Finite Group Databasebeta

← #6565758

Subgroups — #6565758

Order 512.

1879 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3365C212yesmaximal, complemented
2#36595C211yesmaximal, characteristic
3#34328C211yesmaximal, characteristic, complemented
4#29801C212yesmaximal, complemented
5#29796C211yesmaximal, characteristic
6#28751C212yesmaximal, complemented
7#28749C211yesmaximal, characteristic
8#28207C212yesmaximal, complemented
9#28205C211yesmaximal, characteristic, complemented
10#25281C214yesmaximal, complemented
11#25279C212yesmaximal, complemented
12#24635C214yesmaximal, complemented
13#24633C212yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23678C212yesmaximal, complemented
16#22734C211yesmaximal, characteristic, complemented
17#22543C212yesmaximal, complemented
18C2×#559C2212yes—
19C2×#557C2211yescharacteristic
20#3365—216no—
21C2×#488C2214yes—
22C2×#486C2212yes—
23C2×#475C2212yes—
24C2×#478C2214yes—
25C2×#464C2214yes—
26C2×#460C2212yes—
27C2×C8×Q8C2212yes—
28C2×C4×Q16C2212yes—
29C2×C4×SD16C2214yes—
30#2867C2211yescharacteristic
31#2858C2212yes—
32#2548—21no—
33#2548—21no—
34#2548C2212yes—
35#2544C2214yes—
36#2445—22no—
37#2445—22no—
38#2371—22no—
39#2370C2212yes—
40#2366—21no—
41#2366—21no—
42#2365—21no—
43#2365C2211yescharacteristic
44#2356—21no—
45#2356C2212yes—
46#2355C2211yescharacteristic
47#2355C2211yescharacteristic
48#2336—22no—
49#2327—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2212yes—
53C4×#125C2212yes—
54#2295C2211yescharacteristic
55#2295C2212yes—
56#2276—21no—
57C42×Q8C2211yescharacteristic
58#2084C2212yes—
59#2083C2211yescharacteristic
60#2072C2211yescharacteristic
61#2068C2212yes—
62#2065C2214yes—
63#2056C2212yes—
64#2055C2214yes—
65#2043C2212yes—
66#2043C2211yescharacteristic
67#2042C2214yes—
68#2041C2212yes—
69#2041C2212yes—
70#2041C2212yes—
71#2041C2212yes—
72#2016C2211yescharacteristic
73#2015C2214yes—
74#2014C2212yes—
75#2014C2212yes—
76#1983C2214yes—
77#1983C2214yes—
78#1983C2214yes—
79#1982C2212yes—
80#1982C2214yes—
81#1958C2214yes—
82#1957C2212yes—
83#1939C2212yes—
84#1939C2214yes—
85#1927C2214yes—
86#1926C2214yes—
87#1925C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1804C2214yes—
91#1803C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C4×#102C2214yes—
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98#559—28no—
99#559—28no—
100#557—24no—
101#557D418yescomplemented
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127—21no—
106C2×#127—21no—
107C2×#127—21no—
108C2×#127—21no—
109C2×#127C2312yes—
110C2×#127C2312yes—
111C2×#127C2312yes—
112C2×#125—22no—
113C2×#125C2312yes—
114C2×#124—22no—
115C2×#124C2314yes—
116C2×#124C2314yes—
117C2×#123—24no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4C2314yes—
131#488—216no—
132#486—28no—
133#478—216no—
134#475—28no—
135#464—216no—
136#460—28no—
137C8×Q8—28no—
138C4×Q16—28no—
139C4×SD16—216no—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2314yes—
161#394—24no—
162#394—24no—
163#394—24no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390C2314yes—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2312yes—
187#389C2311yescharacteristic
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—21no—
196#387—22no—
197#387—21no—
198#387—22no—
199#387—22no—
200#387—22no—
201#386—24no—
202#386—24no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#383—22no—
206#382—21no—
207#382C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229#376—21no—
230#376—22no—
231#376—22no—
232C43C2312yes—
233C43C2311yescharacteristic
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2314yes—
241#336C2312yes—
242C22×Q8—21no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4C2414yes—
