Finite Group Databasebeta

← #35467171

Subgroups — #35467171

Order 21248.

820 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127983C211yesmaximal, characteristic, complemented
2C2×#19127744C211yesmaximal, characteristic
3#24910466C211yesmaximal, characteristic
4#24909014C211yesmaximal, characteristic, complemented
5#24908959C211yesmaximal, characteristic, complemented
6#24908841C211yesmaximal, characteristic, complemented
7#24908793C211yesmaximal, characteristic, complemented
8#24908229C211yesmaximal, characteristic, complemented
9#24908045C211yesmaximal, characteristic, complemented
10#24907894C211yesmaximal, characteristic
11#24907775C211yesmaximal, characteristic
12#24907716C211yesmaximal, characteristic, complemented
13#24907655C211yesmaximal, characteristic
14#24907648C211yesmaximal, characteristic
15C83×#2439C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581771C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C22×C4⋊Dic83C2211yescharacteristic
20#19127983—24no—
21#19127744—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581715C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32C2×#14581715C2211yescharacteristic
33C2×#14581715C2211yescharacteristic
34C2×C22⋊(C83⋊C8)C2211yescharacteristic
35#19127607C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127598—21no—
38#19127588—21no—
39#19127573—21no—
40C4×C4⋊Dic83C2211yescharacteristic
41C22⋊C4×Dic83—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4⋊Q8×C83C2211yescharacteristic
51C2×#117×C83C2211yescharacteristic
52#396×C83—21no—
53#393×C83—21no—
54#391×C83—21no—
55#389×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#387×C83—21no—
58#382×C83C2211yescharacteristic
59#380×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×Dic166C2311yescharacteristic
62C23×Dic83—21no—
63C2×#13050574—22no—
64C2×#13050574—22no—
65C2×#13050574—22no—
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581771—22no—
71#14581771Q814yescomplemented
72C4×Dic166—22no—
73C4×Dic166—41no—
74C2×C4⋊Dic83—22no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—41no—
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×C22⋊Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86#14581725—22no—
87#14581725—22no—
88#14581723—22no—
89#14581723—22no—
90#14581723—41no—
91#14581723D414yes—
92C4⋊(C83⋊C8)—22no—
93C4⋊(C83⋊C8)—22no—
94#14581715—22no—
95#14581715—22no—
96#14581715—22no—
97#14581715—41no—
98C22⋊(C83⋊C8)—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C22×Q8×C83—21no—
104C23×C4×C83C2311yescharacteristic
105C4⋊Q8×C83—22no—
106C4⋊Q8×C83—41no—
107#117×C83—22no—
108#117×C83—41no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83D411yescharacteristic
119C2×C22⋊C4×C83D411yescharacteristic
120C2×C42×C83—21no—
121C2×C42×C83C2311yescharacteristic
122C2×C42×C83C2311yescharacteristic
123#95×C83—21no—
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83D411yescharacteristic
129#95×C83D411yescharacteristic
130C2×Dic166—21no—
131C2×Dic166—21no—
132C2×Dic166—21no—
133C2×Dic166—21no—
134C2×Dic166—21no—
135C2×Dic166C2×Q812yes—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—42no—
141C22×Dic83—42no—
142C22×Dic83—81no—
143#13050574—28no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8C2×D412yes—
155C2×C83⋊C8#4112yes—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—42no—
161C4⋊Dic83C2×D412yes—
162C4⋊Dic83C2×Q812yes—
163Dic83⋊C4—24no—
164Dic83⋊C4—42no—
165Dic83⋊C4—81no—
166C22⋊Dic83—24no—
167C4×Dic83—42no—
168C4×Dic83—81no—
169C24×C83C2×D411yescharacteristic
170C2×Q8×C83—42no—
171C2×Q8×C83—42no—
172C2×Q8×C83—81no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×Q812yes—
187C22×C4×C83#4112yes—
188C22×C4×C83C2411yescharacteristic
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—42no—
