Finite Group Databasebeta

← #35375839

Subgroups — #35375839

Order 21248.

716 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#13050573C211yesmaximal, characteristic, complemented
2C4×SD16×D83C211yesmaximal, characteristic
3#24916199C211yesmaximal, characteristic
4#24906356C211yesmaximal, characteristic, complemented
5#24906351C211yesmaximal, characteristic, complemented
6#24906307C211yesmaximal, characteristic, complemented
7#24906301C211yesmaximal, characteristic, complemented
8#24906298C211yesmaximal, characteristic, complemented
9C8×#13050577C211yesmaximal, characteristic, complemented
10#24906260C211yesmaximal, characteristic, complemented
11#24906217C211yesmaximal, characteristic, complemented
12#24905890C211yesmaximal, characteristic, complemented
13#24905886C211yesmaximal, characteristic
14#24905742C211yesmaximal, characteristic, complemented
15C83×#1583C211yesmaximal, characteristic, complemented
16C2×SD16×D83C411yescharacteristic
17C2×M16×D83—21no—
18C4×Q8×D83C2211yescharacteristic
19C4×D4×D83C2211yescharacteristic
20#19127831—21no—
21#19127831C2211yescharacteristic, complemented
22#19127828—21no—
23#19127828C2211yescharacteristic, complemented
24Q8×C83⋊C8C2211yescharacteristic, complemented
25#19127792C411yescharacteristic
26C4×#13050579C2211yescharacteristic
27SD16×Dic83C411yescharacteristic
28C4×#13050577C2211yescharacteristic
29#19127788C411yescharacteristic
30C4×#13050578C2211yescharacteristic
31#19127786C411yescharacteristic
32#19127779C2211yescharacteristic
33#19127775—21no—
34#19127770—21no—
35C8×D332C2211yescharacteristic
36#19127766C2211yescharacteristic
37#19127752C2211yescharacteristic
38#106×D83C411yescharacteristic
39#19127723—21no—
40#19127723C2211yescharacteristic
41#19127723C2211yescharacteristic
42#103×D83C411yescharacteristic
43#102×D83C411yescharacteristic
44#19127663—21no—
45C4×#13050573—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127452C2211yescharacteristic, complemented
49#19127451C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127433C2211yescharacteristic, complemented
52Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
53#19127431C2211yescharacteristic, complemented
54#19127430C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic, complemented
58C8×C83⋊C8C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C8×Q8×C83C2211yescharacteristic, complemented
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic, complemented
64Q8⋊C8×C83C2211yescharacteristic, complemented
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic, complemented
67C2×Q8×D83C2×C411yescharacteristic
68C2×D4×D83C2×C411yescharacteristic
69C22×C4×D83—21no—
70C2×#13050579C2×C411yescharacteristic
71C2×#13050578C2×C411yescharacteristic
72C2×#13050577C2×C411yescharacteristic
73SD16×D83—24no—
74C2×#13050574—21no—
75C2×#13050573—21no—
76C2×#13050573—21no—
77C2×#13050573D411yescharacteristic
78C2×#13050573D411yescharacteristic
79M16×D83—44no—
80C2×C8×D83—21no—
81C2×C8×D83—21no—
82C2×C8×D83C2×C411yescharacteristic
83C2×C8×D83C2×C411yescharacteristic
84Q8×Dic83C2×C411yescharacteristic
85C4×Dic166—21no—
86C4×Dic166C2311yescharacteristic