253C23×C4—24no—
254C2×Q16—24no—
255C2×Q16—42no—
256C2×Q16—42no—
257C2×SD16—28no—
258C2×SD16—44no—
259C2×SD16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—22no—
265C4⋊Q8—42no—
266C4⋊Q8C2×D414yes—
267#127—24no—
268#127—24no—
269#127—44no—
270#127—44no—
271#127—44no—
272#127—44no—
273#127—82no—
274#127—82no—
275#125—216no—
276#125—24no—
277#125C2×D418yes—
278#124—28no—
279#124—28no—
280#124—44no—
281#124—44no—
282#124—48no—
283#123—416no—
284C4×Q8—28no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—42no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8C2×D418yes—
294C4×Q8C2×D414yes—
295C4×Q8C2×D414yes—
296C4×Q8#4114yes—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—48no—
300C4×D4—84no—
301C4×D4—84no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358#107—24no—
359#106—28no—
360C4⋊C8—24no—
361C4⋊C8—28no—
362C4⋊C8—44no—
363C4⋊C8—42no—
364#103—24no—
365#103—24no—
366#103—28no—
367#103—42no—
368#103—44no—
369#103—42no—
370#102—28no—
371#102—28no—
372#102—28no—
373#102—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377C4×C8—28no—
378C4×C8—24no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—24no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8#12114yes—
417C2×Q8C22×D412yes—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—48no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464Q16—48no—
465SD16—416no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4#12114yes—
503C4⋊C4#12118yes—
504C4⋊C4C22×D414yes—
505C4⋊C4C22×D412yes—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—28no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42—41no—
525C42—44no—
526C42—44no—
527C42—44no—
528C42—42no—
529C42—82no—
530C42—82no—
531C42#13514yes—
532C42C22×D414yes—
533C42C22×D412yes—
534C42C2×#4114yes—
535C42C2×#4112yes—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C23×D411yescharacteristic
544Q8—28no—
545Q8—24no—
546Q8—42no—
547Q8—48no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552D4—44no—
553D4—44no—
554D4—44no—
555D4—44no—
556D4—88no—
557D4—88no—
558D4—88no—
559D4—164no—
560D4—164no—
561D4—164no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—168no—
608C2×C4C2×#12114yes—
609C2×C4C2×#12114yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4#57512yes—
616C2×C4#57712yes—
617C2×C4C23×D411yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4#58512yes—
620C2×C4#58512yes—
621C8—42no—
622C8—44no—
623C8—42no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22C22×#12111yescharacteristic
632C22#348711yescharacteristic
633C22#350011yescharacteristic
634C22C22×#13511yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57711yescharacteristic
637C22#358511yescharacteristic
638C4—24no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—44no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4#303514yes—
656C4#348712yes—
657C4#352412yes—
658C2—164no—
659C2#3601811yescharacteristic
660C2C2×#303511yescharacteristic
661C2#6043111yescharacteristic
662C2#6188511yescharacteristic
663C2#6190411yescharacteristic
664C2#6253111yescharacteristic
665C2#6259011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3365C2semidirect
3. #34328C2semidirect
4. #29801C2semidirect
6. #28751C2semidirect
8. #28207C2semidirect
9. #28205C2semidirect
10. #25281C2semidirect
11. #25279C2semidirect
12. #24635C2semidirect
13. #24633C2semidirect
14. #23679C2semidirect
15. #23678C2semidirect
16. #22734C2semidirect
17. #22543C2semidirect
101. #557D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33652
#365952
#343282
#298012
#297962
#287512
#287492
#282072
#282052
#252812
#252792
#246352
#246332
#236792
#236782
#227342
#225432