Finite Group Databasebeta

← #35504430

Subgroups — #35504430

Order 21248.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915773C211yesmaximal, characteristic, complemented
2#24909031C211yesmaximal, characteristic, complemented
3#24908984C211yesmaximal, characteristic, complemented
4#24908801C211yesmaximal, characteristic, complemented
5#24908609C211yesmaximal, characteristic, complemented
6#24908055C211yesmaximal, characteristic, complemented
7#24908041C211yesmaximal, characteristic, complemented
8#24907892C211yesmaximal, characteristic
9#24907892C211yesmaximal, characteristic
10#24907724C211yesmaximal, characteristic, complemented
11#24907710C211yesmaximal, characteristic, complemented
12#24907490C211yesmaximal, characteristic, complemented
13#24907486C211yesmaximal, characteristic, complemented
14C2×#19127456C211yesmaximal, characteristic, complemented
15C4×#121×C83C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C23×C83⋊C8C2211yescharacteristic
18C2×#14581784C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27#19127599C2211yescharacteristic
28#19127599C2211yescharacteristic
29#19127585—21no—
30#19127585—21no—
31#19127574—21no—
32#19127574—21no—
33#19127567C2211yescharacteristic
34#19127561C2211yescharacteristic
35#19127561C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127456C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#121×C83C2211yescharacteristic
49C2×C4×D4×C83—21no—
50C2×C4×D4×C83C2211yescharacteristic
51C22×C42×C83C2211yescharacteristic
52#390×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#388×C83—21no—
55#386×C83—21no—
56#380×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C22×C83⋊C8—22no—
60C22×C83⋊C8—22no—
61C22×C83⋊C8—22no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8—41no—
64C22×C83⋊C8—41no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581784—41no—
70#14581784—41no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81#14581725—22no—
82#14581725—41no—
83#14581723—22no—
84#14581723—41no—
85#14581713—22no—
86#14581713—22no—
87#14581713—41no—
88#14581713—41no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C22×D4×C83—21no—
96C22×D4×C83C2311yescharacteristic
97C23×C4×C83—21no—
98C23×C4×C83D411yescharacteristic
99C23×C4×C83D411yescharacteristic
100C23×C4×C83C2311yescharacteristic
101#121×C83—22no—
102#121×C83Q814yes—
103C4×D4×C83—41no—
104C4×D4×C83—41no—
105C4×D4×C83—44no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83—21no—
115C2×C42×C83—21no—
116C2×C42×C83—41no—
117C2×C42×C83—41no—
118C2×C42×C83D411yescharacteristic
119C2×C42×C83D411yescharacteristic
120C2×C42×C83C2311yescharacteristic
121C2×C42×C83C2311yescharacteristic
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—44no—
140C2×C83⋊C8—44no—
141C2×C83⋊C8#4112yes—
142C2×C83⋊C8#4112yes—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151Dic83⋊C4—42no—
152Dic83⋊C4—42no—
153Dic83⋊C4—42no—
154Dic83⋊C4—42no—
155C22⋊Dic83—42no—
156C22⋊Dic83—42no—
157C24×C83—21no—
158C24×C83C2×D411yescharacteristic
159C2×D4×C83—24no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83—21no—
165C2×D4×C83—42no—
166C2×D4×C83—42no—
167C2×D4×C83C2×Q812yes—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—21no—
175C22×C4×C83—22no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—41no—
179C22×C4×C83—41no—
180C22×C4×C83—41no—
181C22×C4×C83—42no—
182C22×C4×C83—42no—
183C22×C4×C83—42no—
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×Q812yes—
190C22×C4×C83#4112yes—
191C22×C4×C83C2411yescharacteristic
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—42no—
198C4⋊C4×C83C2×Q812yes—
199C22⋊C4×C83—24no—
200C22⋊C4×C83—42no—
201C22⋊C4×C83—42no—
202C22⋊C4×C83—42no—
203C22⋊C4×C83—42no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—21no—
208C42×C83—41no—
209C42×C83—41no—
210C42×C83—42no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83—22no—
216C23×C83—41no—
217C23×C83—41no—
218C23×C83—41no—
219C23×C83—41no—
220C23×C83—41no—
221C23×C83—41no—
222C23×C83C22⋊Q812yes—
223C23×C83#13512yes—