87#14581757—21no—
88#14581757C2×C411yescharacteristic
89C4×D332C2311yescharacteristic
90D4×Dic83C2×C411yescharacteristic
91#14581754C2×C411yescharacteristic
92C4×C22⋊D83—21no—
93(C22⋊D83)⋊C4—21no—
94C4⋊C4×D83—21no—
95C4⋊C4×D83C2×C411yescharacteristic
96C4⋊C4×D83C2×C411yescharacteristic
97C22⋊C4×D83—21no—
98C42×D83—21no—
99C42×D83C2311yescharacteristic
100#14581724C2×C411yescharacteristic
101#14581723C2×C411yescharacteristic
102#14581723C2×C411yescharacteristic
103C4⋊(C83⋊C8)—21no—
104C4⋊(C83⋊C8)C2311yescharacteristic
105C4⋊(C83⋊C8)C2311yescharacteristic
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110Dic83⋊C8—21no—
111#14581717C2×C411yescharacteristic
112#14581716C2×C411yescharacteristic
113#14581715C2×C411yescharacteristic
114#14581714C2×C411yescharacteristic
115#14581713C2×C411yescharacteristic
116#14581712C2×C411yescharacteristic
117C22⋊(C83⋊C8)—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8—21no—
120(C2×D83)⋊C8—21no—
121#14581702—21no—
122#14581702—21no—
123#14581702D411yescharacteristic
124#14581702D411yescharacteristic
125C4×C83⋊C8—21no—
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C4×C83⋊C8C2311yescharacteristic
129C8×Dic83C2×C411yescharacteristic
130C8×Dic83C2×C411yescharacteristic
131C2×SD16×C83C2×C411yescharacteristic
132C2×M16×C83—21no—
133C4×Q8×C83C2311yescharacteristic
134C4×D4×C83C2311yescharacteristic
135#106×C83C2×C411yescharacteristic
136C4⋊C8×C83—21no—
137C4⋊C8×C83C2311yescharacteristic
138C4⋊C8×C83C2311yescharacteristic
139#103×C83C2×C411yescharacteristic
140#102×C83C2×C411yescharacteristic
141C22⋊C8×C83—21no—
142C4×C8×C83—21no—
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C4×C8×C83C2311yescharacteristic
146C23×D83—21no—
147C2×Dic166—21no—
148C2×Dic166C22×C411yescharacteristic
149Q8×D83—22no—
150Q8×D83—41no—
151C2×D332C22×C411yescharacteristic
152C2×C22⋊D83—21no—
153D4×D83—22no—
154D4×D83—41no—
155C22×Dic83—21no—
156C2×C4×D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83—21no—
159C2×C4×D83—21no—
160C2×C4×D83—21no—
161C2×C4×D83—42no—
162C2×C4×D83—42no—
163C2×C4×D83C22×C411yescharacteristic
164C2×C4×D83C2×D411yescharacteristic
165C2×C4×D83#4111yescharacteristic
166#13050579—22no—
167#13050578—22no—
168#13050577M1614yescomplemented
169#13050574—42no—
170#13050573—22no—
171#13050573—42no—
172#13050573—42no—
173#13050573SD1614yescomplemented
174C2×C83⋊C8—21no—
175C2×C83⋊C8—21no—
176C2×C83⋊C8—21no—
177C2×C83⋊C8C22×C411yescharacteristic
178C2×C83⋊C8C22×C411yescharacteristic
179C2×C83⋊C8C2×D411yescharacteristic
180C2×C83⋊C8C2×D411yescharacteristic
181C8×D83—22no—
182C8×D83—41no—
183C8×D83—42no—
184C8×D83—42no—
185C4⋊Dic83—21no—
186C4⋊Dic83C22×C411yescharacteristic
187C4⋊Dic83C22×C411yescharacteristic
188Dic83⋊C4—21no—
189Dic83⋊C4—21no—
190Dic83⋊C4—21no—
191Dic83⋊C4—21no—
192C22⋊Dic83—21no—
193(C2×D83)⋊C4—21no—
194(C2×D83)⋊C4—21no—
195C4×Dic83—21no—
196C4×Dic83—21no—
197C4×Dic83—21no—
198C4×Dic83C22×C411yescharacteristic
199C4×Dic83C2×D411yescharacteristic
200C4×Dic83#4111yescharacteristic
201C2×Q8×C83C22×C411yescharacteristic
202C2×D4×C83C22×C411yescharacteristic
203C22×C4×C83—21no—
204SD16×C83—22no—
205M16×C83—42no—
