Finite Group Databasebeta

← #35463071

Subgroups — #35463071

Order 21248.

894 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128104C211yesmaximal, characteristic, complemented
2C2×#107×D83C211yesmaximal, characteristic
3#24911167C211yesmaximal, characteristic
4#24911161C211yesmaximal, characteristic
5#24908938C211yesmaximal, characteristic, complemented
6#24908937C211yesmaximal, characteristic, complemented
7#24908792C211yesmaximal, characteristic, complemented
8#24908195C211yesmaximal, characteristic, complemented
9#24908193C211yesmaximal, characteristic, complemented
10#24907652C211yesmaximal, characteristic
11#24907649C211yesmaximal, characteristic
12#24907621C211yesmaximal, characteristic, complemented
13#24907491C211yesmaximal, characteristic
14#24907233C211yesmaximal, characteristic
15C83×#2057C211yesmaximal, characteristic, complemented
16C22×C8×D83C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581770—21no—
19C2×#14581768—21no—
20C2×C4⋊C4×D83C2211yescharacteristic
21C2×C4⋊C4×D83C2211yescharacteristic
22#19128104—24no—
23C2×#14581726C2211yescharacteristic
24C2×#14581726C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581725C2211yescharacteristic
28C2×#14581725C2211yescharacteristic
29C2×#14581725C2211yescharacteristic
30C2×#14581725C2211yescharacteristic
31#107×D83—24no—
32C2×(C2×D83)⋊C8C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34C2×C8×Dic83C2211yescharacteristic
35#19127635—21no—
36#19127634—21no—
37#19127621—21no—
38#19127620—21no—
39#19127600C2211yescharacteristic
40#19127599C2211yescharacteristic
41#19127597C2211yescharacteristic
42#19127575—21no—
43#19127572—21no—
44#19127570C2211yescharacteristic
45#19127570C2211yescharacteristic
46#19127558C2211yescharacteristic
47#19127558C2211yescharacteristic
48C4×(C2×D83)⋊C4C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127463C2211yescharacteristic
53#19127455C2211yescharacteristic
54C2×C4⋊Q8×C83C2211yescharacteristic
55C2×#107×C83C2211yescharacteristic
56C2×#107×C83C2211yescharacteristic
57#389×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60#336×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×C4×D83—21no—
63C22×C4×D83—21no—
64C22×C4×D83C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C2×C8×D83—22no—
68C2×C8×D83—22no—
69C2×C8×D83Q814yescomplemented
70#14581792—22no—
71#14581792—41no—
72#14581770—44no—
73#14581768—44no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×Dic83⋊C4—21no—
87C2×Dic83⋊C4—21no—
88C4⋊C4×D83—22no—
89C4⋊C4×D83—22no—
90C4⋊C4×D83—41no—
91C4⋊C4×D83—41no—
92C2×(C2×D83)⋊C4—21no—
93C2×(C2×D83)⋊C4—21no—
94C2×(C2×D83)⋊C4D411yescharacteristic
95C2×(C2×D83)⋊C4D411yescharacteristic
96C2×C4×Dic83C2311yescharacteristic
97#14581726—22no—
98#14581726—41no—
99#14581725—22no—
100#14581725—22no—
101#14581725—22no—
102#14581725—22no—
103#14581725—22no—
104#14581725D414yes—
105(C2×D83)⋊C8Q814yescomplemented
106C4×C83⋊C8—22no—
107C8×Dic83—22no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C83⋊#95—21no—
112C83⋊#95—21no—
113C83⋊#95D411yescharacteristic
114C83⋊#95D411yescharacteristic
115C22×Q8×C83—21no—
116C22×C8×C83C2311yescharacteristic
117C22×C8×C83C2311yescharacteristic
118C4⋊Q8×C83—22no—
119C4⋊Q8×C83—41no—
120C2×C4⋊C4×C83—21no—
121C2×C4⋊C4×C83—21no—
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C4⋊C4×C83C2311yescharacteristic
