Finite Group Databasebeta

← #6795849

Subgroups — #6795849

Order 512.

1607 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32198C214yesmaximal, complemented
5#30669C214yesmaximal, complemented
6#29922C212yesmaximal, complemented
7#29206C211yesmaximal, characteristic, complemented
8#28283C211yesmaximal, characteristic, complemented
9#28223C212yesmaximal
10#27625C212yesmaximal, complemented
11#26088C212yesmaximal, complemented
12#25341C212yesmaximal, complemented
13#24751C212yesmaximal, complemented
14#24678C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21959C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#475C2212yes—
20C2×#476C2212yes—
21C2×#465C2214yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2571—21no—
29#2563—21no—
30#2553—22no—
31#2552C2211yescharacteristic
32#2547C2211yescharacteristic
33#2542C2211yescharacteristic
34#2538C2212yes—
35#2537C2211yescharacteristic
36#2497—22no—
37#2496C2212yes—
38#2492C2214yes—
39#2489—21no—
40#2439—22no—
41#2427—21no—
42#2422C2212yes—
43#2395—22no—
44#2369—21no—
45#2355C2212yes—
46#2354—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8—21no—
50C4×#121C2212yes—
51#2285C2212yes—
52#2285C2212yes—
53C42×D4C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2096C2214yes—
56#2077C2214yes—
57#2072C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2058C2214yes—
61#2055C2214yes—
62#2035C2214yes—
63#2035C2214yes—
64#2034C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1981C2212yes—
68#1981C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1948C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2211yescharacteristic
76#1948C2214yes—
77#1939C2212yes—
78#1927C2212yes—
79#1926C2214yes—
80#1919C2214yes—
81#1892C2214yes—
82#1877C2212yes—
83#1850C2214yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1807C2214yes—
87#1807C2214yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96C23×C8C2314yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102#529—48no—
103C2×#126—22no—
104C2×#125—21no—
105C2×#124—21no—
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2314yes—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C42—21no—
124C22×C42C2312yes—
125#476—44no—
126#475—44no—
127#465—48no—
128#461—24no—
129C4×SD16—44no—
130C8×D4—44no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8—28no—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#400—24no—
153#400—22no—
154#399—22no—
155#398—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#396—24no—
160#394—22no—
161#394—22no—
162#393—24no—
163#391—21no—
164#391—21no—
165#391—24no—
166#391C2312yes—
167#391C2314yes—
168#390—22no—
169#390C2312yes—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#388—24no—
180#388—22no—
181#387—24no—
182#387—22no—
183#386—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#382—22no—
194#382—22no—
195#382—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—21no—
201#380C2314yes—
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8—28no—
252C22×C8—44no—
253C22×C8—44no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265#126—48no—
266#125—44no—
267#124—44no—
268#123—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273#121—24no—
274#121—42no—
275#121—44no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—42no—
287C4×D4—41no—
288C4×D4—48no—
289C4×D4SD1618yescomplemented
290#117—48no—
291#117—84no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#106—44no—
356#106—42no—
357#106—44no—
358#106—44no—
359#106—44no—
360C4⋊C8—28no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—22no—
364C4⋊C8—42no—
365C4⋊C8#4114yes—
366#103—24no—
367#103—42no—
368#103—42no—
369#103—44no—
370#103—44no—
371#102—28no—
372#102—42no—
373C22⋊C8—44no—
374C4×C8—24no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—21no—
394C24C22×D412yes—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—82no—
417C2×D4C2×SD1614yes—
418C2×D4H(5,𝔽2)12yes—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—81no—
447C22×C4C2×SD1618yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2511yescharacteristic
458SD16—48no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—88no—
468C2×C8C2×#4114yes—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C2×SD1614yes—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4C2×SD1618yes—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—22no—
527C42—21no—
528C42—41no—
529C42—44no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—84no—
535C42C2×SD1614yes—
536C42C2×#4112yes—
537C42H(5,𝔽2)12yes—
538C422−1+412yes—
539C422−1+412yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C22×SD1614yes—
554C23C23×D411yescharacteristic
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—84no—
560Q8—164no—
561D4—28no—
562D4—22no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—88no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—216no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4#53712yes—
618C2×C4#53712yes—
619C2×C4#54712yes—
620C2×C4#54712yes—
621C2×C4C22×SD1612yes—
622C2×C4C22×SD1614yes—
623C2×C4C22×SD1612yes—
624C2×C4C22×SD1612yes—
625C2×C4#57512yes—
626C2×C4#57712yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C2×C4#58511yescharacteristic
633C8—24no—
634C8—44no—
635C8—42no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22C2×#53711yescharacteristic
651C22C2×#54711yescharacteristic
652C22#348811yescharacteristic
653C22C23×SD1611yescharacteristic
654C22C2×#57511yescharacteristic
655C22C2×#57711yescharacteristic
656C22#358611yescharacteristic
657C4—22no—
658C4—24no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—41no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—81no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4#324614yes—
673C4#333514yes—
674C4#351811yescharacteristic
675C4#351911yescharacteristic
676C4#354411yescharacteristic
677C4#354711yescharacteristic
678C2—44no—
679C2—82no—
680C2#3892211yescharacteristic
681C2C2×#324611yescharacteristic
682C2#6093111yescharacteristic
683C2C2×#333511yescharacteristic
684C2#6125711yescharacteristic
685C2#6194811yescharacteristic
686C2#6198611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #33197C2semidirect
4. #32198C2semidirect
5. #30669C2semidirect
6. #29922C2semidirect
7. #29206C2semidirect
8. #28283C2semidirect
10. #27625C2semidirect
11. #26088C2semidirect
12. #25341C2semidirect
13. #24751C2semidirect
14. #24678C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21959C2semidirect
17. C4×#461C2semidirect
289. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361942
#349322
#331972
#321982
#306692
#299222
#292062
#282832
#282232
#276252
#260882
#253412
#247512
#246782
D4×C4⋊C82
#219592
C4×#4612