Finite Group Databasebeta

← #35375599

Subgroups — #35375599

Order 21248.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916623C211yesmaximal, characteristic
2#24916552C211yesmaximal, characteristic, complemented
3#24906360C211yesmaximal, characteristic, complemented
4#24906354C211yesmaximal, characteristic, complemented
5#24906342C211yesmaximal, characteristic, complemented
6#24906335C211yesmaximal, characteristic, complemented
7#24906335C211yesmaximal, characteristic, complemented
8#24906321C211yesmaximal, characteristic, complemented
9Q16×C83⋊C8C211yesmaximal, characteristic, complemented
10SD16×C83⋊C8C211yesmaximal, characteristic, complemented
11#24906248C211yesmaximal, characteristic, complemented
12#24905923C211yesmaximal, characteristic
13C2×Q8⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
14#24905747C211yesmaximal, characteristic, complemented
15C83×#3108C211yesmaximal, characteristic
16#19127832—21no—
17#19127832—21no—
18#19127832C2211yescharacteristic
19#19127832C2211yescharacteristic
20#19127832C2211yescharacteristic
21Q8×C83⋊C8C2211yescharacteristic
22Q8×C83⋊C8C2211yescharacteristic
23Q8×C83⋊C8C2211yescharacteristic
24#19127779C2211yescharacteristic
25#19127773—21no—
26D4×C83⋊C8—21no—
27D4×C83⋊C8C2211yescharacteristic
28#19127762—21no—
29#19127758—21no—
30#19127758C2211yescharacteristic
31#19127754—21no—
32#19127754C2211yescharacteristic
33#19127728—21no—
34C2×C4⋊(C83⋊C8)C2211yescharacteristic
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
44Q8⋊(C83⋊C8)C2211yescharacteristic
45Q8⋊(C83⋊C8)C2211yescharacteristic
46D4⋊(C83⋊C8)C2211yescharacteristic
47D4⋊(C83⋊C8)C2211yescharacteristic
48#19127423C2211yescharacteristic
49#19127423C2211yescharacteristic
50C8×C83⋊C8C2211yescharacteristic
51C8×C83⋊C8C2211yescharacteristic
52#529×C83C2211yescharacteristic
53C2×C4×Q8×C83C2211yescharacteristic
54#483×C83C411yescharacteristic
55#480×C83C411yescharacteristic
56#479×C83C411yescharacteristic
57#479×C83C411yescharacteristic
58#464×C83C411yescharacteristic
59#464×C83C411yescharacteristic
60#451×C83C411yescharacteristic
61#448×C83C411yescharacteristic
62C4×Q16×C83C2211yescharacteristic
63C4×Q16×C83C2211yescharacteristic
64C4×SD16×C83C2211yescharacteristic
65C4×SD16×C83C2211yescharacteristic
66#431×C83C2211yescharacteristic
67C2×#13050574—21no—
68C22×C83⋊C8—21no—
69C22×C83⋊C8—21no—
70C22×C83⋊C8—21no—
71C4⋊(C83⋊C8)—21no—
72C4⋊(C83⋊C8)—21no—
73C4⋊(C83⋊C8)—21no—
74C4⋊(C83⋊C8)—21no—
75C4⋊(C83⋊C8)—21no—
76C4⋊(C83⋊C8)—21no—
77C4⋊(C83⋊C8)—21no—
78C4⋊(C83⋊C8)D411yescharacteristic
79C4⋊(C83⋊C8)D411yescharacteristic
80C4⋊(C83⋊C8)C2311yescharacteristic
81C4⋊(C83⋊C8)C2311yescharacteristic
82C4⋊(C83⋊C8)C2311yescharacteristic
83C4⋊(C83⋊C8)C2311yescharacteristic
84C22⋊(C83⋊C8)—21no—
85C22⋊(C83⋊C8)—21no—
86C22⋊(C83⋊C8)—21no—
87C22⋊(C83⋊C8)D411yescharacteristic
88C22⋊(C83⋊C8)D411yescharacteristic
89#14581703—21no—
90#14581703—21no—
91#14581703—21no—
92#14581703—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
101C22×Q8×C83C2×C411yescharacteristic
102C2×Q16×C83C2×C411yescharacteristic
103C2×Q16×C83C2×C411yescharacteristic
104C2×SD16×C83C2×C411yescharacteristic
105C2×SD16×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×C83—21no—
