Finite Group Databasebeta

← #35477220

Subgroups — #35477220

Order 21248.

742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24911896C211yesmaximal, characteristic, complemented
2#24908866C211yesmaximal, characteristic, complemented
3#24908851C211yesmaximal, characteristic, complemented
4#24908642C211yesmaximal, characteristic
5#24908642C211yesmaximal, characteristic
6#24908221C211yesmaximal, characteristic, complemented
7#24908193C211yesmaximal, characteristic, complemented
8#24908071C211yesmaximal, characteristic, complemented
9#24908052C211yesmaximal, characteristic, complemented
10#24908041C211yesmaximal, characteristic, complemented
11#24907724C211yesmaximal, characteristic, complemented
12#24907662C211yesmaximal, characteristic, complemented
13#24907558C211yesmaximal, characteristic, complemented
14C4×#14581713C211yesmaximal, characteristic, complemented
15C83×#2363C211yesmaximal, characteristic
16C2×#14581725C2211yescharacteristic
17C2×#14581725C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581723C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×C22⋊(C83⋊C8)C2211yescharacteristic
26C2×C22⋊(C83⋊C8)C2211yescharacteristic
27C2×C4×C83⋊C8C2211yescharacteristic
28#19127643—21no—
29#19127624—21no—
30#19127599C2211yescharacteristic
31#19127599C2211yescharacteristic
32#19127594—21no—
33#19127588—21no—
34#19127571C2211yescharacteristic
35#19127561C2211yescharacteristic
36#19127561C2211yescharacteristic
37C4×C4⋊Dic83C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#121×C83C2211yescharacteristic
49C2×C4×D4×C83C2211yescharacteristic
50C2×#117×C83C2211yescharacteristic
51#390×C83—21no—
52#390×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#388×C83—21no—
55#386×C83—21no—
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C22⋊C4×C83—21no—
58C4×C22⋊C4×C83C2211yescharacteristic
59C22×C83⋊C8C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83C2311yescharacteristic
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×Dic83⋊C4—21no—
70C2×Dic83⋊C4—21no—
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C4×Dic83—21no—
74#14581725—22no—
75#14581725—22no—
76#14581725—41no—
77#14581723—22no—
78C4⋊(C83⋊C8)—22no—
79#14581713—22no—
80#14581713—22no—
81#14581713—22no—
82#14581713—41no—
83C22⋊(C83⋊C8)—22no—
84C22⋊(C83⋊C8)—41no—
85C4×C83⋊C8—41no—
86C83⋊#95—21no—
87C83⋊#95—21no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C22×D4×C83—21no—
92C22×D4×C83C2311yescharacteristic
93C23×C4×C83—21no—
94C23×C4×C83C2311yescharacteristic
95#121×C83—22no—
96#121×C83—22no—
97C4×D4×C83—22no—
98C4×D4×C83—41no—
99#117×C83—22no—
100#117×C83—41no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C22⋊C4×C83—21no—
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83—21no—
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83D411yescharacteristic
111C2×C22⋊C4×C83D411yescharacteristic
112C2×C42×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116#95×C83—21no—
117#95×C83—21no—
118#95×C83D411yescharacteristic
119#95×C83D411yescharacteristic
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C2×C83⋊C8—21no—
125C2×C83⋊C8—21no—
126C2×C83⋊C8—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—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#4112yes—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—81no—
145C4⋊Dic83—81no—
146Dic83⋊C4—42no—
147Dic83⋊C4—81no—
148C22⋊Dic83—81no—
149C22⋊Dic83—81no—
150C4×Dic83—81no—
151C24×C83—21no—
152C24×C83C2×D411yescharacteristic
153C2×D4×C83—24no—
154C2×D4×C83—21no—
155C2×D4×C83—21no—
156C2×D4×C83—21no—
157C2×D4×C83—21no—
158C2×D4×C83—21no—
159C2×D4×C83—81no—
160C2×D4×C83—81no—