206C2×C8×C83—21no—
207C2×C8×C83—21no—
208C2×C8×C83—21no—
209C2×C8×C83C22×C411yescharacteristic
210C2×C8×C83C22×C411yescharacteristic
211C2×C8×C83C2×D411yescharacteristic
212C2×C8×C83C2×D411yescharacteristic
213C4⋊C4×C83—21no—
214C4⋊C4×C83C22×C411yescharacteristic
215C4⋊C4×C83C22×C411yescharacteristic
216C22⋊C4×C83—21no—
217C42×C83—21no—
218C42×C83C2411yescharacteristic
219C23×C83—21no—
220C22×D83—21no—
221C22×D83—42no—
222C22×D83—42no—
223C22×D83—81no—
224C22×D83C4×D411yescharacteristic
225Q8×C83—21no—
226Q8×C83C2×M1612yescomplemented
227D4×C83—21no—
228D4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C22⋊D83—42no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—42no—
242C2×Dic83C4×D411yescharacteristic
243C2×Dic83C4×D411yescharacteristic
244C2×Dic83C4×D411yescharacteristic
245D332—21no—
246D332C2×M1612yes—
247C4×D83—21no—
248C4×D83—21no—
249C4×D83—41no—
250C4×D83—42no—
251C4×D83—42no—
252C4×D83—41no—
253C4×D83—41no—
254C4×D83—42no—
255C4×D83—81no—
256C4×D83C2×SD1612yes—
257C4×D83#13512yes—
258Dic166—21no—
259Dic166—21no—
260Dic166—42no—
261C8×C83—21no—
262C8×C83—21no—
263C8×C83—21no—
264C8×C83—41no—
265C8×C83—41no—
266C8×C83—41no—
267C8×C83C2×SD1612yescomplemented
268C83⋊C8—21no—
269C83⋊C8—21no—
270C83⋊C8—41no—
271C83⋊C8—41no—
272C83⋊C8—41no—
273C83⋊C8C2×M1612yescomplemented
274C83⋊C8C2×SD1612yescomplemented
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83C2×C4×D411yescharacteristic
279C2×D83—21no—
280C2×D83—21no—
281C2×D83—22no—
282C2×D83—41no—
283C2×D83—41no—
284C2×D83—84no—
285C2×D83C4×SD1612yes—
286C4×C83—22no—
287C4×C83—21no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83C22×M1611yescharacteristic
291C4×C83#56811yescharacteristic
292C4×C83C22×SD1611yescharacteristic
293C4×C83C2×#13511yescharacteristic
294Dic83—21no—
295Dic83—21no—
296Dic83—41no—
297Dic83—41no—
298Dic83—41no—
299Dic83C4×SD1612yes—
300M16×SD16—831nomaximal, Sylow
301C2×C83—81no—
302C2×C83D4×M1611yescharacteristic
303C2×C83C2×C4×SD1611yescharacteristic
304C2×C83#295711yescharacteristic
305D83—22no—
306D83—42no—
307Q8×M16—831no—
308C2×C4×SD16—831no—
309D4×M16—831no—
310#1590—831no—
311#1590—831no—
312#1583—831no—
313#1583D8311yescharacteristic, complemented
314#1582—831no—
315#1582—831no—
316C8×SD16—831no—
317C8×SD16—831no—
318#1566—831no—
319#1487—831no—
320#1486—831no—
321C8×M16—831no—
322C83M16×SD1611yescharacteristic, complemented, Sylow
323C22×SD16—831no—
324C22×M16—1661no—
325C2×C4×Q8—831no—
326C2×C4×D4—831no—
327#446—831no—
328#446—831no—
329#446—1661no—
330#446—1661no—
331C8×Q8—831no—
332C8×Q8C2×D8311yescharacteristic, complemented
333C4×SD16—831no—
334C4×SD16—831no—
335C4×SD16—831no—
336C4×SD16—831no—
337C4×SD16—831no—
338C4×SD16—831no—
339C4×SD16—831no—
340C4×SD16C2×D8311yescharacteristic
341#436—831no—
342#436C2×D8311yescharacteristic
343#435—1661no—
344#435—1661no—
345C8×D4—831no—
346C8×D4—831no—
347#431—831no—
348C2×#106—831no—
349#423—831no—
350#423—831no—
351#423—1661no—
352C2×#103—831no—
353C2×#102—831no—
354#407—1661no—
355C4×M16—831no—
356C4×M16—1661no—
357C2×C4×C8—831no—
358#334—831no—
359#334—831no—
360#334—831no—
361#334C2×D8311yescharacteristic, complemented
