Finite Group Databasebeta

← #6481756

Subgroups — #6481756

Order 512.

2038 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic, complemented
3C2×#2357C211yesmaximal, characteristic, complemented
4#33110C212yesmaximal, complemented
5#33109C211yesmaximal, characteristic, complemented
6#33107C211yesmaximal, characteristic, complemented
7#29718C212yesmaximal, complemented
8#29714C211yesmaximal, characteristic, complemented
9#29712C211yesmaximal, characteristic
10#24686C214yesmaximal, complemented
11#24684C214yesmaximal, complemented
12#24636C214yesmaximal, complemented
13#24634C214yesmaximal, complemented
14#21951C211yesmaximal, characteristic, complemented
15#21861C212yesmaximal, complemented
16C2×#529C2212yes—
17C22×C4×D4C2211yescharacteristic
18#3105—216no—
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#478C2214yes—
22C2×#463C2214yes—
23C2×#450C2212yes—
24C2×#447C2212yes—
25C2×C4×SD16C2214yes—
26C2×C4×D8C2214yes—
27C2×#431C2212yes—
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2850C2212yes—
31#2618—21no—
32#2618—21no—
33#2601—22no—
34#2601—22no—
35#2566C2212yescomplemented
36#2565C2212yescomplemented
37#2545C2212yescomplemented
38#2544C2212yescomplemented
39#2357—22no—
40#2357—22no—
41#2357C2214yes—
42#2357C2214yes—
43#2325—21no—
44#2325—21no—
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#382C2211yescharacteristic
50C2×#381—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55#2073C2214yes—
56#2071C2212yes—
57#2071C2212yes—
58#2055C2214yes—
59#2055C2214yes—
60#2054C2214yes—
61#2054C2214yes—
62#2013C2212yes—
63#2012C2214yes—
64#2011C2212yes—
65#1991C2214yescomplemented
66#1983C2214yescomplemented
67#1982C2212yescomplemented
68#1981C2212yescomplemented
69#1958C2214yescomplemented
70#1949C2214yescomplemented
71#1948C2212yescomplemented
72#1947C2212yescomplemented
73#1939C2214yescomplemented
74#1939C2214yescomplemented
75#1928C2214yescomplemented
76#1927C2214yescomplemented
77#1926C2214yescomplemented
78#1925C2214yescomplemented
79#1874C2212yes—
80#1874C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1805C2214yescomplemented
84#1804C2214yescomplemented
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1797C2212yes—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C22×D8C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94#529—28no—
95#529—28no—
96C2×#126—22no—
97C2×#125C2312yes—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—24no—
109C2×C4×D4D418yescomplemented
110C2×C4×D4C2312yes—
111C2×C4×D4C2314yes—
112C2×#117—22no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#484—28no—
129#481—28no—
130#478—216no—
131#463—216no—
132#450—28no—
133#447—28no—
134C4×SD16—216no—
135C4×D8—216no—
136#431—28no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C4×C8C2314yes—
154#398—22no—
155#398—22no—
156#396—22no—
157#396—22no—
158#396—22no—
159#396—22no—
160#394—22no—
161#394—22no—
162#391C2312yes—
163#391C2312yes—
164#390—21no—
165#390—21no—
166#390—22no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2314yes—
173#390C2312yes—
174#389—21no—
175#389—21no—
176#388—24no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382C2312yes—
196#382C2314yes—
197#381—24no—
198#381—24no—
199#380—21no—
200#380—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—24no—
212C4×C22⋊C4—24no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95D411yescharacteristic
218C2×#95D411yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2314yes—
226C25—21no—
227C22×Q8C2412yes—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4C2×D414yes—
236C22×D4C2412yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—28no—
240C23×C4—24no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—82no—
245C23×C4—82no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×D8—28no—
257C2×D8—44no—
258C2×D8—44no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—42no—
262#127—42no—
263#126—48no—
264#125—24no—
265#125—24no—
266#124—24no—
267#124—24no—
268#124—44no—
269#123—48no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272#121—24no—
273#121—24no—
274#121—44no—
275#121—44no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×D4—216no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—48no—
283C4×D4C2×D4116yes—
284#117—48no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—82no—
343C2×C42—82no—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D414yes—
349C2×C42#4114yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354#107—28no—
355#106—28no—
356C4⋊C8—28no—
357C4⋊C8—44no—
358C4⋊C8—42no—
359C4⋊C8—42no—
360#103—28no—
361#103—44no—
362#103—44no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—44no—
367#102—44no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C4×C8—28no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24C22×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4C22×D414yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—216no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—816no—
461C22×C4#12114yes—
462C22×C4#12114yes—
463C22×C4#13514yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2511yescharacteristic
476SD16—416no—
477D8—416no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4C22×D414yes—
510C22⋊C4—216no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—41no—
532C42—41no—
533C42—42no—
534C42—44no—
535C42—82no—
536C42—82no—
537C42—84no—
538C42#12118yes—
539C42C22×D414yes—
540C42C2×#4114yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—24no—
546C23—48no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23C2×#12112yes—
567C23C2×#13512yes—
568C23#57512yes—
569C23C23×D411yescharacteristic
570Q8—44no—
571Q8—44no—
572D4—28no—
573D4—48no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—88no—
578D4—162no—
579D4—162no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—816no—
628C2×C4—88no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4C2×#12114yes—
634C2×C4C2×#12112yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#13512yes—
637C2×C4C2×#13512yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4#58512yes—
641C2×C4#58512yes—
642C8—44no—
643C8—44no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—816no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22#303514yes—
666C22C22×#12111yescharacteristic
667C22#348511yescharacteristic
668C22#348711yescharacteristic
669C22C22×#13511yescharacteristic
670C22C2×#57511yescharacteristic
671C22#358511yescharacteristic
672C22#358511yescharacteristic
673C4—22no—
674C4—22no—
675C4—24no—
676C4—22no—
677C4—24no—
678C4—42no—
679C4—44no—
680C4—42no—
681C4—41no—
682C4—41no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#344912yes—
689C4#349312yes—
690C2—24no—
691C2—84no—
692C2—162no—
693C2#3515011yescharacteristic
694C2C2×#303511yescharacteristic
695C2#6042311yescharacteristic
696C2#6169911yescharacteristic
697C2#6171811yescharacteristic
698C2#6188511yescharacteristic
699C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#29712C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3105C2semidirect
2. #35103C2semidirect
3. C2×#2357C2semidirect
4. #33110C2semidirect
5. #33109C2semidirect
6. #33107C2semidirect
7. #29718C2semidirect
8. #29714C2semidirect
10. #24686C2semidirect
11. #24684C2semidirect
12. #24636C2semidirect
13. #24634C2semidirect
14. #21951C2semidirect
15. #21861C2semidirect
35. #2566C22semidirect
36. #2565C22semidirect
37. #2545C22semidirect
38. #2544C22semidirect
65. #1991C22semidirect
66. #1983C22semidirect
67. #1982C22semidirect
68. #1981C22semidirect
69. #1958C22semidirect
70. #1949C22semidirect
71. #1948C22semidirect
72. #1947C22semidirect
73. #1939C22semidirect
74. #1939C22semidirect
75. #1928C22semidirect
76. #1927C22semidirect
77. #1926C22semidirect
78. #1925C22semidirect
83. #1805C22semidirect
84. #1804C22semidirect
109. C2×C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31052
#351032
C2×#23572
#331102
#331092
#331072
#297182
#297142
#297122
#246862
#246842
#246362
#246342
#219512
#218612