Finite Group Databasebeta

← #35375353

Subgroups — #35375353

Order 21248.

616 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916483C211yesmaximal, characteristic, complemented
2#24916468C211yesmaximal, characteristic, complemented
3#24906354C211yesmaximal, characteristic, complemented
4#24906348C211yesmaximal, characteristic, complemented
5#24906314C211yesmaximal, characteristic, complemented
6#24906307C211yesmaximal, characteristic, complemented
7#24906307C211yesmaximal, characteristic, complemented
8#24906293C211yesmaximal, characteristic, complemented
9Q16×C83⋊C8C211yesmaximal, characteristic, complemented
10SD16×C83⋊C8C211yesmaximal, characteristic, complemented
11#24906243C211yesmaximal, characteristic, complemented
12#24905923C211yesmaximal, characteristic, complemented
13#24905845C211yesmaximal, characteristic, complemented
14#24905744C211yesmaximal, characteristic, complemented
15C83×#3102C211yesmaximal, characteristic
16#19127832C2211yescharacteristic
17Q8×C83⋊C8C2211yescharacteristic
18Q8×C83⋊C8C2211yescharacteristic
19Q8×C83⋊C8C2211yescharacteristic
20#19127779—21no—
21#19127779—21no—
22#19127779—21no—
23#19127779C2211yescharacteristic
24#19127779C2211yescharacteristic
25#19127779C2211yescharacteristic
26#19127773—21no—
27D4×C83⋊C8C2211yescharacteristic
28#19127762—21no—
29#19127758—21no—
30#19127758C2211yescharacteristic
31#19127754C2211yescharacteristic
32#19127657—21no—
33C4×#13050574—21no—
34C4×#13050574C2211yescharacteristic
35C2×C4×C83⋊C8C2211yescharacteristic
36#19127454C2211yescharacteristic
37#19127454C2211yescharacteristic
38#19127452C2211yescharacteristic
39#19127452C2211yescharacteristic
40Q8⋊(C83⋊C8)C2211yescharacteristic
41Q8⋊(C83⋊C8)C2211yescharacteristic
42Q8⋊(C83⋊C8)C2211yescharacteristic
43Q8⋊(C83⋊C8)C2211yescharacteristic
44D4⋊(C83⋊C8)C2211yescharacteristic
45D4⋊(C83⋊C8)C2211yescharacteristic
46D4⋊(C83⋊C8)C2211yescharacteristic
47D4⋊(C83⋊C8)C2211yescharacteristic
48#19127423C2211yescharacteristic
49#19127423C2211yescharacteristic
50C8×C83⋊C8C2211yescharacteristic
51C8×C83⋊C8C2211yescharacteristic
52#529×C83C2211yescharacteristic
53C4×#41×C83C2211yescharacteristic
54#485×C83C411yescharacteristic
55#482×C83C411yescharacteristic
56#479×C83C411yescharacteristic
57#478×C83C411yescharacteristic
58#464×C83C411yescharacteristic
59#463×C83C411yescharacteristic
60#452×C83C411yescharacteristic
61#449×C83C411yescharacteristic
62C4×Q16×C83C2211yescharacteristic
63C4×SD16×C83C2211yescharacteristic
64C4×SD16×C83C2211yescharacteristic
65C4×D8×C83C2211yescharacteristic
66#431×C83C2211yescharacteristic
67C2×#13050574—21no—
68C2×#13050574—21no—
69C2×#13050574—21no—
70C2×#13050574D411yescharacteristic
71C2×#13050574D411yescharacteristic
72C22×C83⋊C8—21no—
73C22×C83⋊C8—21no—
74C4⋊(C83⋊C8)—21no—
75C4⋊(C83⋊C8)—21no—
76C4⋊(C83⋊C8)—21no—
77C4⋊(C83⋊C8)—21no—
78C4⋊(C83⋊C8)C2311yescharacteristic
79C4⋊(C83⋊C8)C2311yescharacteristic
80C4⋊(C83⋊C8)C2311yescharacteristic
81C4⋊(C83⋊C8)C2311yescharacteristic
82C22⋊(C83⋊C8)—21no—
83C22⋊(C83⋊C8)—21no—
84C22⋊(C83⋊C8)—21no—
85C22⋊(C83⋊C8)—21no—
86C22⋊(C83⋊C8)—21no—
87C22⋊(C83⋊C8)—21no—
88#14581703—21no—
89#14581703—21no—
90#14581703D411yescharacteristic
91#14581703D411yescharacteristic
92C4×C83⋊C8—21no—
93C4×C83⋊C8—21no—
94C4×C83⋊C8—21no—
95C4×C83⋊C8—21no—
96C4×C83⋊C8—21no—
97C4×C83⋊C8C2311yescharacteristic
98C4×C83⋊C8C2311yescharacteristic
99C4×C83⋊C8C2311yescharacteristic
100C4×C83⋊C8C2311yescharacteristic
101C2×#41×C83C2×C411yescharacteristic
102C2×Q16×C83C2×C411yescharacteristic