362Q8⋊C8—831no—
363Q8⋊C8—831no—
364Q8⋊C8—831no—
365Q8⋊C8C2×D8311yescharacteristic, complemented
366D4⋊C8—831no—
367D4⋊C8—831no—
368D4⋊C8—831no—
369D4⋊C8C2×D8311yescharacteristic
370#322—831no—
371#322C2×D8311yescharacteristic, complemented
372C82—831no—
373C82—831no—
374C22×Q8—831no—
375C22×D4—831no—
376C23×C4—1661no—
377C2×SD16—831no—
378C2×SD16—831no—
379C2×SD16—831no—
380C2×SD16—1664no—
381C2×SD16C4×D8311yescharacteristic
382C2×M16—21no—
383C2×M16—3324no—
384C2×M16—831no—
385C2×M16—831no—
386C2×M16—1661no—
387C2×M16—1661no—
388C2×M16—1661no—
389C22×C8—831no—
390C22×C8—831no—
391C22×C8—1661no—
392C22×C8—1661no—
393C4×Q8—831no—
394C4×Q8—831no—
395C4×Q8—831no—
396C4×Q8—1661no—
397C4×Q8—1661no—
398C4×Q8C22×D8311yescharacteristic
399C4×D4—831no—
400C4×D4—831no—
401C4×D4—831no—
402C4×D4—1661no—
403C4×D4—1661no—
404C4×D4C22×D8311yescharacteristic
405C2×C4⋊C4—831no—
406C2×C4⋊C4—831no—
407C2×C4⋊C4—1661no—
408C2×C22⋊C4—1661no—
409C2×C42—831no—
410C2×C42—1661no—
411#106—831no—
412#106—831no—
413#106—831no—
414#106C4×D8311yescharacteristic
415C4⋊C8—21no—
416C4⋊C8—831no—
417C4⋊C8—831no—
418C4⋊C8—1661no—
419C4⋊C8—1661no—
420C4⋊C8—1661no—
421C4⋊C8—1661no—
422C4⋊C8—1661no—
423C4⋊C8—1661no—
424C4⋊C8C22×D8311yescharacteristic
425C4⋊C8C22×D8311yescharacteristic
426#103—831no—
427#103—831no—
428#103—831no—
429#103C4×D8311yescharacteristic
430#102—831no—
431#102—831no—
432#102—831no—
433#102C4×D8311yescharacteristic
434C22⋊C8—21no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8—1661no—
439#97—831no—
440#97—831no—
441#97—1661no—
442#97—1661no—
443C4×C8—21no—
444C4×C8—831no—
445C4×C8—831no—
446C4×C8—831no—
447C4×C8—831no—
448C4×C8—831no—
449C4×C8—1661no—
450C4×C8C22×D8311yescharacteristic
451C4×C8C22×D8311yescharacteristic
452C4×C8C22×D8311yescharacteristic
453C24—1661no—
454C2×Q8—3321no—
455C2×Q8—831no—
456C2×Q8—1661no—
457C2×Q8—1662no—
458C2×Q8C2×C4×D8311yescharacteristic
459C2×D4—3321no—
460C2×D4—831no—
461C2×D4—1661no—
462C2×D4—1662no—
463C2×D4C2×C4×D8311yescharacteristic
464C22×C4—21no—
465C22×C4—3322no—
466C22×C4—3322no—
467C22×C4—831no—
468C22×C4—831no—
469C22×C4—831no—
470C22×C4—1661no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476SD16—22no—
477SD16—834no—
478SD16—1662no—
479SD16—1662no—
480M16—42no—
481M16—3322no—
482M16—3322no—
483M16—3322no—
484M16—834no—
485M16—1662no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—3321no—
490C2×C8—3322no—
491C2×C8—3322no—
492C2×C8—831no—
493C2×C8—831no—
494C2×C8—831no—
495C2×C8—831no—
496C2×C8—1662no—
497C2×C8—1661no—
498C2×C8—1661no—
499C2×C8—1661no—
500C2×C8C2×C4×D8311yescharacteristic
501C2×C8C2×C4×D8311yescharacteristic
502C2×C8D4×D8311yescharacteristic
503C2×C8D4×D8311yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—831no—
506C4⋊C4—831no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4—1661no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4C2×C4×D8311yescharacteristic
513C4⋊C4C2×C4×D8311yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—1661no—
516C22⋊C4—1661no—
517C22⋊C4—1661no—
518C42—21no—
519C42—831no—
520C42—831no—
521C42—831no—
522C42—1661no—
523C42—1661no—
524C42—1661no—
