescharacteristic
283C22×C83#53812yes—
284C22×C83#53812yes—
285C22×C83#53912yes—
286C22×C83#53912yes—
287C22×C83#54012yes—
288C22×C83#54012yes—
289C22×C83#54212yes—
290C22×C83#54212yes—
291C22×C83#54312yes—
292C22×C83#54412yes—
293C2×D83—24no—
294C2×D83—24no—
295C2×D83—24no—
296C2×D83—44no—
297C2×D83—44no—
298C2×D83—42no—
299C2×D83—42no—
300C2×D83—42no—
301C2×D83—42no—
302C2×D83—44no—
303C2×D83—44no—
304C4×C83—24no—
305C4×C83—44no—
306C4×C83—42no—
307C4×C83—42no—
308C4×C83—82no—
309Dic83—24no—
310Dic83—44no—
311Dic83—42no—
312Dic83—42no—
313Dic83—82no—
314#14295—831nomaximal, Sylow
315C2×C83#229611yescharacteristic
316C2×C83#231511yescharacteristic
317C2×C83#231711yescharacteristic
318C2×C83#233512yes—
319C2×C83#233512yes—
320C2×C83#263511yescharacteristic
321C2×C83#263511yescharacteristic
322C2×C83#263811yescharacteristic
323C2×C83#263811yescharacteristic
324C2×C83#266512yes—
325C2×C83#266712yes—
326D83—44no—
327C2×#381—831no—
328C2×#380—831no—
329C2×C4×C22⋊C4—831no—
330#1444—831no—
331#1444—831no—
332#1442—831no—
333#1442D8311yescharacteristic, complemented
334#1439—832no—
335#1439—832no—
336#1436—832no—
337#1436—832no—
338C83#1429511yescharacteristic, complemented, Sylow
339C24×C4—831no—
340C22×C4⋊C4—832no—
341C22×C4⋊C4—832no—
342C22×C4⋊C4—831no—
343C22×C4⋊C4C2×D8311yescharacteristic
344C22×C22⋊C4—832no—
345C22×C22⋊C4—832no—
346C22×C22⋊C4—831no—
347C22×C22⋊C4—831no—
348C22×C42—831no—
349C22×C42—831no—
350#381—838no—
351#380—1664no—
352C4×C22⋊C4—1664no—
353C2×#95—832no—
354C2×#95—832no—
355C2×#95—832no—
356C2×#95—832no—
357C2×#95—831no—
358C2×#95—832no—
359C2×#95—832no—
360C2×#95—831no—
361C2×#95C2×D8312yes—
362C2×#95C2×D8311yescharacteristic
363C2×#95C2×D8312yes—
364C2×#95C2×D8311yescharacteristic
365C25—831no—
366C23×C4—3322no—
367C23×C4—3322no—
368C23×C4—834no—
369C23×C4—834no—
370C23×C4—834no—
371C23×C4—831no—
372C23×C4—831no—
373C23×C4—831no—
374C23×C4—832no—
375C23×C4—832no—
376C23×C4C22×D8312yes—
377C23×C4C22×D8312yes—
378C23×C4C22×D8311yescharacteristic
379C23×C4C22×D8311yescharacteristic
380C23×C4C22×D8311yescharacteristic
381C2×C4⋊C4—22no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—3322no—
384C2×C4⋊C4—3322no—
385C2×C4⋊C4—3322no—
386C2×C4⋊C4—3322no—
387C2×C4⋊C4—3322no—
388C2×C4⋊C4—1664no—
389C2×C4⋊C4—1664no—
390C2×C4⋊C4—1662no—
391C2×C22⋊C4—3322no—
392C2×C22⋊C4—3322no—
393C2×C22⋊C4—3322no—
394C2×C22⋊C4—3322no—
395C2×C22⋊C4—838no—
396C2×C22⋊C4—1664no—
397C2×C22⋊C4—1662no—
398C2×C22⋊C4—1662no—
399C2×C22⋊C4—1662no—
400C2×C22⋊C4—1662no—
401C2×C22⋊C4—1662no—
402C2×C22⋊C4—1662no—
403C2×C42—3322no—
404C2×C42—3322no—
405C2×C42—1662no—
406C2×C42—1662no—
407#95—22no—
408#95—24no—
409#95—42no—
410#95—3322no—
411#95—3322no—
412#95—3322no—
413#95—838no—
414#95—834no—
415#95—1664no—
416#95—1662no—
417#95—1664no—
418#95C4×D8314yes—
419C24—832no—
420C24—832no—
421C24—832no—
422C24—1661no—
423C24—1661no—
424C24—1661no—
425C24—1661no—
426C24—1662no—
427C24—1662no—
428C24—1662no—
429C24—1662no—
430C24C23×D8311yescharacteristic
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
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—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—3328no—
455C22×C4—3328no—
456C22×C4—3324no—
457C22×C4—3324no—
458C22×C4—8316no—
459C22×C4—832no—
460C22×C4—832no—
461C22×C4—832no—
462C22×C4—834no—
463C22×C4—832no—
464C22×C4—832no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1662no—
475C22×C4—1661no—
476C22×C4—1662no—
477C22×C4—1661no—
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×C4—1662no—
487C22×C4—1662no—