103C2×SD16×C83C2×C411yescharacteristic
104C2×SD16×C83C2×C411yescharacteristic
105C2×D8×C83C2×C411yescharacteristic
106#126×C83—21no—
107#125×C83C2×C411yescharacteristic
108#124×C83C2×C411yescharacteristic
109#123×C83—21no—
110C22⋊Q8×C83C2×C411yescharacteristic
111#121×C83C2×C411yescharacteristic
112C4×Q8×C83C2×C411yescharacteristic
113C4×Q8×C83C2311yescharacteristic
114C4×Q8×C83C2311yescharacteristic
115C4×D4×C83—21no—
116C4×D4×C83—21no—
117C4×D4×C83—21no—
118C4×D4×C83C2×C411yescharacteristic
119C4×D4×C83C2311yescharacteristic
120C4×D4×C83C2311yescharacteristic
121#117×C83—21no—
122#117×C83—21no—
123#117×C83C2×C411yescharacteristic
124C2×C42×C83—21no—
125C2×C42×C83C2311yescharacteristic
126#107×C83C2×C411yescharacteristic
127#107×C83C2×C411yescharacteristic
128#106×C83C2×C411yescharacteristic
129#106×C83C2×C411yescharacteristic
130C4⋊C8×C83C2×C411yescharacteristic
131C4⋊C8×C83C2×C411yescharacteristic
132#103×C83C2×C411yescharacteristic
133#103×C83C2×C411yescharacteristic
134#103×C83C2×C411yescharacteristic
135#103×C83C2×C411yescharacteristic
136#102×C83C2×C411yescharacteristic
137#102×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83C2×C411yescharacteristic
141C22⋊C8×C83C2×C411yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144#13050574—22no—
145#13050574—22no—
146#13050574—42no—
147#13050574—42no—
148#13050574—42no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—42no—
156C2×C83⋊C8—42no—
157C2×C83⋊C8C2×D411yescharacteristic
158C2×C83⋊C8C2×D411yescharacteristic
159C2×C83⋊C8C2×D411yescharacteristic
160C2×C83⋊C8C2×D411yescharacteristic
161#41×C83—22no—
162#41×C83—41no—
163C2×Q8×C83C22×C411yescharacteristic
164C2×Q8×C83C22×C411yescharacteristic
165C2×D4×C83—21no—
166C2×D4×C83—21no—
167C2×D4×C83C22×C411yescharacteristic
168C2×D4×C83C22×C411yescharacteristic
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83C22×C411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83#4111yescharacteristic
176Q16×C83—22no—
177SD16×C83—22no—
178SD16×C83—41no—
179D8×C83—41no—
180C2×C8×C83C22×C411yescharacteristic
181C2×C8×C83C22×C411yescharacteristic
182C2×C8×C83C22×C411yescharacteristic
183C2×C8×C83C22×C411yescharacteristic
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×C83C22×C411yescharacteristic
191C4⋊C4×C83C22×C411yescharacteristic
192C4⋊C4×C83C22×C411yescharacteristic
193C4⋊C4×C83C22×C411yescharacteristic
194C22⋊C4×C83—21no—
195C22⋊C4×C83—21no—
196C22⋊C4×C83—21no—
197C22⋊C4×C83—21no—
198C22⋊C4×C83—21no—
199C22⋊C4×C83—21no—
200C22⋊C4×C83—21no—
201C22⋊C4×C83—21no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83C22×C411yescharacteristic
207C42×C83C2×D411yescharacteristic
208C42×C83#4111yescharacteristic
209C42×C83C2411yescharacteristic
210C23×C83—21no—
211C23×C83—21no—
212C23×C83C4×D411yescharacteristic
213Q8×C83—21no—
214Q8×C83—21no—
215Q8×C83—21no—
216Q8×C83#13112yes—
217D4×C83—21no—
218D4×C83—21no—
219D4×C83—21no—
220D4×C83—21no—
221D4×C83—42no—
222D4×C83—81no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—41no—
234C2×C4×C83—42no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—81no—
240C2×C4×C83C4×D411yescharacteristic
241C2×C4×C83C4×D411yescharacteristic
242C2×C4×C83C4×D411yescharacteristic
243C2×C4×C83C23×C411yescharacteristic
244C2×C4×C83C22×D411yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C8×C83—21no—
247C8×C83—21no—
248C8×C83—21no—
249C8×C83—21no—
250C83⋊C8—22no—
251C83⋊C8—21no—
252C83⋊C8—21no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C83⋊C8—41no—
257C83⋊C8—41no—
258C83⋊C8#13512yes—
259C83⋊C8#13512yes—