161C2×D4×C83#4112yes—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—42no—
172C22×C4×C83—42no—
173C22×C4×C83—81no—
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2411yescharacteristic
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—81no—
185C4⋊C4×C83#4112yes—
186C4⋊C4×C83#4112yes—
187C22⋊C4×C83—24no—
188C22⋊C4×C83—22no—
189C22⋊C4×C83—22no—
190C22⋊C4×C83—42no—
191C22⋊C4×C83—42no—
192C22⋊C4×C83—42no—
193C22⋊C4×C83—81no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—42no—
199C42×C83—81no—
200C23×C83—22no—
201C23×C83—21no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83—41no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83—41no—
214C23×C83C22×D411yescharacteristic
215D4×C83—22no—
216D4×C83—22no—
217D4×C83—41no—
218D4×C83—41no—
219D4×C83—84no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—22no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—84no—
247C2×C4×C83#13512yes—
248C2×C4×C83#13712yes—
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83H(5,𝔽2)11yescharacteristic
254C2×C4×C832−1+411yescharacteristic
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C83⋊C8—22no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83—42no—
275C22×C83—41no—
276C22×C83—42no—
277C22×C83—41no—
278C22×C83—42no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—81no—
282C22×C83—81no—
283C22×C83—81no—
284C22×C83—81no—
285C22×C83#52511yescharacteristic
286C22×C83#54611yescharacteristic
287C22×C83#54711yescharacteristic
288C22×C83C2×#13511yescharacteristic
289C22×C83C2×#13711yescharacteristic
290C22×C83#57611yescharacteristic
291C22×C83#57611yescharacteristic
292C4×C83—22no—
293C4×C83—22no—
294C4×C83—41no—
295C4×C83—42no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83—42no—
299C4×C83—81no—
300C4×C83#52911yescharacteristic
301C4×C83#53911yescharacteristic
302C4×C83#55311yescharacteristic
303C4×C83#55611yescharacteristic
304Dic83—81no—
305Dic83—81no—
306Dic83—81no—
307Dic83—81no—
308#27041—831nomaximal, Sylow
309C2×C83—81no—
310C2×C83—82no—
311C2×C83#246611yescharacteristic
312C2×C83#310711yescharacteristic
313C2×C83#312211yescharacteristic
314C2×C83#330511yescharacteristic
315C2×C83#331411yescharacteristic
316C2×C83#333911yescharacteristic
317C2×C83#335711yescharacteristic
318#2497—831no—
319#2363D8311yescharacteristic
320#2071—831no—
321#2067—831no—
322#2033—831no—
323#2033—831no—
324#1947—831no—
325#1942—831no—
326#1923—831no—
327#1921—831no—
328#1919—831no—
329#1864—831no—
330#1854—831no—
331#1825—831no—
332C4×#102—831no—
333C83#2704111yescharacteristic, complemented, Sylow
334C2×#121C2×D8311yescharacteristic
335C2×C4×D4C2×D8311yescharacteristic
336C2×#117C2×D8311yescharacteristic
337C2×#107—831no—
338C2×#107—831no—
339C2×#107—831no—
340C2×#106—831no—
341C2×C4⋊C8—831no—
342C2×#102—831no—
343C2×#102—831no—
344C2×#102—831no—
345C2×#102—831no—
346C2×C22⋊C8—831no—
347C2×C22⋊C8—831no—
348C2×C4×C8—831no—
349#400—1661no—
350#396—1661no—
351#390—21no—
352#390C2×D8311yescharacteristic
353#389—831no—
354#389—831no—
355#389C2×D8311yescharacteristic
356#388—21no—
357#388—1661no—
358#387—1661no—
359#386—21no—
360#384—831no—
361#380—831no—
362#380—831no—
363C4×C4⋊C4—831no—
364C4×C4⋊C4C2×D8311yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C2×D8311yescharacteristic
367#340—831no—
368#340—831no—
369#340—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#340—831no—
375#336—831no—
376#336—831no—
377C22×D4—21no—
378C22×D4C22×D8311yescharacteristic
379C23×C4—21no—
380C23×C4C22×D8311yescharacteristic
381C22×C8—831no—
382C22×C8—831no—
383C22×C8—831no—
384C22×C8—831no—
385#121—22no—
386#121—22no—
387C4×D4—22no—
388C4×D4—41no—
389#117—22no—
