Finite Group Databasebeta

← #6481757

Subgroups — #6481757

Order 512.

1886 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic
3C2×#2356C211yesmaximal, characteristic, complemented
4#33111C212yesmaximal, complemented
5#33108C211yesmaximal, characteristic, complemented
6#33106C211yesmaximal, characteristic
7#29719C212yesmaximal, complemented
8#29715C211yesmaximal, characteristic, complemented
9#29713C211yesmaximal, characteristic
10#24687C214yesmaximal, complemented
11#24685C214yesmaximal, complemented
12#24637C214yesmaximal, complemented
13#24635C214yesmaximal, complemented
14#21951C211yesmaximal, characteristic, complemented
15#21861C212yesmaximal, complemented
16C2×#529C2212yes—
17C22×C4×Q8C2211yescharacteristic
18#3108—216no—
19C2×#483C2212yes—
20C2×#480C2212yes—
21C2×#479C2214yes—
22C2×#464C2214yes—
23C2×#451C2212yes—
24C2×#448C2212yes—
25C2×C4×Q16C2214yes—
26C2×C4×SD16C2214yes—
27C2×#431C2212yes—
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2850C2212yes—
31#2618—21no—
32#2618—21no—
33#2601—22no—
34#2601—22no—
35#2566C2212yes—
36#2565C2212yes—
37#2545C2212yes—
38#2544C2212yes—
39#2356—22no—
40#2356—22no—
41#2356C2214yes—
42#2356C2214yes—
43#2325—21no—
44#2325—21no—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49C2×#383—21no—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×#376—21no—
55#2074C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#2056C2214yes—
59#2056C2214yes—
60#2055C2214yes—
61#2055C2214yes—
62#2016C2212yes—
63#2015C2214yes—
64#2014C2212yes—
65#1991C2214yes—
66#1983C2214yes—
67#1982C2212yes—
68#1981C2212yes—
69#1958C2214yes—
70#1949C2214yes—
71#1948C2212yes—
72#1947C2212yes—
73#1939C2214yes—
74#1939C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1926C2214yes—
78#1925C2214yes—
79#1874C2212yes—
80#1874C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1805C2214yes—
84#1804C2214yes—
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1797C2212yes—
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C22×SD16C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93#529—28no—
94#529—28no—
95C2×#126—22no—
96C2×#125C2312yes—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8C2312yes—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—24no—
106C2×C4×Q8D418yescomplemented
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2314yes—
109C2×C4×D4—22no—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#483—28no—
125#480—28no—
126#479—216no—
127#464—216no—
128#451—28no—
129#448—28no—
130C4×Q16—216no—
131C4×SD16—216no—
132#431—28no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C4×C8C2314yes—
150#398—22no—
151#398—22no—
152#396—22no—
153#396—22no—
154#396—22no—
155#396—22no—
156#394—22no—
157#394—22no—
158#391—22no—
159#391—21no—
160#391—21no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2314yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—24no—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#383—24no—
191#383—24no—
192#382—22no—
193#382C2312yes—
194#382C2314yes—
195#380—21no—
196#380—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207#376—24no—
208#376—24no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
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—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C2×D414yes—
231C22×Q8C2412yes—
232C22×Q8C2412yes—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×Q16—28no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—42no—
258#127—42no—
259#126—48no—
260#125—24no—
261#125—24no—
262#124—24no—
263#124—24no—
264#124—44no—
265#123—48no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268#121—24no—
269#121—24no—
270#121—44no—
271#121—44no—
272C4×Q8—216no—
273C4×Q8—28no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8C2×D4116yes—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—48no—
280#117—48no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C42—28no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D414yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342#107—28no—
343#106—28no—
344C4⋊C8—28no—
345C4⋊C8—44no—
346C4⋊C8—42no—
347C4⋊C8—42no—
348#103—28no—
349#103—28no—
350#103—28no—
351#103—44no—
352#103—44no—
353#102—28no—
354#102—44no—
355#102—44no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C4×C8—28no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—48no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—48no—
394C2×Q8C22×D414yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4#13514yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462Q16—416no—
463SD16—416no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C4⋊C4—216no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4C22×D414yes—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—216no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—42no—
515C42—48no—
516C42—44no—
517C42—44no—
518C42—44no—
519C42#12118yes—
520C42C22×D414yes—
521C42C2×#4114yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23C2×#12112yes—
534C23C2×#13512yes—
535C23#57512yes—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—48no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542D4—44no—
543D4—44no—
544D4—88no—
545D4—162no—
546D4—162no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—88no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4C2×#12114yes—
604C2×C4C2×#12112yes—
605C2×C4C2×#12114yes—
606C2×C4C2×#13512yes—
607C2×C4C2×#13512yes—
608C2×C4C23×D411yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4#58512yes—
611C2×C4#58512yes—
612C8—44no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22#303514yes—
628C22C22×#12111yescharacteristic
629C22#348511yescharacteristic
630C22#348711yescharacteristic
631C22C22×#13511yescharacteristic
632C22C2×#57511yescharacteristic
633C22#358611yescharacteristic
634C22#358611yescharacteristic
635C4—22no—
636C4—22no—
637C4—24no—
638C4—22no—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—44no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#344912yes—
652C4#349312yes—
653C2—24no—
654C2—162no—
655C2#3515011yescharacteristic
656C2C2×#303511yescharacteristic
657C2#6042411yescharacteristic
658C2#6169611yescharacteristic
659C2#6171611yescharacteristic
660C2#6188211yescharacteristic
661C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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×#3108C2semidirect
3. C2×#2356C2semidirect
4. #33111C2semidirect
5. #33108C2semidirect
7. #29719C2semidirect
8. #29715C2semidirect
10. #24687C2semidirect
11. #24685C2semidirect
12. #24637C2semidirect
13. #24635C2semidirect
14. #21951C2semidirect
15. #21861C2semidirect
106. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31082
#351032
C2×#23562
#331112
#331082
#331062
#297192
#297152
#297132
#246872
#246852
#246372
#246352
#219512
#218612