260C22×C83—21no—
261C22×C83—21no—
262C22×C83—21no—
263C22×C83—41no—
264C22×C83—41no—
265C22×C83—41no—
266C22×C83—41no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83C2×C4×D411yescharacteristic
270C4×C83—21no—
271C4×C83—22no—
272C4×C83—21no—
273C4×C83—21no—
274C4×C83—21no—
275C4×C83—41no—
276C4×C83—41no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83—41no—
280C4×C83C2×#13111yescharacteristic
281C4×C83#56811yescharacteristic
282C4×C83C2×#13511yescharacteristic
283C4×C83C2×#13511yescharacteristic
284#17999—831nomaximal, Sylow
285C2×C83—41no—
286C2×C83—81no—
287C2×C83—81no—
288C2×C83#293311yescharacteristic
289C2×C83C4×#13511yescharacteristic
290C2×C83#295611yescharacteristic
291#3102D8311yescharacteristic
292#2971—831no—
293#2969—831no—
294#1590—831no—
295#1589—831no—
296#1584—831no—
297#1583—831no—
298#1583—831no—
299#1581—831no—
300C8×Q16—831no—
301C8×SD16—831no—
302#1572—831no—
303#1494—831no—
304#1476—831no—
305C8×M16—831no—
306C83#1799911yescharacteristic, complemented, Sylow
307#529C2×D8311yescharacteristic
308C4×#41C2×D8311yescharacteristic
309#485Dic8311yescharacteristic
310#482Dic8311yescharacteristic
311#479Dic8311yescharacteristic
312#478Dic8311yescharacteristic
313#464Dic8311yescharacteristic
314#463Dic8311yescharacteristic
315#452Dic8311yescharacteristic
316#449Dic8311yescharacteristic
317#446—831no—
318C8×Q8—831no—
319C8×Q8—831no—
320C8×Q8—831no—
321C4×Q16C2×D8311yescharacteristic
322C4×SD16C2×D8311yescharacteristic
323C4×SD16C2×D8311yescharacteristic
324C4×D8C2×D8311yescharacteristic
325#436—831no—
326#436—831no—
327#436—831no—
328#436—1661no—
329#436—1661no—
330#436—1661no—
331#435—1661no—
332C8×D4—831no—
333#433—1661no—
334#432—831no—
335#432—1661no—
336#431—831no—
337#431C2×D8311yescharacteristic
338#405—1661no—
339C4×M16—831no—
340C4×M16—1661no—
341C2×C4×C8—831no—
342#335—831no—
343#335—831no—
344#334—831no—
345#334—831no—
346Q8⋊C8—831no—
347Q8⋊C8—831no—
348Q8⋊C8—831no—
349Q8⋊C8—831no—
350D4⋊C8—831no—
351D4⋊C8—831no—
352D4⋊C8—831no—
353D4⋊C8—831no—
354#322—831no—
355#322—831no—
356C82—831no—
357C82—831no—
358C2×#41C2×Dic8311yescharacteristic
359C2×Q16C2×Dic8311yescharacteristic
360C2×SD16C2×Dic8311yescharacteristic
361C2×SD16C2×Dic8311yescharacteristic
362C2×D8C2×Dic8311yescharacteristic
363C2×M16—831no—
364C2×M16—831no—
365C2×M16—1661no—
366C2×M16—1661no—
367C2×M16—1661no—
368C22×C8—1661no—
369C22×C8—1661no—
370#126—21no—
371#125C2×Dic8311yescharacteristic
372#124C2×Dic8311yescharacteristic
373#123—21no—
374C22⋊Q8C2×Dic8311yescharacteristic
375#121C2×Dic8311yescharacteristic
376C4×Q8C2×Dic8311yescharacteristic
377C4×Q8C22×D8311yescharacteristic
378C4×Q8C22×D8311yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C2×Dic8311yescharacteristic
383C4×D4C22×D8311yescharacteristic
384C4×D4C22×D8311yescharacteristic
385#117—21no—
386#117—21no—
387#117C2×Dic8311yescharacteristic
388C2×C42—21no—
389C2×C42C22×D8311yescharacteristic
390#107C2×Dic8311yescharacteristic
391#107C2×Dic8311yescharacteristic
392#106C2×Dic8311yescharacteristic
393#106C2×Dic8311yescharacteristic
394C4⋊C8—831no—
395C4⋊C8—831no—
396C4⋊C8—831no—
397C4⋊C8—831no—
398C4⋊C8—1661no—
399C4⋊C8—1661no—
400C4⋊C8—1661no—
401C4⋊C8—1661no—
402C4⋊C8C2×Dic8311yescharacteristic
403C4⋊C8C2×Dic8311yescharacteristic
404#103C2×Dic8311yescharacteristic
405#103C2×Dic8311yescharacteristic
406#103C2×Dic8311yescharacteristic
407#103C2×Dic8311yescharacteristic
408#102C2×Dic8311yescharacteristic
409#102C2×Dic8311yescharacteristic
410#102C2×Dic8311yescharacteristic
