Finite Group Databasebeta

← #6495530

Subgroups — #6495530

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33317C211yesmaximal, characteristic, complemented
4#33299C211yesmaximal, characteristic, complemented
5#33234C211yesmaximal, characteristic, complemented
6#33233C211yesmaximal, characteristic, complemented
7#29852C212yesmaximal, complemented
8#29851C212yesmaximal
9#29194C214yesmaximal, complemented
10#28659C212yesmaximal, complemented
11#28658C212yesmaximal, complemented
12#27893C214yesmaximal, complemented
13#25679C212yesmaximal, complemented
14#25678C212yesmaximal, complemented
15#22226C212yesmaximal, complemented
16#22225C212yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#491C2212yes—
22C2×#490C2212yes—
23C2×#489C2212yes—
24C2×#489C2212yes—
25C2×#432C2212yes—
26#2864C2212yes—
27#2863—21no—
28#2863—21no—
29#2862C2211yescharacteristic
30#2861C2212yes—
31#2860—21no—
32#2860—21no—
33#2859C2211yescharacteristic
34#2859C2211yescharacteristic
35#2858C2211yescharacteristic
36#2857C2211yescharacteristic
37#2857C2211yescharacteristic
38#2857C2211yescharacteristic
39#2856C2211yescharacteristic
40#2855—21no—
41#2855—21no—
42#2854—21no—
43#2854—21no—
44#2851C2212yes—
45#2850C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#125C2212yes—
51C4×#124C2212yes—
52C4×C22⋊Q8—21no—
53C4×C22⋊Q8—21no—
54C4×#121—21no—
55C4×#121—21no—
56#2295—21no—
57#2295—21no—
58#2293—21no—
59#2293—21no—
60#2283C2212yes—
61C2×C43C2211yescharacteristic
62#2068C2214yes—
63#2067C2214yes—
64#2066C2214yes—
65#2065C2214yes—
66#2036C2214yes—
67#2035C2214yes—
68#2034C2214yes—
69#2033C2214yes—
70#1992C2212yes—
71#1992C2212yes—
72#1992C2212yes—
73#1992C2212yes—
74#1961C2212yes—
75#1960C2212yes—
76#1959C2212yes—
77#1959C2212yes—
78#1958C2214yes—
79#1958C2214yes—
80#1928C2214yes—
81#1927C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1865C2214yes—
85#1864C2214yes—
86#1863C2214yes—
87#1862C2214yes—
88#1847C2212yes—
89#1847C2212yes—
90#1846C2214yes—
91#1835C2214yes—
92#1767C2212yes—
93C4×C4⋊C8C2212yes—
94#1749C2212yes—
95C4×C22⋊C8C2212yes—
96#1745C2212yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103#529—44no—
104#529—44no—
105C2×#126—22no—
106C2×#125C2312yes—
107C2×#125C2312yes—
108C2×#124C2312yes—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#501—44no—
133#501—44no—
134#491—28no—
135#490—28no—
136#489—44no—
137#489—44no—
138#432—28no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390C2312yes—
178#390C2312yes—
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#389—21no—
194#389—21no—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#386—22no—
216#386—22no—
217#385—22no—
218#385—22no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#383—22no—
224#382C2312yes—
225#382C2312yes—
226#382C2312yes—
227#382C2312yes—
228#381—22no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380C2312yes—
234#380C2312yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246#376—22no—
247C43D411yescharacteristic
248C43D411yescharacteristic
249C43D411yescharacteristic
250C43D411yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#336C2314yes—
260#336C2314yes—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8C2412yes—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2412yes—
268C23×C4—22no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C22×C8C2414yes—
277C22×C8C2414yes—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281#127—24no—
282#126—48no—
283#125—24no—
284#125—24no—
285#125—24no—
286#125—24no—
287#124—24no—
288#124—24no—
289#124—28no—
290#123—48no—
291C22⋊Q8—42no—
292C22⋊Q8—42no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295#121—42no—
296#121—42no—
297#121—44no—
298#121—44no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—48no—
308#117—48no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
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×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42—42no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—44no—
389#107—44no—
390#106—44no—
391#106—44no—
392C4⋊C8—28no—
393#103—28no—
394#103—28no—
395#103—44no—
396#103—44no—
397#102—28no—
398#102—28no—
399#102—44no—
400#102—44no—
401C22⋊C8—28no—
402#97—24no—
403#97—24no—
404#97—24no—
405#97#4118yes—
406C4×C8—24no—
407C4×C8—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
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—
428C24—22no—
429C24C22×D411yescharacteristic
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
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—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8C2×#4118yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42C2×#4114yes—
578C42H(5,𝔽2)14yes—
579C422−1+414yes—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23#57512yes—
594C23C23×D411yescharacteristic
595Q8—44no—
596Q8—44no—
597Q8—84no—
598Q8—84no—
599D4—44no—
600D4—44no—
601D4—84no—
602D4—84no—
603D4—88no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
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—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
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—88no—
663C2×C4#53212yes—
664C2×C4#53212yes—
665C2×C4#53512yes—
666C2×C4#53512yes—
667C2×C4#53612yes—
668C2×C4#53612yes—
669C2×C4#57512yes—
670C2×C4#57512yes—
671C2×C4#57512yes—
672C2×C4C22×#4112yes—
673C2×C4C2×H(5,𝔽2)12yes—
674C2×C4C2×2−1+412yes—
675C8—28no—
676C8—44no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22C2×#53211yescharacteristic
692C22C2×#53511yescharacteristic
693C22C2×#53611yescharacteristic
694C22C2×#57511yescharacteristic
695C22C2×#57511yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57511yescharacteristic
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4#351114yes—
719C2—42no—
720C2—162no—
721C2#3529811yescharacteristic
722C2#6070211yescharacteristic
723C2#6070211yescharacteristic
724C2#6080511yescharacteristic
725C2#6080511yescharacteristic
726C2#6085411yescharacteristic
727C2#6085411yescharacteristic
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. #33317C2semidirect
4. #33299C2semidirect
5. #33234C2semidirect
6. #33233C2semidirect
7. #29852C2semidirect
9. #29194C2semidirect
10. #28659C2semidirect
11. #28658C2semidirect
12. #27893C2semidirect
13. #25679C2semidirect
14. #25678C2semidirect
15. #22226C2semidirect
16. #22225C2semidirect
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
#333172
#332992
#332342
#332332
#298522
#298512
#291942
#286592
#286582
#278932
#256792
#256782
#222262
#222252
C4×#4322