488C22×C4C2×C4×D8314yes—
489C22×C4C2×C4×D8312yes—
490C22×C4D4×D8312yes—
491C22×C4Q8×D8312yes—
492C22×C4#1305058912yes—
493C22×C4#1305059112yes—
494C4⋊C4—44no—
495C4⋊C4—3324no—
496C4⋊C4—3328no—
497C4⋊C4—3328no—
498C22⋊C4—3328no—
499C22⋊C4—3328no—
500C22⋊C4—1668no—
501C22⋊C4—1668no—
502C42—3324no—
503C42—3324no—
504C23—3322no—
505C23—3322no—
506C23—3322no—
507C23—3322no—
508C23—3322no—
509C23—3322no—
510C23—3322no—
511C23—3322no—
512C23—834no—
513C23—1664no—
514C23—1664no—
515C23—1662no—
516C23—1662no—
517C23—1664no—
518C23—1664no—
519C23—1662no—
520C23—1662no—
521C23—1662no—
522C23—1662no—
523C23—1662no—
524C23—1662no—
525C23—1662no—
526C23—1662no—
527C23C22×C4×D8311yescharacteristic
528C23C2×D4×D8311yescharacteristic
529C23C2×Q8×D8311yescharacteristic
530C23#41×D8311yescharacteristic
531C23#41×D8311yescharacteristic
532C23C2×#1305058911yescharacteristic
533C23C2×#1305059111yescharacteristic
534C23#1458183712yes—
535C23#1458183812yes—
536C23#1458183812yes—
537C23#1458183912yes—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—33216no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3322no—
572C2×C4—3322no—
573C2×C4—3322no—
574C2×C4—3322no—
575C2×C4—3322no—
576C2×C4—3322no—
577C2×C4—3322no—
578C2×C4—3322no—
579C2×C4—3322no—
580C2×C4—3322no—
581C2×C4—3322no—
582C2×C4—3322no—
583C2×C4—3322no—
584C2×C4—834no—
585C2×C4—1664no—
586C2×C4—1664no—
587C2×C4—1664no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4—1664no—
591C2×C4—1664no—
592C2×C4—1664no—
593C2×C4—1664no—
594C2×C4—1662no—
595C2×C4—1662no—
596C2×C4C4⋊C4×D8314yes—
597C22—3324no—
598C22—3324no—
599C22—3322no—
600C22—3322no—
601C22—3324no—
602C22—3322no—
603C22—3322no—
604C22—3324no—
605C22—1664no—
606C22—1664no—
607C22—1664no—
608C22C2×C4⋊C4×D8311yescharacteristic
609C22C4×#1305058911yescharacteristic
610C22C4×#1305059111yescharacteristic
611C22#1912817312yes—
612C22#1912818112yes—
613C22#1912821111yescharacteristic
614C22#1912821411yescharacteristic
615C22#1912825611yescharacteristic
616C22#1912825711yescharacteristic
617C22#1912826111yescharacteristic
618C22#1912826211yescharacteristic
619C22#1912826911yescharacteristic
620C22#1912827111yescharacteristic
621C22#1912828712yes—
622C22#1912829012yes—
623C22#1912829512yes—
624C22#1912830012yes—
625C22#1912830412yes—
626C22#1912830712yes—
627C22#1912831512yes—
628C22#1912831712yes—
629C22#1912832612yes—
630C22#1912833112yes—
631C4—24no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—3324no—
637C4—3322no—
638C4—3322no—
639C4—6642no—
640C4—1664no—
641C2—3324no—
642C2#2490994411yescharacteristic
643C2#2491007411yescharacteristic
644C2#2491008811yescharacteristic
645C2#2491025212yes—
646C2#2491026012yes—
647C2#2491335511yescharacteristic
648C2#2491335611yescharacteristic
649C2#2491338511yescharacteristic
650C2#2491338611yescharacteristic
651C2#2491364012yes—
652C2#2491366212yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1442C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenter#1442D83
hypocenterC83#14295
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #24905728C2semidirect
5. #24905726C2semidirect
6. #24905719C2semidirect
9. #24905697C2semidirect
10. #24905694C2semidirect
11. C83×#1442C2semidirect
22. #19127563C4semidirect
136. C2×C4×D83C4⋊C4semidirect
333. #1442D83semidirect
338. C83#14295semidirect

Maximal subgroups

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

subgroupindex
C2×#191275632
C2×#191275582
C2×#191275482
#249057282
#249057262
#249057192
#249057092
#249057072
#249056972
#249056942
C83×#14422
#1429583