224C23×C83AGL(1,ℤ/8)12yes—
225C23×C83C22×D411yescharacteristic
226D4×C83—22no—
227D4×C83—22no—
228D4×C83—48no—
229D4×C83—41no—
230D4×C83—41no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—22no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—22no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—21no—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—44no—
250C2×C4×C83—44no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83—84no—
260C2×C4×C83C22⋊Q812yes—
261C2×C4×C83#13512yes—
262C2×C4×C83C22×Q811yescharacteristic
263C2×C4×C83C2×#4111yescharacteristic
264C2×C4×C83C2×#4111yescharacteristic
265C2×C4×C83C2×#4111yescharacteristic
266C2×C4×C83H(5,𝔽2)11yescharacteristic
267C2×C4×C83H(5,𝔽2)11yescharacteristic
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83—41no—
276C2×Dic83—41no—
277C2×Dic83—41no—
278C2×Dic83—41no—
279C2×Dic83—41no—
280C83⋊C8—22no—
281C83⋊C8—22no—
282C83⋊C8—41no—
283C83⋊C8—41no—
284C22×C83—22no—
285C22×C83—22no—
286C22×C83—22no—
287C22×C83—22no—
288C22×C83—22no—
289C22×C83—22no—
290C22×C83—42no—
291C22×C83—42no—
292C22×C83—42no—
293C22×C83—81no—
294C22×C83—81no—
295C22×C83—81no—
296C22×C83—81no—
297C22×C83#47814yes—
298C22×C83C2×C22⋊Q811yescharacteristic
299C22×C83#53211yescharacteristic
300C22×C83#53411yescharacteristic
301C22×C83C2×#13511yescharacteristic
302C22×C83C2×#13511yescharacteristic
303C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C83#57511yescharacteristic
305C4×C83—22no—
306C4×C83—22no—
307C4×C83—42no—
308C4×C83—41no—
309C4×C83—41no—
310C4×C83—41no—
311C4×C83—81no—
312C4×C83—81no—
313C4×C83#55111yescharacteristic
314C4×C83#55111yescharacteristic
315C4×C83#55411yescharacteristic
316C4×C83#55411yescharacteristic
317Dic83—81no—
318Dic83—81no—
319Dic83—81no—
320Dic83—81no—
321#29161—831nomaximal, Sylow
322C2×C83—24no—
323C2×C83—82no—
324C2×C83#256411yescharacteristic
325C2×C83C2×#47811yescharacteristic
326C2×C83#307611yescharacteristic
327C2×C83#316811yescharacteristic
328C2×C83#318011yescharacteristic
329C2×C83#318811yescharacteristic
330C2×C83#319111yescharacteristic
331#2851—831no—
332C4×#121D8311yescharacteristic
333#2101—831no—
334#2092—831no—
335#2059—831no—
336#2025—831no—
337#1921—831no—
338#1919—831no—
339#1893—831no—
340#1893—831no—
341#1864—831no—
342#1862—831no—
343#1808—831no—
344#1807—831no—
345C2×#336—831no—
346C83#2916111yescharacteristic, complemented, Sylow
347C23×C8—831no—
348C23×C8—831no—
349C2×#125—831no—
350C2×#121C2×D8311yescharacteristic
351C2×C4×D4—21no—
352C2×C4×D4C2×D8311yescharacteristic
353C22×C42C2×D8311yescharacteristic
354C2×#107—831no—
355C2×#107—831no—
356C2×#106—831no—
357C2×#106—831no—
358C2×#102—831no—
359C2×#102—831no—
360C2×#102—831no—
361C2×#102—831no—
362#390C2×D8311yescharacteristic
363#389—831no—
364#389—831no—
365#389C2×D8311yescharacteristic
366#388—21no—
367#387—1661no—
368#387—1661no—
369#386—21no—
370#385—1661no—
371#385—1661no—
372#382—831no—
373#380—831no—
374#380—831no—
375#380C2×D8311yescharacteristic
376C4×C4⋊C4C2×D8311yescharacteristic
377C4×C22⋊C4—21no—
378#340—831no—
379#340—831no—
380#340—831no—
381#340—831no—
382#340—831no—
383#340—831no—
384#340—831no—
385#340—831no—
386#336—831no—
387#336—831no—
388#336—831no—
389#336—831no—
390C22×D4—21no—
391C22×D4C22×D8311yescharacteristic
392C23×C4—21no—
393C23×C4C22⋊D8311yescharacteristic
394C23×C4C22⋊D8311yescharacteristic
395C23×C4C22×D8311yescharacteristic
396C22×C8—3321no—
397C22×C8—3321no—
398C22×C8—831no—
399C22×C8—831no—
400C22×C8—831no—
401C22×C8—831no—
402C22×C8—1662no—
403C22×C8—1662no—
404C22×C8—1662no—
405C22×C8—1662no—
406#125—3321no—
407#125—3321no—
408#121—22no—
409#121Dic16614yes—
410C4×D4—41no—
411C4×D4—41no—
412C4×D4—44no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—831no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—831no—
418C2×C4⋊C4—1661no—
419C2×C4⋊C4—1661no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4—1661no—
422C2×C4⋊C4C22×D8311yescharacteristic
423C2×C4⋊C4C22×D8311yescharacteristic
424C2×C4⋊C4C22×D8311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1661no—
