Finite Group Databasebeta

← #6485376

Subgroups — #6485376

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33282C212yesmaximal, complemented
4#33250C212yesmaximal, complemented
5#30319C212yesmaximal
6#30318C212yesmaximal, complemented
7#29501C212yesmaximal, complemented
8#29498C212yesmaximal, complemented
9#28753C211yesmaximal, characteristic, complemented
10#28723C211yesmaximal, characteristic, complemented
11#28692C211yesmaximal, characteristic, complemented
12#28447C211yesmaximal, characteristic, complemented
13#25678C212yesmaximal, complemented
14#25604C212yesmaximal, complemented
15#24657C214yesmaximal, complemented
16#22150C214yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#501C2212yes—
20C2×#497C2212yes—
21C2×#491C2212yes—
22C2×#489C2212yes—
23C2×#464C2214yes—
24C2×#432C2212yes—
25#2864—21no—
26#2863—21no—
27#2863C2212yes—
28#2862C2211yescharacteristic
29#2861C2212yes—
30#2860—21no—
31#2860—21no—
32#2859C2211yescharacteristic
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2857C2211yescharacteristic
38#2856C2211yescharacteristic
39#2855—21no—
40#2855C2212yes—
41#2854—21no—
42#2854—21no—
43#2851—21no—
44#2850C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C4×#125—21no—
50C4×#124C2212yes—
51C4×C22⋊Q8—21no—
52C4×C22⋊Q8C2212yes—
53C4×#121—21no—
54C4×#121—21no—
55#2295—21no—
56#2295C2212yes—
57#2293—21no—
58#2293—21no—
59#2283—21no—
60C2×C43C2211yescharacteristic
61#2084C2212yes—
62#2083C2212yes—
63#2072C2212yes—
64#2072C2212yes—
65#2043C2212yes—
66#2041C2212yes—
67#2036C2214yes—
68#2034C2214yes—
69#2016C2212yes—
70#2014C2212yes—
71#2001C2214yes—
72#2000C2214yes—
73#1992C2212yes—
74#1992C2212yes—
75#1983C2214yes—
76#1973C2214yes—
77#1971C2214yes—
78#1969C2214yes—
79#1968C2214yes—
80#1959C2212yes—
81#1959C2212yes—
82#1949C2214yes—
83#1936C2214yes—
84#1935C2214yes—
85#1934C2214yes—
86#1933C2214yes—
87#1878C2214yes—
88#1877C2214yes—
89#1865C2214yes—
90#1862C2214yes—
91#1805C2214yes—
92#1767C2211yescharacteristic
93#1767C2211yescharacteristic
94C4×C4⋊C8C2212yes—
95#1749C2212yes—
96C4×C22⋊C8C2212yes—
97#1745C2212yes—
98C4×#97C2212yes—
99C22×Q16C2314yes—
100C22×SD16C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105#529—44no—
106#529—44no—
107C2×#126—22no—
108C2×#125—21no—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#124C2312yes—
112C2×#123—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#501—44no—
135#497—44no—
136#491—44no—
137#489—44no—
138#464—216no—
139#432—28no—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2314yes—
156C2×C4×C8C2314yes—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#386—22no—
214#386—22no—
215#385—22no—
216#385—22no—
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#383—22no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382C2312yes—
226#381—22no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380C2312yes—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4C2312yes—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244#376—22no—
245C43D411yescharacteristic
246C43D411yescharacteristic
247C43D411yescharacteristic
248C43D411yescharacteristic
249C43C2311yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#336C2312yes—
258#336C2312yes—
259#336C2314yes—
260C22×Q8—21no—
261C22×Q8C2412yes—
262C22×Q8C2412yes—
263C22×D4—22no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4C2412yes—
267C23×C4—22no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2411yescharacteristic
275C2×Q16—28no—
276C2×Q16—44no—
277C2×Q16—44no—
278C2×SD16—28no—
279C2×SD16—44no—
280C2×SD16—44no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—42no—
288#127—42no—
289#126—48no—
290#125—24no—
291#125—24no—
292#125—44no—
293#124—28no—
294#124—24no—
295#124—24no—
296#123—48no—
297C22⋊Q8—44no—
298C22⋊Q8—42no—
299C22⋊Q8—42no—
300C22⋊Q8—44no—
301#121—42no—
302#121—42no—
303#121—44no—
304#121—44no—
305C4×Q8—28no—
306C4×Q8—42no—
307C4×Q8—42no—
308C4×Q8—44no—
309C4×D4—42no—
310C4×D4—42no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—48no—
314#117—48no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—42no—
364C2×C42—42no—
365C2×C42—42no—
366C2×C42—42no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—44no—
396#106—44no—
397C4⋊C8—24no—
398C4⋊C8#4118yes—
399#103—28no—
400#103—44no—
401#103—44no—
402#103—44no—
403#103—44no—
404#102—28no—
405#102—44no—
406#102—44no—
407C22⋊C8—28no—
408#97—28no—
409#97—24no—
410#97—24no—
411C4×C8—44no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432C24—22no—
433C24C22×D411yescharacteristic
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4#13714yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C2511yescharacteristic
504Q16—88no—
505SD16—88no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8C2×#4118yes—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—21no—
572C42—21no—
573C42—22no—
574C42—22no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42C2×#4114yes—
587C42H(5,𝔽2)12yes—
588C42H(5,𝔽2)12yes—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23C2×#13712yes—
603C23C23×D411yescharacteristic
604Q8—44no—
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—84no—
609D4—44no—
610D4—44no—
611D4—84no—
612D4—84no—
613D4—88no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—88no—
674C2×C4#53212yes—
675C2×C4#53212yes—
676C2×C4#53512yes—
677C2×C4#53512yes—
678C2×C4C2×#13712yes—
679C2×C4C2×#13712yes—
680C2×C4#57512yes—
681C2×C4#57512yes—
682C2×C4#57512yes—
683C2×C4C22×#4112yes—
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4#58512yes—
687C8—28no—
688C8—44no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22C2×#53211yescharacteristic
704C22C2×#53511yescharacteristic
705C22#344911yescharacteristic
706C22C22×#13711yescharacteristic
707C22C2×#57511yescharacteristic
708C22C2×#57511yescharacteristic
709C22C2×#57511yescharacteristic
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#351412yes—
731C4#351712yes—
732C2—42no—
733C2—162no—
734C2#3519111yescharacteristic
735C2#6067611yescharacteristic
736C2#6070111yescharacteristic
737C2#6080411yescharacteristic
738C2#6080511yescharacteristic
739C2#6095911yescharacteristic
740C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #33282C2semidirect
4. #33250C2semidirect
6. #30318C2semidirect
7. #29501C2semidirect
8. #29498C2semidirect
9. #28753C2semidirect
10. #28723C2semidirect
11. #28692C2semidirect
12. #28447C2semidirect
13. #25678C2semidirect
14. #25604C2semidirect
15. #24657C2semidirect
16. #22150C2semidirect
17. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#332822
#332502
#303192
#303182
#295012
#294982
#287532
#287232
#286922
#284472
#256782
#256042
#246572
#221502
C4×#4322