124C2×C4⋊C4×C83C2311yescharacteristic
125C2×C4⋊C4×C83C2311yescharacteristic
126C2×C42×C83C2311yescharacteristic
127#107×C83—22no—
128#107×C83—41no—
129#95×C83—21no—
130C23×D83C2×D411yescharacteristic
131C2×Dic166—42no—
132C2×Dic166—42no—
133C2×Dic166—81no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83C2×D411yescharacteristic
139C22×Dic83C2×D411yescharacteristic
140C22×Dic83C2×D411yescharacteristic
141C2×C4×D83—21no—
142C2×C4×D83—21no—
143C2×C4×D83—21no—
144C2×C4×D83—21no—
145C2×C4×D83—42no—
146C2×C4×D83—42no—
147C2×C4×D83—42no—
148C2×C4×D83—42no—
149C2×C4×D83—81no—
150C2×C4×D83—81no—
151C2×C4×D83C2×Q812yes—
152C2×C4×D83C2×Q812yes—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8C2×D412yes—
156C2×C83⋊C8C2×Q812yes—
157C2×C83⋊C8C2×Q812yes—
158C2×C83⋊C8#4112yes—
159C8×D83—28no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83—21no—
165C4⋊Dic83—21no—
166C4⋊Dic83—21no—
167C4⋊Dic83—42no—
168C4⋊Dic83—42no—
169C4⋊Dic83C2×D412yes—
170Dic83⋊C4—42no—
171Dic83⋊C4—42no—
172Dic83⋊C4—42no—
173Dic83⋊C4—42no—
174Dic83⋊C4—42no—
175Dic83⋊C4—42no—
176Dic83⋊C4—81no—
177Dic83⋊C4—81no—
178(C2×D83)⋊C4—42no—
179(C2×D83)⋊C4—42no—
180(C2×D83)⋊C4D814yes—
181(C2×D83)⋊C4Q1614yes—
182C4×Dic83—21no—
183C4×Dic83—21no—
184C2×Q8×C83—42no—
185C2×Q8×C83—42no—
186C2×Q8×C83—81no—
187C22×C4×C83—21no—
188C22×C4×C83—21no—
189C22×C4×C83—21no—
190C22×C4×C83—21no—
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2411yescharacteristic
194C2×C8×C83—21no—
195C2×C8×C83—21no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83C2×Q812yes—
199C2×C8×C83C2×Q812yes—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83—21no—
207C4⋊C4×C83—42no—
208C4⋊C4×C83—42no—
209C4⋊C4×C83C2×D412yes—
210C42×C83—21no—
211C42×C83—21no—
212C23×C83C22×D411yescharacteristic
213C22×D83—21no—
214C22×D83—21no—
215C22×D83—21no—
216C22×D83—21no—
217C22×D83C4⋊Q812yes—
218C22×D83C2×D812yes—
219C22×D83C2×Q1612yes—
220Q8×C83—84no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—22no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
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×C83C2×D812yes—
238C2×C4×C83C2×Q1612yes—
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C22×Q811yescharacteristic
241C2×C4×C83C22×Q811yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83H(5,𝔽2)11yescharacteristic
244C2×C4×C832−1+411yescharacteristic
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251C2×Dic83—22no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
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×Dic83C4⋊Q812yes—
264C2×Dic83C2×D812yes—
265C2×Dic83C2×Q1612yes—
266C4×D83—22no—
267C4×D83—22no—
268C4×D83—22no—
269C4×D83—22no—
270C4×D83—84no—
271C4×D83—84no—
272Dic166—84no—
273C8×C83—22no—
274C8×C83—22no—
275C83⋊C8—22no—
276C83⋊C8C22⋊Q814yescomplemented
277C22×C83C2×C4⋊Q811yescharacteristic
278C22×C83D4211yescharacteristic
279C22×C83#54711yescharacteristic
280C22×C83C22×D811yescharacteristic
281C22×C83C22×Q1611yescharacteristic
282C22×C83#57611yescharacteristic
283C22×C83#57811yescharacteristic
284C2×D83—22no—
285C2×D83—22no—
286C2×D83—22no—
287C2×D83—22no—
288C2×D83—22no—
289C2×D83—22no—
290C2×D83#50014yescomplemented
291C4×C83—22no—
292C4×C83—22no—
293C4×C83—42no—
294C4×C83—81no—
295C4×C83—81no—
296C4×C83—81no—
297C4×C83C2×C22⋊Q811yescharacteristic
298C4×C83#54411yescharacteristic