113C4×Q8×C83—21no—
114C4×Q8×C83C2×C411yescharacteristic
115C4×Q8×C83C2×C411yescharacteristic
116C4×Q8×C83C2311yescharacteristic
117C4×Q8×C83C2311yescharacteristic
118C4×Q8×C83C2311yescharacteristic
119C4×D4×C83—21no—
120C4×D4×C83C2311yescharacteristic
121#117×C83—21no—
122C2×C4⋊C4×C83—21no—
123C2×C4⋊C4×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#103×C83C2×C411yescharacteristic
137#103×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—81no—
145C2×C83⋊C8—24no—
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—81no—
153C2×C83⋊C8—81no—
154C2×C83⋊C8C2×D411yescharacteristic
155C2×C83⋊C8C2×D411yescharacteristic
156C2×C83⋊C8C2×D411yescharacteristic
157C2×C83⋊C8C2×D411yescharacteristic
158C2×Q8×C83—21no—
159C2×Q8×C83—22no—
160C2×Q8×C83—41no—
161C2×Q8×C83C22×C411yescharacteristic
162C2×Q8×C83C22×C411yescharacteristic
163C2×Q8×C83C22×C411yescharacteristic
164C2×D4×C83—21no—
165C2×D4×C83C22×C411yescharacteristic
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83C22×C411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83#4111yescharacteristic
172Q16×C83—22no—
173Q16×C83—41no—
174SD16×C83—22no—
175SD16×C83—41no—
176C2×C8×C83C22×C411yescharacteristic
177C2×C8×C83C22×C411yescharacteristic
178C2×C8×C83C22×C411yescharacteristic
179C2×C8×C83C22×C411yescharacteristic
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
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×C83C22×C411yescharacteristic
190C4⋊C4×C83C22×C411yescharacteristic
191C4⋊C4×C83C22×C411yescharacteristic
192C4⋊C4×C83C22×C411yescharacteristic
193C22⋊C4×C83—21no—
194C22⋊C4×C83—21no—
195C22⋊C4×C83—21no—
196C22⋊C4×C83—21no—
197C22⋊C4×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83C22×C411yescharacteristic
204C42×C83C2×D411yescharacteristic
205C42×C83#4111yescharacteristic
206C42×C83C2411yescharacteristic
207C23×C83—21no—
208C23×C83C4×D411yescharacteristic
209Q8×C83—21no—
210Q8×C83—21no—
211Q8×C83—21no—
212Q8×C83—21no—
213Q8×C83—21no—
214Q8×C83—42no—
215Q8×C83#13112yes—
216D4×C83—21no—
217D4×C83—21no—
218D4×C83—42no—
219C2×C4×C83—21no—
220C2×C4×C83—22no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
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—42no—
232C2×C4×C83—42no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83C4×D411yescharacteristic
236C2×C4×C83C4×D411yescharacteristic
237C2×C4×C83C4×D411yescharacteristic
238C2×C4×C83C23×C411yescharacteristic
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C8×C83—21no—
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—21no—
245C83⋊C8—22no—
246C83⋊C8—22no—
247C83⋊C8—21no—
248C83⋊C8—21no—
249C83⋊C8—21no—
250C83⋊C8—21no—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C22×C83—21no—
256C22×C83—21no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83#44312yes—
261C22×C83C2×C4×D411yescharacteristic
262C4×C83—21no—
263C4×C83—22no—
264C4×C83—21no—
265C4×C83—21no—
266C4×C83—21no—
267C4×C83—41no—
268C4×C83—41no—
269C4×C83—41no—
270C4×C83—41no—
271C4×C83—41no—
272C4×C83—41no—
273C4×C83C2×#13111yescharacteristic
274C4×C83#56811yescharacteristic
275C4×C83#57711yescharacteristic
276C4×C83#57811yescharacteristic
277#18017—831nomaximal, Sylow
278C2×C83—22no—
279C2×C83—81no—