411#102C2×Dic8311yescharacteristic
412C22⋊C8—1661no—
413C22⋊C8—1661no—
414C22⋊C8—1661no—
415C22⋊C8—1661no—
416C22⋊C8—1661no—
417C22⋊C8—1661no—
418C22⋊C8C2×Dic8311yescharacteristic
419C22⋊C8C2×Dic8311yescharacteristic
420#97—831no—
421#97—831no—
422#97—1661no—
423#97—1661no—
424C4×C8—831no—
425C4×C8—831no—
426C4×C8—831no—
427C4×C8—831no—
428C4×C8—1661no—
429C4×C8—1661no—
430C4×C8—1661no—
431C4×C8—1661no—
432C4×C8—1661no—
433C4×C8C22×D8311yescharacteristic
434C4×C8C22×D8311yescharacteristic
435#41—22no—
436#41—41no—
437C2×Q8C22×Dic8311yescharacteristic
438C2×Q8C22×Dic8311yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C22×Dic8311yescharacteristic
442C2×D4C22×Dic8311yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4C22×Dic8311yescharacteristic
448C22×C4D4×D8311yescharacteristic
449C22×C4#1305058911yescharacteristic
450Q16—22no—
451SD16—22no—
452SD16—41no—
453D8—41no—
454M16—3322no—
455M16—3322no—
456M16—3322no—
457M16—1662no—
458M16—1662no—
459C2×C8—3322no—
460C2×C8—3322no—
461C2×C8—831no—
462C2×C8—831no—
463C2×C8—831no—
464C2×C8—831no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8C22×Dic8311yescharacteristic
472C2×C8C22×Dic8311yescharacteristic
473C2×C8C22×Dic8311yescharacteristic
474C2×C8C22×Dic8311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4C22×Dic8311yescharacteristic
482C4⋊C4C22×Dic8311yescharacteristic
483C4⋊C4C22×Dic8311yescharacteristic
484C4⋊C4C22×Dic8311yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42C22×Dic8311yescharacteristic
498C42D4×D8311yescharacteristic
499C42#1305058911yescharacteristic
500C42C23×D8311yescharacteristic
501C23—21no—
502C23—21no—
503C23D4×Dic8311yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8#1458179912yes—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—81no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4D4×Dic8311yescharacteristic
532C2×C4D4×Dic8311yescharacteristic
533C2×C4D4×Dic8311yescharacteristic
534C2×C4C23×Dic8311yescharacteristic
535C2×C4C2×D4×D8311yescharacteristic
536C2×C4C2×#1305058911yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—3321no—
542C8—3321no—
543C8—3321no—
544C8—3321no—
545C8—3321no—
546C8—832no—
547C8—832no—
548C8—1662no—
549C8—1661no—
550C8—1661no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22C2×D4×Dic8311yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4C2×#1458179911yescharacteristic
572C4#1912846811yescharacteristic
573C4#135×D8311yescharacteristic
574C4#135×D8311yescharacteristic
575C2—41no—
576C2—81no—
577C2—81no—
578C2#2491619011yescharacteristic
579C2#135×Dic8311yescharacteristic
580C2#2491637311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#3102C2
socleC22×C83C2×C4×D4
prime radical#449×C83C4
cosocleC42×C83C24
hypercenter#3102D83
hypocenterC83#17999
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3102

Complements

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

NH ≅ G/Ntype
1. #24916483C2semidirect
2. #24916468C2semidirect
3. #24906354C2semidirect
4. #24906348C2semidirect
5. #24906314C2semidirect
6. #24906307C2semidirect
7. #24906307C2semidirect
8. #24906293C2semidirect
9. Q16×C83⋊C8C2semidirect
10. SD16×C83⋊C8C2semidirect
11. #24906243C2semidirect
12. #24905923C2semidirect
13. #24905845C2semidirect
14. #24905744C2semidirect
306. C83#17999semidirect

Maximal subgroups

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

subgroupindex
#249164832
#249164682
#249063542
#249063482
#249063142
#249063072
#249062932
Q16×C83⋊C82
SD16×C83⋊C82
#249062432
#249059232
#249058452
#249057442
C83×#31022
#1799983