Finite Group Databasebeta

← #35466556

Subgroups — #35466556

Order 21248.

742 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912636C211yesmaximal, characteristic
2#24908900C211yesmaximal, characteristic, complemented
3#24908894C211yesmaximal, characteristic, complemented
4#24908894C211yesmaximal, characteristic, complemented
5#24908889C211yesmaximal, characteristic, complemented
6#24908435C211yesmaximal, characteristic, complemented
7#24908415C211yesmaximal, characteristic, complemented
8#24908262C211yesmaximal, characteristic
9#24908229C211yesmaximal, characteristic, complemented
10#24907662C211yesmaximal, characteristic, complemented
11#24907656C211yesmaximal, characteristic, complemented
12#24907558C211yesmaximal, characteristic, complemented
13#24907549C211yesmaximal, characteristic, complemented
14#24907361C211yesmaximal, characteristic, complemented
15C83×#2352C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581725C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581723C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×C22⋊(C83⋊C8)C2211yescharacteristic
23C2×C22⋊(C83⋊C8)C2211yescharacteristic
24C2×C22⋊(C83⋊C8)C2211yescharacteristic
25C2×C22⋊(C83⋊C8)C2211yescharacteristic
26C2×#14581703C2211yescharacteristic
27#19127613—21no—
28#19127613—21no—
29#19127604—21no—
30#19127566—21no—
31#19127561C2211yescharacteristic
32#19127561C2211yescharacteristic
33#19127561C2211yescharacteristic
34#19127561C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48C2×C4×D4×C83C2211yescharacteristic
49C2×#117×C83C2211yescharacteristic
50C22×C4⋊C4×C83C2211yescharacteristic
51#388×C83—21no—
52#387×C83—21no—
53#386×C83—21no—
54#382×C83C2211yescharacteristic
55#380×C83C2211yescharacteristic
56#380×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C22×Dic166—21no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581792—22no—
65#14581792—22no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C22⋊Dic83—21no—
75C2×C4×Dic83—21no—
76#14581725—22no—
77#14581725—41no—
78#14581723—22no—
79#14581723—41no—
80C4⋊(C83⋊C8)—22no—
81C22⋊(C83⋊C8)—22no—
82C22⋊(C83⋊C8)—22no—
83C22⋊(C83⋊C8)—41no—
84C22⋊(C83⋊C8)—41no—
85#14581703—22no—
86C83⋊#95—21no—
87C83⋊#95—21no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C22×D4×C83D411yescharacteristic
92C23×C4×C83—21no—
93C23×C4×C83C2311yescharacteristic
94C23×C4×C83C2311yescharacteristic
95C4×D4×C83—22no—
96C4×D4×C83—22no—
97#117×C83—22no—
98#117×C83—41no—
99C2×C4⋊C4×C83—22no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—41no—
102C2×C4⋊C4×C83D411yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83C2311yescharacteristic
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
115#95×C83—21no—
116#95×C83—21no—
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120C2×Dic166—42no—
121C2×Dic166—81no—
122C2×Dic166—81no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—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—21no—
138C2×C83⋊C8#4112yes—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—42no—
148C22⋊Dic83—81no—
149C22⋊Dic83—81no—
150C22⋊Dic83—81no—
151C22⋊Dic83—81no—
152C4×Dic83—42no—
153C24×C83C2×D411yescharacteristic
154C24×C83C2×D411yescharacteristic
155C2×D4×C83—22no—
156C2×D4×C83—22no—
157C2×D4×C83—41no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
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—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—42no—
175C22×C4×C83—42no—
176C22×C4×C83—81no—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2411yescharacteristic
182C4⋊C4×C83—22no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83#4112yes—
192C22⋊C4×C83—22no—
193C22⋊C4×C83—22no—
194C22⋊C4×C83—42no—
195C22⋊C4×C83—42no—
196C22⋊C4×C83—81no—
197C42×C83—42no—
198C42×C83—81no—
199C42×C83#4112yes—
200C42×C83#4112yes—
201C23×C83—21no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83#13712yes—
215C23×C83C22×D411yescharacteristic
216D4×C83—44no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—22no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
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—41no—
247C2×C4×C83—84no—
248C2×C4×C83#12312yes—
249C2×C4×C83#12312yes—
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83C2×#4111yescharacteristic
254C2×C4×C83H(5,𝔽2)11yescharacteristic
255C2×C4×C832−1+411yescharacteristic
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—
267C2×Dic83—41no—
268Dic166—84no—
269C83⋊C8—22no—
270C83⋊C8—41no—
271C83⋊C8—41no—
272C83⋊C8—41no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—41no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—41no—
286C22×C83—41no—
287C22×C83C2×#12311yescharacteristic
288C22×C83#54611yescharacteristic
289C22×C83#54711yescharacteristic
290C22×C83C2×#13711yescharacteristic
291C22×C83#57511yescharacteristic
292C22×C83#57611yescharacteristic
293C22×C83#57711yescharacteristic
294C4×C83—22no—
295C4×C83—22no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83—42no—
299C4×C83—81no—
300C4×C83—81no—
301C4×C83—81no—
302C4×C83#52911yescharacteristic
303C4×C83#54211yescharacteristic
304C4×C83#55611yescharacteristic
305C4×C83#55911yescharacteristic
306Dic83—81no—
307Dic83—81no—
308Dic83—81no—
309Dic83—81no—
310#26313—831nomaximal, Sylow
311C2×C83—42no—
312C2×C83—81no—
313C2×C83#244711yescharacteristic
314C2×C83#309511yescharacteristic
315C2×C83#309611yescharacteristic
316C2×C83#331411yescharacteristic