198C4⋊C4×C83—81no—
199C4⋊C4×C83C2×D412yes—
200C22⋊C4×C83—22no—
201C22⋊C4×C83—41no—
202C22⋊C4×C83—42no—
203C22⋊C4×C83—81no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—21no—
208C42×C83—42no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83C22⋊Q812yes—
216C23×C83C22×D411yescharacteristic
217Q8×C83—84no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—21no—
227C2×C4×C83—22no—
228C2×C4×C83—21no—
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—21no—
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×C83C22⋊Q812yes—
248C2×C4×C83#13712yes—
249C2×C4×C83C22×D411yescharacteristic
250C2×C4×C83C22×Q811yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83H(5,𝔽2)11yescharacteristic
254C2×C4×C832−1+411yescharacteristic
255C2×Dic83—22no—
256C2×Dic83—22no—
257C2×Dic83—22no—
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—84no—
268Dic166—22no—
269Dic166—22no—
270Dic166—41no—
271Dic166—41no—
272C83⋊C8—22no—
273C83⋊C8—22no—
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×C83C2×C22⋊Q811yescharacteristic
284C22×C83D4211yescharacteristic
285C22×C83#54711yescharacteristic
286C22×C83C2×#13711yescharacteristic
287C22×C83#57511yescharacteristic
288C22×C83#57711yescharacteristic
289C22×C83#57711yescharacteristic
290C4×C83—22no—
291C4×C83—22no—
292C4×C83—41no—
293C4×C83—41no—
294C4×C83—42no—
295C4×C83—41no—
296C4×C83—81no—
297C4×C83—81no—
298C4×C83—81no—
299C4×C83#52511yescharacteristic
300C4×C83#53911yescharacteristic
301C4×C83D4×Q811yescharacteristic
302C4×C83#55411yescharacteristic
303Dic83—42no—
304Dic83—42no—
305Dic83—42no—
306Dic83—81no—
307#26356—831nomaximal, Sylow
308C2×C83—42no—
309C2×C83#244911yescharacteristic
310C2×C83#307811yescharacteristic
311C2×C83#308411yescharacteristic
312C2×C83#326611yescharacteristic
313C2×C83#327511yescharacteristic
314C2×C83#334811yescharacteristic
315C2×C83#335611yescharacteristic
316C2×#475—831no—
317C2×#428—831no—
318#2439D8311yescharacteristic
319#2360—831no—
320#2097—831no—
321#2087—831no—
322#2066—831no—
323#2058—831no—
324#1949—831no—
325#1920—831no—
326#1894—831no—
327#1873—831no—
328#1863—831no—
329#1853—831no—
330#1851—831no—
331C83#2635611yescharacteristic, complemented, Sylow
332C22×M16—831no—
333C2×C4⋊Q8C2×D8311yescharacteristic
334C2×C22⋊Q8—831no—
335C2×C4×Q8—831no—
336C2×#117C2×D8311yescharacteristic
337C22×C4⋊C4—831no—
338#475—1664no—
339#428—1664no—
340C2×#107—831no—
341C2×#107—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×#106—831no—
345C2×#106—831no—
346C2×C4⋊C8—831no—
347C2×C4⋊C8—831no—
348C2×#103—831no—
349C2×#103—831no—
350C2×#103—831no—
351C2×#103—831no—
352C2×C22⋊C8—831no—
353#396—21no—
354#393—21no—
355#391—21no—
356#391—831no—
357#389—831no—
358#389—1661no—
359#389C2×D8311yescharacteristic
360#389C2×D8311yescharacteristic
361#387—21no—
362#387—1661no—
363#385—1661no—
364#382C2×D8311yescharacteristic
365#380C2×D8311yescharacteristic
366C4×C4⋊C4—831no—
367C4×C22⋊C4—1661no—
368C4×C22⋊C4C2×D8311yescharacteristic
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×Q8—21no—
378C22×Q8—831no—
379C23×C4—1661no—
380C23×C4C22×D8311yescharacteristic
381C2×M16—1662no—
382C2×M16—1662no—
383C2×M16—1662no—
384C22×C8—831no—
385C22×C8—831no—
386C22×C8—831no—
387C22×C8—831no—
388C4⋊Q8—22no—
389C4⋊Q8—41no—
390C22⋊Q8—834no—
391C22⋊Q8—1662no—
392C4×Q8—3321no—
393C4×Q8—1662no—
394#117—22no—
395#117—41no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—3321no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1662no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4C22×D8311yescharacteristic
409C2×C4⋊C4C22×D8311yescharacteristic
410C2×C4⋊C4C22×D8311yescharacteristic
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1661no—
415C2×C22⋊C4C22⋊D8311yescharacteristic
