Finite Group Databasebeta

← #35477055

Subgroups — #35477055

Order 21248.

812 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24911885C211yesmaximal, 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#24908034C211yesmaximal, characteristic, complemented
10#24908031C211yesmaximal, characteristic, complemented
11#24907710C211yesmaximal, characteristic, complemented
12#24907651C211yesmaximal, characteristic, complemented
13#24907549C211yesmaximal, characteristic, complemented
14C4×#14581713C211yesmaximal, characteristic, complemented
15C83×#2361C211yesmaximal, characteristic
16C2×#14581785—21no—
17C2×#14581725C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×C22⋊(C83⋊C8)C2211yescharacteristic
27C2×C22⋊(C83⋊C8)C2211yescharacteristic
28C2×C4×C83⋊C8C2211yescharacteristic
29#19127641—21no—
30#19127626—21no—
31#19127599C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127586—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×C83—21no—
50C2×C4×D4×C83C2211yescharacteristic
51C22×C4⋊C4×C83C2211yescharacteristic
52#390×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#388×C83—21no—
55#386×C83—21no—
56#385×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59C22×C83⋊C8C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63#14581785—44no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×Dic83⋊C4—21no—
71C2×Dic83⋊C4—21no—
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C4×Dic83—21no—
75#14581725—22no—
76#14581725—22no—
77#14581725—22no—
78#14581725—41no—
79C4⋊(C83⋊C8)—22no—
80#14581713—22no—
81#14581713—22no—
82#14581713—22no—
83#14581713—41no—
84C22⋊(C83⋊C8)—22no—
85C22⋊(C83⋊C8)—41no—
86C4×C83⋊C8—41no—
87C83⋊#95—21no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C22×D4×C83—21no—
93C22×D4×C83C2311yescharacteristic
94C23×C4×C83—21no—
95C23×C4×C83—21no—
96C23×C4×C83C2311yescharacteristic
97#121×C83—22no—
98#121×C83—22no—
99C4×D4×C83—22no—
100C4×D4×C83—41no—
101C4×D4×C83—44no—
102C2×C4⋊C4×C83—22no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—41no—
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83D411yescharacteristic
114C2×C22⋊C4×C83D411yescharacteristic
115C2×C42×C83—21no—
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118#95×C83—21no—
119#95×C83—21no—
120#95×C83D411yescharacteristic
121#95×C83D411yescharacteristic
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—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—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8#4112yes—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—42no—
147C4⋊Dic83—42no—
148Dic83⋊C4—42no—
149Dic83⋊C4—42no—
150C22⋊Dic83—42no—
151C22⋊Dic83—42no—
152C4×Dic83—42no—
153C24×C83—21no—
154C24×C83C2×D411yescharacteristic
155C2×D4×C83—24no—
156C2×D4×C83—21no—
157C2×D4×C83—21no—
158C2×D4×C83—21no—
159C2×D4×C83—21no—
160C2×D4×C83—21no—
161C2×D4×C83—42no—
162C2×D4×C83—42no—
163C2×D4×C83#4112yes—
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—21no—
172C22×C4×C83—21no—
173C22×C4×C83—42no—
174C22×C4×C83—42no—
175C22×C4×C83—42no—
176C22×C4×C83—42no—
177C22×C4×C83—81no—
178C22×C4×C83—81no—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2411yescharacteristic
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83#4112yes—
192C4⋊C4×C83#4112yes—
193C22⋊C4×C83—24no—
194C22⋊C4×C83—41no—
195C22⋊C4×C83—42no—
196C22⋊C4×C83—42no—
197C22⋊C4×C83—81no—
198C22⋊C4×C83D814yes—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—42no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—22no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83—41no—
214C23×C83—41no—
215C23×C83—41no—
216C23×C83—41no—
