Finite Group Databasebeta

← #6495504

Subgroups — #6495504

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33311C211yesmaximal, characteristic, complemented
4#33291C211yesmaximal, characteristic, complemented
5#33226C211yesmaximal, characteristic, complemented
6#33226C211yesmaximal, characteristic, complemented
7#29846C214yesmaximal, complemented
8#29193C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic, complemented
10#29191C211yesmaximal, characteristic, complemented
11#29190C211yesmaximal, characteristic, complemented
12#28657C214yesmaximal, complemented
13#27892C211yesmaximal, characteristic, complemented
14#27891C211yesmaximal, characteristic
15#27890C211yesmaximal, characteristic
16#27889C211yesmaximal, characteristic, complemented
17#25675C214yesmaximal, complemented
18#22222C214yesmaximal, complemented
19C4×#431C211yesmaximal, characteristic, complemented
20C2×#529C2212yes—
21C2×#499C2214yes—
22C2×#488C2214yes—
23C2×#488C2214yes—
24C2×#431C2212yes—
25#2864—21no—
26#2864—21no—
27#2861C2212yes—
28#2860—21no—
29#2860—21no—
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2859C2211yescharacteristic
34#2859C2211yescharacteristic
35#2858C2211yescharacteristic
36#2857C2211yescharacteristic
37#2856C2211yescharacteristic
38#2854—21no—
39#2854—21no—
40#2851—21no—
41#2851—21no—
42#2851C2212yes—
43#2851C2212yes—
44#2850C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#125—21no—
48C4×#125—21no—
49C4×#125C2212yes—
50C4×#125C2212yes—
51C4×#124C2212yes—
52C4×#121—21no—
53C4×#121—21no—
54#2296—21no—
55#2296—21no—
56#2293—21no—
57#2293—21no—
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×C43C2211yescharacteristic
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2034C2214yes—
68#2033C2214yes—
69#1991C2214yes—
70#1991C2214yes—
71#1958C2214yes—
72#1958C2214yes—
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1928C2214yes—
82#1927C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1865C2214yes—
86#1864C2214yes—
87#1863C2214yes—
88#1862C2214yes—
89#1846C2214yes—
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1844C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2212yes—
101C4×C22⋊C8C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1744C2212yes—
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109#529—48no—
110C2×#126—22no—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#124C2312yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117—21no—
130C2×#117—22no—
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#499—48no—
139#488—216no—
140#488—48no—
141#431—28no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2312yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166#391C2312yes—
167#391C2312yes—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390C2312yes—
177#390C2312yes—
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#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#388—22no—
208#388—22no—
209#388—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#386—22no—
219#386—22no—
220#385—22no—
221#385—22no—
222#385—22no—
223#385—22no—
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#382—21no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382C2312yes—
235#382C2312yes—
236#382C2312yes—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380C2312yes—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—22no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4—22no—
248C4×C4⋊C4—22no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C22⋊C4—22no—
254C4×C22⋊C4—22no—
255C43D411yescharacteristic
256C43D411yescharacteristic
257C43D411yescharacteristic
258C43D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8C2412yes—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2412yes—
276C23×C4—22no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2411yescharacteristic
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290#127—22no—
291#127—22no—
292#126—48no—
293#125—28no—
294#125—28no—
295#125—28no—
296#125—44no—
297#125—44no—
298#124—28no—
299#124—28no—
300#123—48no—
301C22⋊Q8—44no—
302#121—44no—
303#121—44no—
304#121—44no—
305C4×Q8—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—48no—
309C4×D4—44no—
310#117—48no—
311#117—44no—
312#117—44no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—42no—
368C2×C42—42no—
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×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—44no—
398#107—44no—
399#106—44no—
400#106—44no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403#103—28no—
404#103—28no—
405#103—44no—
406#103—44no—
407#102—28no—
408#102—28no—
409#102—44no—
410#102—44no—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C4×C8—24no—
414C4×C8—24no—
415C4×C8—24no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8#4118yes—
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—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439C24—22no—
440C24C22×D411yescharacteristic
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
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—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4#13514yes—
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
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8C2×#4114yes—
515C2×C8C2×#4114yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—21no—
581C42—21no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42#13514yes—
590C42C2×#4114yes—
591C42H(5,𝔽2)14yes—
592C422−1+412yes—
593C422−1+412yes—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23C2×#13512yes—
608C23C23×D411yescharacteristic
609Q8—44no—
610Q8—44no—
611D4—44no—
612D4—44no—
613D4—88no—
614D4—84no—
615D4—84no—
616C2×C4—28no—
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—28no—
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—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—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—88no—
672C2×C4#53212yes—
673C2×C4#53212yes—
674C2×C4#53512yes—
675C2×C4#53512yes—
676C2×C4#53612yes—
677C2×C4#53612yes—
678C2×C4C2×#13512yes—
679C2×C4C2×#13512yes—
680C2×C4C2×#13512yes—
681C2×C4C2×#13512yes—
682C2×C4C2×#13512yes—
683C2×C4#57512yes—
684C2×C4#57512yes—
685C2×C4C22×#4112yes—
686C2×C4C2×H(5,𝔽2)12yes—
687C2×C4C2×2−1+411yescharacteristic
688C2×C4C2×2−1+411yescharacteristic
689C8—24no—
690C8—24no—
691C8—42no—
692C8—42no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22C2×#53211yescharacteristic
708C22C2×#53511yescharacteristic
709C22C2×#53611yescharacteristic
710C22C22×#13511yescharacteristic
711C22C22×#13511yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57511yescharacteristic
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4#323214yes—
733C4#323314yes—
734C4#351112yes—
735C4#351112yes—
736C2—42no—
737C2—162no—
738C2#3529811yescharacteristic
739C2#6067311yescharacteristic
740C2#6069311yescharacteristic
741C2#6078211yescharacteristic
742C2#6078211yescharacteristic
743C2C2×#323211yescharacteristic
744C2C2×#323311yescharacteristic
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. #36640C2semidirect
3. #33311C2semidirect
4. #33291C2semidirect
5. #33226C2semidirect
6. #33226C2semidirect
7. #29846C2semidirect
8. #29193C2semidirect
9. #29192C2semidirect
10. #29191C2semidirect
11. #29190C2semidirect
12. #28657C2semidirect
13. #27892C2semidirect
16. #27889C2semidirect
17. #25675C2semidirect
18. #22222C2semidirect
19. C4×#431C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#366312
#333112
#332912
#332262
#298462
#291932
#291922
#291912
#291902
#286572
#278922
#278912
#278902
#278892
#256752
#222222
C4×#4312