525C42C23×D8311yescharacteristic
526C23—21no—
527C23—3322no—
528C23—3322no—
529C23—831no—
530C23—6641no—
531C23—1661no—
532Q8—21no—
533Q8—3322no—
534Q8—1661no—
535Q8—1661no—
536Q8M16×D8312yescomplemented
537D4—21no—
538D4—21no—
539D4—3322no—
540D4—832no—
541D4—1661no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—3322no—
548C2×C4—3321no—
549C2×C4—3322no—
550C2×C4—3322no—
551C2×C4—3322no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—831no—
555C2×C4—832no—
556C2×C4—832no—
557C2×C4—831no—
558C2×C4—831no—
559C2×C4—6641no—
560C2×C4—1661no—
561C2×C4—1661no—
562C2×C4—1661no—
563C2×C4—1661no—
564C2×C4—1661no—
565C2×C4C22×C4×D8311yescharacteristic
566C2×C4C2×D4×D8311yescharacteristic
567C2×C4#41×D8311yescharacteristic
568C8—21no—
569C8—21no—
570C8—21no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—3321no—
575C8—3321no—
576C8—3321no—
577C8—832no—
578C8—832no—
579C8—1661no—
580C8—1661no—
581C8SD16×D8312yescomplemented
582C22—41no—
583C22—41no—
584C22—41no—
585C22—3321no—
586C22—3321no—
587C22—832no—
588C22—6644no—
589C22—1662no—
590C22—1661no—
591C22—1661no—
592C22C4×D4×D8311yescharacteristic
593C4—21no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—3321no—
598C4—3321no—
599C4—3321no—
600C4—832no—
601C4—1661no—
602C4—1661no—
603C4C2×M16×D8311yescharacteristic
604C4#1912846611yescharacteristic
605C4C2×SD16×D8311yescharacteristic
606C4#135×D8311yescharacteristic
607C2—81no—
608C2—3322no—
609C2—1662no—
610C2#2491619611yescharacteristic
611C2C4×SD16×D8311yescharacteristic
612C2#2491637711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1583C2
socleC22×C83C2×C4×D4
prime radicalC2×D4×D83C2×C4
cosocleC42×C83C24
hypercenter#1583D83
hypocenterC83M16×SD16
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
1. Q8×#13050573C2semidirect
4. #24906356C2semidirect
5. #24906351C2semidirect
6. #24906307C2semidirect
7. #24906301C2semidirect
8. #24906298C2semidirect
9. C8×#13050577C2semidirect
10. #24906260C2semidirect
11. #24906217C2semidirect
12. #24905890C2semidirect
14. #24905742C2semidirect
15. C83×#1583C2semidirect
21. #19127831C22semidirect
23. #19127828C22semidirect
24. Q8×C83⋊C8C22semidirect
48. #19127452C22semidirect
49. #19127451C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. Q8⋊(C83⋊C8)C22semidirect
53. #19127431C22semidirect
57. #19127423C22semidirect
58. C8×C83⋊C8C22semidirect
59. C8×C83⋊C8C22semidirect
60. C8×Q8×C83C22semidirect
63. #334×C83C22semidirect
64. Q8⋊C8×C83C22semidirect
66. #322×C83C22semidirect
168. #13050577M16semidirect
173. #13050573SD16semidirect
226. Q8×C83C2×M16semidirect
267. C8×C83C2×SD16semidirect
273. C83⋊C8C2×M16semidirect
274. C83⋊C8C2×SD16semidirect
313. #1583D83semidirect
322. C83M16×SD16semidirect
332. C8×Q8C2×D83semidirect
361. #334C2×D83semidirect
365. Q8⋊C8C2×D83semidirect
371. #322C2×D83semidirect
536. Q8M16×D83semidirect
581. C8SD16×D83semidirect

Maximal subgroups

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

subgroupindex
Q8×#130505732
C4×SD16×D832
#249161992
#249063562
#249063512
#249063072
#249063012
#249062982
C8×#130505772
#249062602
#249062172
#249058902
#249058862
#249057422
C83×#15832
M16×SD1683