280C2×C83#293311yescharacteristic
281C2×C83#294711yescharacteristic
282C2×C83C2×#44311yescharacteristic
283#3108D8311yescharacteristic
284#2994—831no—
285#2984—831no—
286#1591—831no—
287#1590—831no—
288#1588—831no—
289#1587—831no—
290#1587—831no—
291#1585—831no—
292C8×Q16—831no—
293C8×SD16—831no—
294#1573—831no—
295#1494—831no—
296C2×Q8⋊C8—831no—
297#1450—831no—
298C83#1801711yescharacteristic, complemented, Sylow
299#529C2×D8311yescharacteristic
300C2×C4×Q8C2×D8311yescharacteristic
301#483Dic8311yescharacteristic
302#480Dic8311yescharacteristic
303#479Dic8311yescharacteristic
304#479Dic8311yescharacteristic
305#464Dic8311yescharacteristic
306#464Dic8311yescharacteristic
307#451Dic8311yescharacteristic
308#448Dic8311yescharacteristic
309#446—831no—
310#446—831no—
311#446—831no—
312#446—1661no—
313#446—1661no—
314C8×Q8—831no—
315C8×Q8—831no—
316C8×Q8—831no—
317C4×Q16C2×D8311yescharacteristic
318C4×Q16C2×D8311yescharacteristic
319C4×SD16C2×D8311yescharacteristic
320C4×SD16C2×D8311yescharacteristic
321#436—831no—
322#435—1661no—
323C8×D4—831no—
324C8×D4—1661no—
325#433—1661no—
326#432—831no—
327#432—1661no—
328#431—831no—
329#431—1661no—
330#431C2×D8311yescharacteristic
331#424—1661no—
332C2×C4⋊C8—831no—
333C2×C4×C8—831no—
334#335—831no—
335#335—831no—
336#334—831no—
337#334—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8—831no—
342Q8⋊C8—831no—
343Q8⋊C8—831no—
344D4⋊C8—831no—
345D4⋊C8—831no—
346#322—831no—
347#322—831no—
348C82—831no—
349C82—831no—
350C22×Q8C2×Dic8311yescharacteristic
351C2×Q16C2×Dic8311yescharacteristic
352C2×Q16C2×Dic8311yescharacteristic
353C2×SD16C2×Dic8311yescharacteristic
354C2×SD16C2×Dic8311yescharacteristic
355C2×M16—1661no—
356C22×C8—1661no—
357C22×C8—1661no—
358C22×C8—1661no—
359#126—21no—
360#125C2×Dic8311yescharacteristic
361#124C2×Dic8311yescharacteristic
362#123—21no—
363C22⋊Q8C2×Dic8311yescharacteristic
364#121C2×Dic8311yescharacteristic
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C2×Dic8311yescharacteristic
368C4×Q8C2×Dic8311yescharacteristic
369C4×Q8C22×D8311yescharacteristic
370C4×Q8C22×D8311yescharacteristic
371C4×Q8C22×D8311yescharacteristic
372C4×D4—21no—
373C4×D4C22×D8311yescharacteristic
374#117—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C2×Dic8311yescharacteristic
377C2×C42—21no—
378C2×C42C22×D8311yescharacteristic
379#107C2×Dic8311yescharacteristic
380#107C2×Dic8311yescharacteristic
381#106C2×Dic8311yescharacteristic
382#106C2×Dic8311yescharacteristic
383C4⋊C8—831no—
384C4⋊C8—831no—
385C4⋊C8—831no—
386C4⋊C8—831no—
387C4⋊C8—831no—
388C4⋊C8—831no—
389C4⋊C8—1661no—
390C4⋊C8—1661no—
391C4⋊C8—1661no—
392C4⋊C8—1661no—
393C4⋊C8—1661no—
394C4⋊C8—1661no—
395C4⋊C8—1661no—
396C4⋊C8C2×Dic8311yescharacteristic
397C4⋊C8C2×Dic8311yescharacteristic
398#103C2×Dic8311yescharacteristic
399#103C2×Dic8311yescharacteristic
400#103C2×Dic8311yescharacteristic
401#103C2×Dic8311yescharacteristic
402#103C2×Dic8311yescharacteristic
403#103C2×Dic8311yescharacteristic
404#102C2×Dic8311yescharacteristic
405#102C2×Dic8311yescharacteristic
406C22⋊C8—831no—
407C22⋊C8—831no—
408C22⋊C8—1661no—
409C22⋊C8—1661no—
410C22⋊C8—1661no—
411C22⋊C8C2×Dic8311yescharacteristic
412C22⋊C8C2×Dic8311yescharacteristic
413#97—1661no—