299C4×C83D4×Q811yescharacteristic
300C4×C83#55411yescharacteristic
301Dic83—22no—
302Dic83—22no—
303Dic83—22no—
304Dic83—42no—
305Dic83—81no—
306Dic83—81no—
307Dic83—81no—
308#26065—831nomaximal, Sylow
309C2×C83#244011yescharacteristic
310C2×C83C2×#50011yescharacteristic
311C2×C83#316411yescharacteristic
312C2×C83D4×D811yescharacteristic
313C2×C83D4×Q1611yescharacteristic
314C2×C83#333411yescharacteristic
315C2×C83#333811yescharacteristic
316D83—24no—
317C2×#500—831no—
318C22×#107—831no—
319#2440—831no—
320#2440—831no—
321#2083—831no—
322#2083—831no—
323#2057—831no—
324#2057D8311yescharacteristic, complemented
325#1942—831no—
326#1942—831no—
327#1852—831no—
328#1852—831no—
329#1845—831no—
330#1808—831no—
331C8×C22⋊C4—831no—
332C83#2606511yescharacteristic, complemented, Sylow
333C23×C8—831no—
334C2×C4⋊Q8—831no—
335C2×C4⋊Q8C2×D8311yescharacteristic
336C2×C22⋊Q8—1661no—
337C2×C22⋊Q8—1661no—
338C22×C4⋊C4—831no—
339C22×C4⋊C4—831no—
340#500—1664no—
341C2×#107—831no—
342C2×#107—831no—
343C2×#107—831no—
344C2×#107—831no—
345C2×#107—831no—
346C2×#107—831no—
347C2×#107—831no—
348C2×#107—831no—
349C2×#107—1664no—
350C2×#107C2×D8311yescharacteristic
351C2×#107C2×D8311yescharacteristic
352C2×C22⋊C8—831no—
353C2×C4×C8—831no—
354C2×C4×C8—831no—
355#398—1661no—
356#398—1661no—
357#395—1661no—
358#395—1661no—
359#389—831no—
360#389—831no—
361#389—831no—
362#389C2×D8311yescharacteristic
363#385—1661no—
364#385—1661no—
365#384—831no—
366#384—831no—
367#380—831no—
368#380—831no—
369C4×C22⋊C4—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#340C2×D8311yescharacteristic
375#340C2×D8311yescharacteristic
376#336—831no—
377#336C2×D8311yescharacteristic
378C22×Q8—21no—
379C22×Q8—1661no—
380C23×C4—831no—
381C23×C4—1661no—
382C23×C4—1661no—
383C22×C8—831no—
384C22×C8—831no—
385C22×C8—834no—
386C22×C8—1662no—
387C22×C8—1662no—
388C22×C8C22×D8311yescharacteristic
389C22×C8C22×D8311yescharacteristic
390C4⋊Q8—22no—
391C4⋊Q8—41no—
392C4⋊Q8—3321no—
393C4⋊Q8—1662no—
394C22⋊Q8—3324no—
395C22⋊Q8—3324no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—3321no—
399C2×C4⋊C4—3321no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—1662no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4—1661no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1662no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4C22×D8311yescharacteristic
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C4⋊C4C22×D8311yescharacteristic
419C2×C4⋊C4C22×D8311yescharacteristic
420C2×C22⋊C4—831no—
421C2×C22⋊C4—831no—
422C2×C22⋊C4—1661no—
423C2×C22⋊C4—1661no—
424C2×C42—831no—
425C2×C42C22×D8311yescharacteristic
426#107—22no—
427#107—41no—
428#107—3321no—
429#107—834no—
430#107—1662no—
431#107—1662no—
432#107—1662no—
433#107—1662no—
434#107—1662no—
435#107—1662no—
436C22⋊C8—834no—
437C4×C8—1662no—
438C4×C8—1662no—
439#95—21no—
440#95—831no—
441#95—831no—
442#95—1661no—
443#95—1661no—
444#95—1661no—
445#95—1661no—
446#95—1661no—
447C24—831no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—81no—
451C2×Q8—3322no—
452C2×Q8—3322no—
453C2×Q8—6641no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—3322no—
459C22×C4—3322no—
460C22×C4—3322no—
461C22×C4—3322no—
462C22×C4—831no—
463C22×C4—831no—
464C22×C4—831no—
465C22×C4—832no—
466C22×C4—832no—
467C22×C4—6641no—
468C22×C4—6641no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1661no—