217C23×C83C2×D812yes—
218C23×C83C22×D411yescharacteristic
219D4×C83—22no—
220D4×C83—22no—
221D4×C83—48no—
222D4×C83—41no—
223D4×C83—41no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—22no—
230C2×C4×C83—21no—
231C2×C4×C83—22no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
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—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—84no—
251C2×C4×C83—84no—
252C2×C4×C83C2×D812yes—
253C2×C4×C83C2×D812yes—
254C2×C4×C83C2×#4111yescharacteristic
255C2×C4×C83C2×#4111yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C2×C4×C83C2×#4111yescharacteristic
258C2×C4×C83H(5,𝔽2)11yescharacteristic
259C2×C4×C832−1+411yescharacteristic
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C83⋊C8—22no—
273C83⋊C8—41no—
274C83⋊C8—41no—
275C83⋊C8—41no—
276C22×C83—22no—
277C22×C83—22no—
278C22×C83—22no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—41no—
282C22×C83—41no—
283C22×C83—42no—
284C22×C83—42no—
285C22×C83—42no—
286C22×C83—81no—
287C22×C83—81no—
288C22×C83—81no—
289C22×C83—81no—
290C22×C83#52511yescharacteristic
291C22×C83#54611yescharacteristic
292C22×C83#54711yescharacteristic
293C22×C83C22×D811yescharacteristic
294C22×C83#57511yescharacteristic
295C22×C83#57711yescharacteristic
296C22×C83#57811yescharacteristic
297C4×C83—22no—
298C4×C83—22no—
299C4×C83—41no—
300C4×C83—42no—
301C4×C83—41no—
302C4×C83—42no—
303C4×C83—41no—
304C4×C83—81no—
305C4×C83#52911yescharacteristic
306C4×C83#53911yescharacteristic
307C4×C83#55311yescharacteristic
308C4×C83#55611yescharacteristic
309Dic83—81no—
310Dic83—81no—
311Dic83—81no—
312Dic83—81no—
313#27030—831nomaximal, Sylow
314C2×C83—42no—
315C2×C83—82no—
316C2×C83#246611yescharacteristic
317C2×C83#310011yescharacteristic
318C2×C83#312011yescharacteristic
319C2×C83#329411yescharacteristic
320C2×C83#331311yescharacteristic
321C2×C83#333411yescharacteristic
322C2×C83#335111yescharacteristic
323#2496—831no—
324#2361D8311yescharacteristic
325#2071—831no—
326#2067—831no—
327#2033—831no—
328#2033—831no—
329#1947—831no—
330#1942—831no—
331#1923—831no—
332#1918—831no—
333#1918—831no—
334#1862—831no—
335#1852—831no—
336#1823—831no—
337C4×#102—831no—
338C83#2703011yescharacteristic, complemented, Sylow
339C2×#126—1661no—
340C2×#121C2×D8311yescharacteristic
341C2×C4×D4—21no—
342C2×C4×D4C2×D8311yescharacteristic
343C22×C4⋊C4C2×D8311yescharacteristic
344C2×#107—831no—
345C2×#107—831no—
346C2×#107—831no—
347C2×#107—831no—
348C2×C4⋊C8—831no—
349C2×#102—831no—
350C2×#102—831no—
351C2×#102—831no—
352C2×#102—831no—
353C2×C22⋊C8—831no—
354C2×C22⋊C8—831no—
355C2×C4×C8—831no—
356#400—1661no—
357#397—1661no—
358#390C2×D8311yescharacteristic
359#389—831no—
360#389—831no—
361#389C2×D8311yescharacteristic
362#388—21no—
363#387—1661no—
364#386—21no—
365#385—21no—
366#384—831no—
367#380—831no—
368#380—831no—
369C4×C4⋊C4—831no—
370C4×C4⋊C4C2×D8311yescharacteristic
371C4×C22⋊C4C2×D8311yescharacteristic
372#340—831no—
373#340—831no—
374#340—831no—
375#340—831no—
376#340—831no—
377#340—831no—
378#340—831no—
379#340—831no—
380#336—831no—
381#336—831no—
382C22×D4—21no—
383C22×D4C22×D8311yescharacteristic
384C23×C4—21no—
385C23×C4—21no—
386C23×C4C22×D8311yescharacteristic
387C22×C8—831no—
388C22×C8—831no—
389C22×C8—831no—
390C22×C8—831no—
391#126—3324no—
392#121—22no—
393#121—22no—
394C4×D4—22no—
395C4×D4—41no—
396C4×D4—44no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—41no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—831no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4C22×D8311yescharacteristic
410C2×C4⋊C4C22×D8311yescharacteristic