414#97—1661no—
415#97—1661no—
416#97—1661no—
417C4×C8—831no—
418C4×C8—831no—
419C4×C8—831no—
420C4×C8—831no—
421C4×C8—1661no—
422C4×C8—1661no—
423C4×C8—1661no—
424C4×C8—1661no—
425C4×C8C22×D8311yescharacteristic
426C4×C8C22×D8311yescharacteristic
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8C22×Dic8311yescharacteristic
431C2×Q8C22×Dic8311yescharacteristic
432C2×Q8C22×Dic8311yescharacteristic
433C2×D4—21no—
434C2×D4C22×Dic8311yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4C22×Dic8311yescharacteristic
439C22×C4D4×D8311yescharacteristic
440C22×C4#1305058911yescharacteristic
441Q16—22no—
442Q16—41no—
443SD16—22no—
444SD16—41no—
445M16—6641no—
446C2×C8—831no—
447C2×C8—831no—
448C2×C8—831no—
449C2×C8—831no—
450C2×C8—6641no—
451C2×C8—6641no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1664no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8C22×Dic8311yescharacteristic
460C2×C8C22×Dic8311yescharacteristic
461C2×C8C22×Dic8311yescharacteristic
462C2×C8C22×Dic8311yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4C22×Dic8311yescharacteristic
473C4⋊C4C22×Dic8311yescharacteristic
474C4⋊C4C22×Dic8311yescharacteristic
475C4⋊C4C22×Dic8311yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42C22×Dic8311yescharacteristic
487C42D4×D8311yescharacteristic
488C42#1305058911yescharacteristic
489C42C23×D8311yescharacteristic
490C23—21no—
491C23D4×Dic8311yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8#1458179912yes—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4D4×Dic8311yescharacteristic
519C2×C4D4×Dic8311yescharacteristic
520C2×C4D4×Dic8311yescharacteristic
521C2×C4C23×Dic8311yescharacteristic
522C2×C4C2×D4×D8311yescharacteristic
523C2×C4C2×#1305058911yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—3321no—
529C8—3321no—
530C8—3321no—
531C8—3321no—
532C8—1662no—
533C8—1662no—
534C8—1661no—
535C8—1661no—
536C8—1661no—
537C8—1661no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1912781812yes—
544C22C2×D4×Dic8311yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4C2×#1458179911yescharacteristic
557C4#1912846811yescharacteristic
558C4#1912851011yescharacteristic
559C4#1912851711yescharacteristic
560C2—22no—
561C2—81no—
562C2#2491619011yescharacteristic
563C2#2491630711yescharacteristic
564C2C2×#1912781811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#3108C2
socleC22×C83C2×C4×D4
prime radical#121×C83C2×C4
cosocleC42×C83C24
hypercenter#3108D83
hypocenterC83#18017
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3108

Complements

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

NH ≅ G/Ntype
2. #24916552C2semidirect
3. #24906360C2semidirect
4. #24906354C2semidirect
5. #24906342C2semidirect
6. #24906335C2semidirect
7. #24906335C2semidirect
8. #24906321C2semidirect
9. Q16×C83⋊C8C2semidirect
10. SD16×C83⋊C8C2semidirect
11. #24906248C2semidirect
13. C2×Q8⋊(C83⋊C8)C2semidirect
14. #24905747C2semidirect
298. C83#18017semidirect

Maximal subgroups

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

subgroupindex
#249166232
#249165522
#249063602
#249063542
#249063422
#249063352
#249063212
Q16×C83⋊C82
SD16×C83⋊C82
#249062482
#249059232
C2×Q8⋊(C83⋊C8)2
#249057472
C83×#31082
#1801783