477C22×C4D4×D8311yescharacteristic
478C22×C4D4×D8311yescharacteristic
479C22×C4C23×D8311yescharacteristic
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—832no—
485C2×C8—832no—
486C2×C8—832no—
487C2×C8—832no—
488C2×C8—1668no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8Q8×D8312yes—
492C2×C8Q8×D8312yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—3322no—
503C4⋊C4—3322no—
504C4⋊C4—3322no—
505C4⋊C4—3322no—
506C4⋊C4—3322no—
507C4⋊C4—3322no—
508C4⋊C4—3322no—
509C4⋊C4—3322no—
510C4⋊C4—832no—
511C4⋊C4—6641no—
512C4⋊C4—6641no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4—1661no—
519C4⋊C4—1661no—
520C4⋊C4D4×D8312yes—
521C22⋊C4—3322no—
522C22⋊C4—3322no—
523C22⋊C4—834no—
524C22⋊C4—834no—
525C42—21no—
526C42—21no—
527C42—1661no—
528C42—1661no—
529C23—832no—
530C23—832no—
531C23—832no—
532C23—1661no—
533C23—1661no—
534C23—1661no—
535C23—1661no—
536C23C2×D4×D8311yescharacteristic
537Q8—84no—
538Q8—6644no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—832no—
567C2×C4—832no—
568C2×C4—832no—
569C2×C4—6644no—
570C2×C4—6644no—
571C2×C4—1662no—
572C2×C4—1661no—
573C2×C4—1661no—
574C2×C4—1661no—
575C2×C4—1661no—
576C2×C4—1661no—
577C2×C4—1661no—
578C2×C4—1662no—
579C2×C4—1662no—
580C2×C4—1662no—
581C2×C4—1662no—
582C2×C4D8×D8312yes—
583C2×C4Q16×D8312yes—
584C2×C4C2×D4×D8311yescharacteristic
585C2×C4C2×Q8×D8311yescharacteristic
586C2×C4C2×Q8×D8311yescharacteristic
587C2×C4#41×D8311yescharacteristic
588C2×C4#1458183711yescharacteristic
589C2×C4#1458184011yescharacteristic
590C8—22no—
591C8—22no—
592C8—834no—
593C8—1662no—
594C22—834no—
595C22—1662no—
596C22—1662no—
597C22—1662no—
598C22—1662no—
599C22—1662no—
600C22—1662no—
601C22C4⋊Q8×D8311yescharacteristic
602C22#1912833311yescharacteristic
603C22#1912835311yescharacteristic
604C22C2×D8×D8311yescharacteristic
605C22C2×Q16×D8311yescharacteristic
606C22#1912850511yescharacteristic
607C22#1912851711yescharacteristic
608C4—22no—
609C4—22no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—3322no—
615C4—6641no—
616C4—6641no—
617C4—6641no—
618C4—1662no—
619C4—1662no—
620C4—1662no—
621C4C22⋊Q8×D8311yescharacteristic
622C4#1912832811yescharacteristic
623C4#1912836611yescharacteristic
624C4#1912840511yescharacteristic
625C2—1664no—
626C2#2491115811yescharacteristic
627C2#500×D8311yescharacteristic
628C2#2491799811yescharacteristic
629C2#2491913311yescharacteristic
630C2#2491920811yescharacteristic
631C2#2491980911yescharacteristic
632C2#2491985411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2057C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
hypercenter#2057D83
hypocenterC83#26065
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
1. C2×#19128104C2semidirect
5. #24908938C2semidirect
6. #24908937C2semidirect
7. #24908792C2semidirect
8. #24908195C2semidirect
9. #24908193C2semidirect
12. #24907621C2semidirect
15. C83×#2057C2semidirect
69. C2×C8×D83Q8semidirect
105. (C2×D83)⋊C8Q8semidirect
276. C83⋊C8C22⋊Q8semidirect
290. C2×D83#500semidirect
324. #2057D83semidirect
332. C83#26065semidirect

Maximal subgroups

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

subgroupindex
C2×#191281042
C2×#107×D832
#249111672
#249111612
#249089382
#249089372
#249087922
#249081952
#249081932
#249076522
#249076492
#249076212
#249074912
#249072332
C83×#20572
#2606583