390#117—41no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4C22×D8311yescharacteristic
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1661no—
409C2×C22⋊C4—1661no—
410C2×C22⋊C4C22⋊D8311yescharacteristic
411C2×C22⋊C4C22⋊D8311yescharacteristic
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—1661no—
415C2×C42C22×D8311yescharacteristic
416C2×C42C22×D8311yescharacteristic
417#107—3321no—
418#107—1662no—
419#107—1662no—
420#106—1662no—
421C4⋊C8—1662no—
422#102—3321no—
423#102—1662no—
424#102—1662no—
425#102—1662no—
426C22⋊C8—3321no—
427C22⋊C8—1662no—
428C4×C8—3321no—
429#95—21no—
430#95—21no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95C22⋊D8311yescharacteristic
437#95C22⋊D8311yescharacteristic
438C24—21no—
439C24C2×C22⋊D8311yescharacteristic
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4#1305059012yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4C2×C22⋊D8311yescharacteristic
468C22×C4C2×C22⋊D8311yescharacteristic
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4C23×D8311yescharacteristic
471C2×C8—832no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—81no—
488C4⋊C4—3322no—
489C4⋊C4—6641no—
490C4⋊C4—6641no—
491C4⋊C4—6641no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4#1305058912yes—
501C4⋊C4#1305059012yes—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—6641no—
510C22⋊C4—6641no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—81no—
517C42—6641no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×C22⋊D8311yescharacteristic
533D4—22no—
534D4—22no—
535D4—41no—
536D4—41no—
537D4—84no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4#1458181212yes—
578C2×C4#1458182212yes—
579C2×C4#41×D8311yescharacteristic
580C2×C4C2×#1305058911yescharacteristic
581C2×C4C2×#1305059011yescharacteristic
582C2×C4C2×#1305059011yescharacteristic
583C2×C4#1458183711yescharacteristic
584C2×C4#1458183911yescharacteristic
585C8—3321no—
586C8—3321no—
587C8—3321no—
588C8—1662no—
589C22—41no—
590C22—41no—
591C22—42no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—42no—
598C22—41no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1912821211yescharacteristic
604C22#1912834511yescharacteristic
605C22#1912835011yescharacteristic
606C22C2×#1458181211yescharacteristic
607C22C2×#1458182211yescharacteristic
608C22#1912850411yescharacteristic
609C22#1912850411yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—6641no—
619C4—6641no—
620C4—6641no—
621C4—6641no—
622C4#1912823111yescharacteristic
623C4#1912829311yescharacteristic
624C4#1912840111yescharacteristic
625C4#1912842111yescharacteristic
626C2—81no—
627C2—82no—
628C2#2491149311yescharacteristic
629C2#2491753411yescharacteristic
630C2#2491770611yescharacteristic
631C2#2491941311yescharacteristic
632C2#2491954411yescharacteristic
633C2#2491985911yescharacteristic
634C2#2492008311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2363C2
socleC23×C83C22×D4
prime radicalC2×#121×C83C22
hypercenter#2363D83
hypocenterC83#27041
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. #24911896C2semidirect
2. #24908866C2semidirect
3. #24908851C2semidirect
6. #24908221C2semidirect
7. #24908193C2semidirect
8. #24908071C2semidirect
9. #24908052C2semidirect
10. #24908041C2semidirect
11. #24907724C2semidirect
12. #24907662C2semidirect
13. #24907558C2semidirect
14. C4×#14581713C2semidirect
333. C83#27041semidirect

Maximal subgroups

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

subgroupindex
#249118962
#249088662
#249088512
#249086422
#249082212
#249081932
#249080712
#249080522
#249080412
#249077242
#249076622
#249075582
C4×#145817132
C83×#23632
#2704183