431C2×C22⋊C4—1661no—
432C2×C42—21no—
433C2×C42—41no—
434C2×C42—41no—
435C2×C42C22⋊D8311yescharacteristic
436C2×C42C22⋊D8311yescharacteristic
437C2×C42C22×D8311yescharacteristic
438C2×C42C22×D8311yescharacteristic
439#107—3321no—
440#107—1662no—
441#106—3321no—
442#106—1662no—
443#102—3321no—
444#102—3321no—
445#102—1662no—
446#102—1662no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—1661no—
451#95—1661no—
452#95—1661no—
453#95—1661no—
454#95—1661no—
455#95—1661no—
456C24—21no—
457C24C2×C22⋊D8311yescharacteristic
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×Dic16612yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—1661no—
484C22×C4—1661no—
485C22×C4—1661no—
486C22×C4—1661no—
487C22×C4C2×C22⋊D8311yescharacteristic
488C22×C4C2×C22⋊D8311yescharacteristic
489C22×C4C2×C22⋊D8311yescharacteristic
490C22×C4C2×C22⋊D8311yescharacteristic
491C22×C4C2×C22⋊D8311yescharacteristic
492C22×C4C2×Dic16612yes—
493C22×C4#1305059012yes—
494C22×C4C23×D8311yescharacteristic
495C2×C8—3324no—
496C2×C8—3324no—
497C2×C8—832no—
498C2×C8—832no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C2×C8—1661no—
503C2×C8—1661no—
504C2×C8—1661no—
505C2×C8—1661no—
506C2×C8—1661no—
507C2×C8—1661no—
508C2×C8—1661no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—3322no—
516C4⋊C4—3322no—
517C4⋊C4—3322no—
518C4⋊C4—3322no—
519C4⋊C4—1661no—
520C4⋊C4—1661no—
521C4⋊C4—1661no—
522C4⋊C4—1661no—
523C4⋊C4—1661no—
524C4⋊C4—1661no—
525C4⋊C4—1661no—
526C4⋊C4—1661no—
527C4⋊C4C2×Dic16612yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—3322no—
534C22⋊C4—3322no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—41no—
540C42—41no—
541C42—42no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23#1458177112yes—
554C23#1458181212yes—
555C23#1458181812yes—
556C23C22×C22⋊D8311yescharacteristic
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—3321no—
592C2×C4—3321no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—3321no—
597C2×C4—3321no—
598C2×C4—3321no—
599C2×C4—3321no—
600C2×C4—3321no—
601C2×C4—3321no—
602C2×C4—3321no—
603C2×C4#1458177112yes—
604C2×C4#1458181212yes—
605C2×C4C22×Dic16611yescharacteristic
606C2×C4#41×D8311yescharacteristic
607C2×C4#41×D8311yescharacteristic
608C2×C4C2×#1305059011yescharacteristic
609C2×C4#1458183711yescharacteristic
610C2×C4#1458183711yescharacteristic
611C8—3321no—
612C8—3321no—
613C8—1662no—
614C8—1662no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#1912800114yes—
629C22C2×#1458177111yescharacteristic
630C22#1912824711yescharacteristic
631C22#1912826011yescharacteristic
632C22C2×#1458181211yescharacteristic
633C22C2×#1458181211yescharacteristic
634C22C2×#1458181811yescharacteristic
635C22#1912849911yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—6641no—
645C4—6641no—
646C4—6641no—
647C4—6641no—
648C4#1912838011yescharacteristic
649C4#1912838011yescharacteristic
650C4#1912841011yescharacteristic
651C4#1912841011yescharacteristic
652C2—24no—
653C2—82no—
654C2#2491263911yescharacteristic
655C2C2×#1912800111yescharacteristic
656C2#2491725111yescharacteristic
657C2#2491802711yescharacteristic
658C2#2491815511yescharacteristic
659C2#2491822811yescharacteristic
660C2#2491825111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC4×#121×C83C2
socleC23×C83C22×D4
prime radicalC2×#121×C83C22
hypercenterC4×#121D83
hypocenterC83#29161
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #24915773C2semidirect
2. #24909031C2semidirect
3. #24908984C2semidirect
4. #24908801C2semidirect
5. #24908609C2semidirect
6. #24908055C2semidirect
7. #24908041C2semidirect
10. #24907724C2semidirect
11. #24907710C2semidirect
12. #24907490C2semidirect
13. #24907486C2semidirect
14. C2×#19127456C2semidirect
346. C83#29161semidirect

Maximal subgroups

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

subgroupindex
#249157732
#249090312
#249089842
#249088012
#249086092
#249080552
#249080412
#249078922
#249077242
#249077102
#249074902
#249074862
C2×#191274562
C4×#121×C832
#2916183