317C2×C83#332111yescharacteristic
318C2×C83#334411yescharacteristic
319C2×C83#334711yescharacteristic
320#2563—831no—
321#2352D8311yescharacteristic
322#2077—831no—
323#2076—831no—
324#2076—831no—
325#2075—831no—
326#1993—831no—
327#1988—831no—
328#1955—831no—
329#1949—831no—
330#1854—831no—
331#1853—831no—
332#1825—831no—
333#1823—831no—
334#1776—831no—
335C83#2631311yescharacteristic, complemented, Sylow
336C2×C4⋊Q8—831no—
337C2×C4×D4C2×D8311yescharacteristic
338C2×#117C2×D8311yescharacteristic
339C22×C4⋊C4C2×D8311yescharacteristic
340C2×#107—831no—
341C2×#107—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×C4⋊C8—831no—
345C2×C22⋊C8—831no—
346C2×C22⋊C8—831no—
347C2×C22⋊C8—831no—
348C2×C22⋊C8—831no—
349C2×#97—831no—
350#393—1661no—
351#393—1661no—
352#390—1661no—
353#388—21no—
354#387—21no—
355#386—21no—
356#382—1661no—
357#382C2×D8311yescharacteristic
358#380—831no—
359#380—831no—
360#380—831no—
361#380—831no—
362#380C2×D8311yescharacteristic
363#380C2×D8311yescharacteristic
364C4×C4⋊C4—831no—
365C4×C4⋊C4C2×D8311yescharacteristic
366C4×C22⋊C4—21no—
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#340—831no—
376#340—831no—
377#340—831no—
378#340—831no—
379C22×Q8—1661no—
380C22×D4C22⋊D8311yescharacteristic
381C23×C4—21no—
382C23×C4C22×D8311yescharacteristic
383C23×C4C22×D8311yescharacteristic
384C22×C8—831no—
385C22×C8—831no—
386C22×C8—831no—
387C22×C8—831no—
388C4⋊Q8—1662no—
389C4⋊Q8—1662no—
390C4×D4—22no—
391C4×D4—22no—
392#117—22no—
393#117—41no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4C22⋊D8311yescharacteristic
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C4⋊C4C22×D8311yescharacteristic
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1661no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4—1661no—
413C2×C22⋊C4—1661no—
414C2×C22⋊C4C22⋊D8311yescharacteristic
415C2×C22⋊C4C22⋊D8311yescharacteristic
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—1661no—
419C2×C42C22×D8311yescharacteristic
420#107—3321no—
421#107—1662no—
422#106—3321no—
423#106—1662no—
424C4⋊C8—1662no—
425C22⋊C8—3321no—
426C22⋊C8—3321no—
427C22⋊C8—1662no—
428C22⋊C8—1662no—
429#97—1662no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440C24C2×C22⋊D8311yescharacteristic
441C24C2×C22⋊D8311yescharacteristic
442C2×Q8—3322no—
443C2×Q8—6641no—
444C2×Q8—6641no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—41no—
448C22×C4—21no—
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—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4C2×C22⋊D8311yescharacteristic
472C22×C4C2×C22⋊D8311yescharacteristic
473C22×C4C2×C22⋊D8311yescharacteristic
474C22×C4C2×C22⋊D8311yescharacteristic
475C22×C4C23×D8311yescharacteristic
476C2×C8—832no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—3322no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4#1305058912yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—81no—
512C22⋊C4—6641no—
513C22⋊C4—6641no—
514C22⋊C4—6641no—
515C22⋊C4—6641no—
516C42—42no—
517C42—81no—
518C42—3322no—
519C42#1305059012yes—
520C42#1305059012yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23#1458182212yes—
535C23C22×C22⋊D8311yescharacteristic
536Q8—6644no—
537D4—44no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
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—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
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—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4#1458177512yes—
582C2×C4#1458177512yes—
583C2×C4#41×D8311yescharacteristic
584C2×C4C2×#1305058911yescharacteristic
585C2×C4C2×#1305059011yescharacteristic
586C2×C4C2×#1305059011yescharacteristic
587C2×C4#1458183711yescharacteristic
588C2×C4#1458183911yescharacteristic
589C8—3321no—
590C8—3321no—
591C8—3321no—
592C8—1662no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1458177511yescharacteristic
608C22#1912834511yescharacteristic
609C22#1912835011yescharacteristic
610C22C2×#1458182211yescharacteristic
611C22#1912849911yescharacteristic
612C22#1912850411yescharacteristic
613C22#1912850911yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—6641no—
623C4—6641no—
624C4—6641no—
625C4—6641no—
626C4#1912823311yescharacteristic
627C4#1912831611yescharacteristic
628C4#1912842111yescharacteristic
629C4#1912843411yescharacteristic
630C2—42no—
631C2—81no—
632C2#2491124411yescharacteristic
633C2#2491741111yescharacteristic
634C2#2491742111yescharacteristic
635C2#2491954411yescharacteristic
636C2#2491964911yescharacteristic
637C2#2491991811yescharacteristic
638C2#2491995711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2352C2
socleC23×C83C22×D4
prime radicalC22×D4×C83D4
hypercenter#2352D83
hypocenterC83#26313
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2352

Complements

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

NH ≅ G/Ntype
2. #24908900C2semidirect
3. #24908894C2semidirect
4. #24908894C2semidirect
5. #24908889C2semidirect
6. #24908435C2semidirect
7. #24908415C2semidirect
9. #24908229C2semidirect
10. #24907662C2semidirect
11. #24907656C2semidirect
12. #24907558C2semidirect
13. #24907549C2semidirect
14. #24907361C2semidirect
335. C83#26313semidirect

Maximal subgroups

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

subgroupindex
#249126362
#249089002
#249088942
#249088892
#249084352
#249084152
#249082622
#249082292
#249076622
#249076562
#249075582
#249075492
#249073612
C83×#23522
#2631383