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1661no—
417C2×C22⋊C4—1661no—
418C2×C22⋊C4C22⋊D8311yescharacteristic
419C2×C22⋊C4C22⋊D8311yescharacteristic
420C2×C42—21no—
421C2×C42—1661no—
422C2×C42C22×D8311yescharacteristic
423C2×C42C22×D8311yescharacteristic
424#107—3321no—
425#107—1662no—
426#107—1662no—
427#107—1662no—
428C4⋊C8—1662no—
429#102—3321no—
430#102—1662no—
431#102—1662no—
432#102—1662no—
433C22⋊C8—3321no—
434C22⋊C8—1662no—
435C4×C8—3321no—
436#95—21no—
437#95—21no—
438#95—1661no—
439#95—1661no—
440#95—1661no—
441#95—1661no—
442#95—1661no—
443#95C22⋊D8311yescharacteristic
444#95C22⋊D8311yescharacteristic
445C24—21no—
446C24C2×C22⋊D8311yescharacteristic
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4#1305059012yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4C2×C22⋊D8311yescharacteristic
476C22×C4C2×C22⋊D8311yescharacteristic
477C22×C4C2×C22⋊D8311yescharacteristic
478C22×C4C2×C22⋊D8311yescharacteristic
479C22×C4C2×C22⋊D8311yescharacteristic
480C22×C4C23×D8311yescharacteristic
481C2×C8—832no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—3322no—
500C4⋊C4—3322no—
501C4⋊C4—3322no—
502C4⋊C4—3322no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4—1661no—
510C4⋊C4—1661no—
511C4⋊C4#1305058912yes—
512C4⋊C4#1305059012yes—
513C22⋊C4—24no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—3322no—
519C22⋊C4—3322no—
520C22⋊C4#1305057514yes—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—3322no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×#1305057512yes—
541C23C22×C22⋊D8311yescharacteristic
542D4—22no—
543D4—22no—
544D4—48no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—3321no—
587C2×C4C2×#1305057512yes—
588C2×C4C2×#1305057512yes—
589C2×C4#41×D8311yescharacteristic
590C2×C4C2×#1305058911yescharacteristic
591C2×C4C2×#1305059011yescharacteristic
592C2×C4C2×#1305059011yescharacteristic
593C2×C4#1458183711yescharacteristic
594C2×C4#1458183911yescharacteristic
595C8—3321no—
596C8—3321no—
597C8—3321no—
598C8—1662no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#1912821211yescharacteristic
614C22#1912834511yescharacteristic
615C22#1912835011yescharacteristic
616C22C22×#1305057511yescharacteristic
617C22#1912849911yescharacteristic
618C22#1912851211yescharacteristic
619C22#1912851611yescharacteristic
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—81no—
628C4—6641no—
629C4—6641no—
630C4—6641no—
631C4—6641no—
632C4#1912823111yescharacteristic
633C4#1912829311yescharacteristic
634C4#1912840111yescharacteristic
635C4#1912842111yescharacteristic
636C2—42no—
637C2—82no—
638C2#2491149311yescharacteristic
639C2#2491746311yescharacteristic
640C2#2491767711yescharacteristic
641C2#2491927611yescharacteristic
642C2#2491952911yescharacteristic
643C2#2491980611yescharacteristic
644C2#2492001711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2361C2
socleC23×C83C22×D4
prime radicalC2×#121×C83C22
hypercenter#2361D83
hypocenterC83#27030
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #24911885C2semidirect
2. #24908866C2semidirect
3. #24908851C2semidirect
6. #24908221C2semidirect
7. #24908193C2semidirect
8. #24908071C2semidirect
9. #24908034C2semidirect
10. #24908031C2semidirect
11. #24907710C2semidirect
12. #24907651C2semidirect
13. #24907549C2semidirect
14. C4×#14581713C2semidirect
338. C83#27030semidirect

Maximal subgroups

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

subgroupindex
#249118852
#249088662
#249088512
#249086422
#249082212
#249081932
#249080712
#249080342
#249080312
#249077102
#249076512
#249075492
C4×#145817132
C83×#23612
#2703083