416C2×C22⋊C4C22⋊D8311yescharacteristic
417C2×C42—21no—
418C2×C42—1661no—
419C2×C42—1661no—
420C2×C42C22×D8311yescharacteristic
421C2×C42C22×D8311yescharacteristic
422#107—1662no—
423#107—1662no—
424#106—3321no—
425#106—834no—
426#106—1662no—
427#106—1662no—
428C4⋊C8—1662no—
429C4⋊C8—1662no—
430#103—3321no—
431#103—1662no—
432#103—1662no—
433#103—1662no—
434C22⋊C8—1662no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—1661no—
441#95—1661no—
442#95—1661no—
443#95—1661no—
444#95C22⋊D8311yescharacteristic
445#95C22⋊D8311yescharacteristic
446C24C2×C22⋊D8311yescharacteristic
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—81no—
450C2×Q8—832no—
451C2×Q8—1661no—
452C2×Q8—1661no—
453C2×Q8—1661no—
454C2×Q8—1661no—
455C2×Q8—1661no—
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—3322no—
465C22×C4—3322no—
466C22×C4—6641no—
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×C4C2×C22⋊D8311yescharacteristic
476C22×C4Q8×D8312yes—
477C22×C4#1305059112yes—
478C22×C4C23×D8311yescharacteristic
479M16—1668no—
480C2×C8—832no—
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—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—3322no—
503C4⋊C4—3322no—
504C4⋊C4—832no—
505C4⋊C4—832no—
506C4⋊C4—6641no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4—1664no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4D4×D8312yes—
513C22⋊C4—22no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—1664no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—42no—
523C42—3322no—
524C42—6641no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23#1458176812yes—
532C23C22×C22⋊D8311yescharacteristic
533Q8—84no—
534Q8—3321no—
535Q8—3321no—
536Q8—1662no—
537Q8—1662no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
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—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—6644no—
577C2×C4—1662no—
578C2×C4—1662no—
579C2×C4—1662no—
580C2×C4#1458176812yes—
581C2×C4#1458182212yes—
582C2×C4C2×D4×D8311yescharacteristic
583C2×C4C2×Q8×D8311yescharacteristic
584C2×C4#41×D8311yescharacteristic
585C2×C4C2×#1305059111yescharacteristic
586C2×C4#1458183711yescharacteristic
587C2×C4#1458183911yescharacteristic
588C8—3321no—
589C8—3321no—
590C8—1662no—
591C8—1662no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22C2×#1458176811yescharacteristic
600C22D4×C22⋊D8311yescharacteristic
601C22#1912835011yescharacteristic
602C22C2×#1458182211yescharacteristic
603C22#1912849911yescharacteristic
604C22#1912850911yescharacteristic
605C22#1912851211yescharacteristic
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—3322no—
616C4—3322no—
617C4—3322no—
618C4—6641no—
619C4#1912821411yescharacteristic
620C4#1912830011yescharacteristic
621C4#1912836411yescharacteristic
622C4#1912840511yescharacteristic
623C2—42no—
624C2#2491126811yescharacteristic
625C2#2491726411yescharacteristic
626C2#2491732111yescharacteristic
627C2#2491896911yescharacteristic
628C2#2491909511yescharacteristic
629C2#2491997811yescharacteristic
630C2#2492007211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2439C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2439D83
hypocenterC83#26356
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
1. C2×#19127983C2semidirect
4. #24909014C2semidirect
5. #24908959C2semidirect
6. #24908841C2semidirect
7. #24908793C2semidirect
8. #24908229C2semidirect
9. #24908045C2semidirect
12. #24907716C2semidirect
71. #14581771Q8semidirect
331. C83#26356semidirect

Maximal subgroups

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

subgroupindex
C2×#191279832
C2×#191277442
#249104662
#249090142
#249089592
#249088412
#249087932
#249082292
#249080452
#249078942
#249077752
#249077162
#249076552
#249076482
C83×#24392
#2635683