Finite Group Databasebeta

← #6486986

Subgroups — #6486986

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33268C214yesmaximal, complemented
4#30358C212yesmaximal, complemented
5#30319C212yesmaximal
6#29579C212yesmaximal, complemented
7#29561C212yesmaximal, complemented
8#29482C214yesmaximal, complemented
9#29355C211yesmaximal, characteristic, complemented
10#29319C211yesmaximal, characteristic, complemented
11#29130C211yesmaximal, characteristic, complemented
12#28382C211yesmaximal, characteristic, complemented
13#24709C212yesmaximal, complemented
14#24659C212yesmaximal, complemented
15#22146C212yesmaximal, complemented
16#22110C212yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#474C2212yes—
20C2×#461C2212yes—
21C2×#432C2212yes—
22#2864—21no—
23#2863—21no—
24#2863C2212yes—
25#2862C2211yescharacteristic
26#2861—21no—
27#2860—21no—
28#2860C2212yes—
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2856C2211yescharacteristic
36#2855—21no—
37#2855C2212yes—
38#2854—21no—
39#2854C2212yes—
40#2851—21no—
41#2850—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125—21no—
47C4×#124—21no—
48C4×C22⋊Q8—21no—
49C4×C22⋊Q8C2212yes—
50C4×#121—21no—
51C4×#121C2212yes—
52#2295—21no—
53#2295C2212yes—
54#2293—21no—
55#2293C2212yes—
56#2283—21no—
57C2×C43C2211yescharacteristic
58#2100C2212yes—
59#2095C2212yes—
60#2084C2212yes—
61#2076C2214yes—
62#2072C2212yes—
63#2057C2212yes—
64#2041C2212yes—
65#2036C2214yes—
66#2034C2214yes—
67#2024C2212yes—
68#2014C2212yes—
69#2000C2214yes—
70#1988C2212yes—
71#1981C2212yes—
72#1973C2214yes—
73#1973C2214yes—
74#1968C2214yes—
75#1948C2212yes—
76#1947C2212yes—
77#1944C2214yes—
78#1943C2214yes—
79#1942C2214yes—
80#1936C2214yes—
81#1936C2214yes—
82#1935C2214yes—
83#1935C2214yes—
84#1934C2214yes—
85#1933C2214yes—
86#1925C2214yes—
87#1918C2214yes—
88#1880C2214yes—
89#1865C2214yes—
90#1802C2212yes—
91#1802C2212yes—
92#1767C2211yescharacteristic
93#1767C2211yescharacteristic
94C4×C4⋊C8C2212yes—
95#1749C2212yes—
96C4×C22⋊C8C2212yes—
97#1745C2212yes—
98C4×#97C2212yes—
99C22×SD16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104#529—48no—
105C2×#126—22no—
106C2×#125—21no—
107C2×#125—21no—
108C2×#124—21no—
109C2×#124—21no—
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#474—28no—
133#461—28no—
134#432—28no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2314yes—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#383—22no—
216#382—21no—
217#382—21no—
218#382—21no—
219#382—21no—
220#381—22no—
221#380—21no—
222#380—21no—
223#380—21no—
224#380—21no—
225#380C2312yes—
226#380C2312yes—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238#376—22no—
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8C2412yes—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C2412yes—
263C23×C4—22no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C22×C8C2414yes—
275C22×C8C2414yes—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—42no—
279#127—42no—
280#126—48no—
281#125—44no—
282#125—44no—
283#124—44no—
284#124—44no—
285#123—48no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289#121—42no—
290#121—42no—
291#121—44no—
292#121—44no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—48no—
302#117—48no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
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×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—41no—
354C2×C42—41no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—28no—
384#107—44no—
385#106—44no—
386C4⋊C8—24no—
387C4⋊C8#4118yes—
388#103—28no—
389#103—44no—
390#103—44no—
391#102—28no—
392#102—44no—
393#102—44no—
394C22⋊C8—28no—
395#97—28no—
396#97—42no—
397#97—42no—
398C4×C8—44no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
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—
419C24—22no—
420C24C22×D411yescharacteristic
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
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—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4#13714yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2511yescharacteristic
492SD16—88no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8C2×#4118yes—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊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—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—22no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42AGL(1,ℤ/8)14yes—
576C42C2×#4114yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23C2×#13712yes—
591C23C23×D411yescharacteristic
592Q8—44no—
593Q8—44no—
594Q8—84no—
595Q8—84no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—84no—
601D4—84no—
602D4—88no—
603C2×C4—21no—
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—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
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×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4C2×#13712yes—
669C2×C4C2×#13712yes—
670C2×C4#57512yes—
671C2×C4#57512yes—
672C2×C4C22×#4112yes—
673C2×C4#58511yescharacteristic
674C2×C4#58512yes—
675C2×C4#58511yescharacteristic
676C8—28no—
677C8—44no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22C2×#53211yescharacteristic
693C22#344911yescharacteristic
694C22#344911yescharacteristic
695C22C22×AGL(1,ℤ/8)11yescharacteristic
696C22C22×#13711yescharacteristic
697C22C2×#57511yescharacteristic
698C22C2×#57511yescharacteristic
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4#317114yes—
719C4#352112yes—
720C4#352412yes—
721C2—42no—
722C2—162no—
723C2#3521211yescharacteristic
724C2C2×#317111yescharacteristic
725C2#6070111yescharacteristic
726C2#6096011yescharacteristic
727C2#6096211yescharacteristic
728C2#6096711yescharacteristic
729C2#6096911yescharacteristic
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. #33268C2semidirect
4. #30358C2semidirect
6. #29579C2semidirect
7. #29561C2semidirect
8. #29482C2semidirect
9. #29355C2semidirect
10. #29319C2semidirect
11. #29130C2semidirect
12. #28382C2semidirect
13. #24709C2semidirect
14. #24659C2semidirect
15. #22146C2semidirect
16. #22110C2semidirect
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
#332682
#303582
#303192
#295792
#295612
#294822
#293552
#293192
#291302
#283822
#247092
#246592
#221462
#221102
C4×#4322