Finite Group Databasebeta

← #7623337

Subgroups — #7623337

Order 512.

1707 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60673C211yesmaximal, characteristic, complemented
2C2×#3147C211yesmaximal, characteristic, complemented
3#33778C214yesmaximal, complemented
4#33777C214yesmaximal, complemented
5#33776C212yesmaximal, complemented
6#33775C211yesmaximal, characteristic, complemented
7#33774C211yesmaximal, characteristic, complemented
8#33773C212yesmaximal, complemented
9#33772C211yesmaximal, characteristic, complemented
10#33771C211yesmaximal, characteristic, complemented
11#33713C212yesmaximal, complemented
12#33697C212yesmaximal, complemented
13#33579C214yesmaximal, complemented
14#33516C214yesmaximal, complemented
15C2×#2162C211yesmaximal, characteristic, complemented
16C22×#135—21no—
17C22×#135C2212yes—
18#3444—21no—
19C2×#532—21no—
20C2×#508C2218yescomplemented
21#3170C2211yescharacteristic
22#3170C2211yescharacteristic, complemented
23#3169C2211yescharacteristic
24#3169C2211yescharacteristic, complemented
25#3168—21no—
26#3168—21no—
27#3168C2212yes—
28#3168C2212yescomplemented
29#3167—21no—
30#3167—21no—
31#3160C2211yescharacteristic, complemented
32#3147—21no—
33#3147—21no—
34#3147—21no—
35#3147—21no—
36#3147C2212yes—
37#3147C2212yes—
38#3147C2212yes—
39#3147C2212yes—
40#3147C2211yescharacteristic, complemented
41C2×#495—21no—
42C2×#495C2212yes—
43C2×#494C2211yescharacteristic
44C2×#493C2211yescharacteristic
45C2×#492—21no—
46#3131—21no—
47#3050—21no—
48#3050C2212yes—
49C4×#135—21no—
50C4×#135C2212yescomplemented
51C22×C4×C8C2211yescharacteristic
52#2249C2218yescomplemented
53#2248C2212yescomplemented
54#2248C2212yescomplemented
55#2247C2212yescomplemented
56#2246C2212yescomplemented
57#2242C2218yescomplemented
58#2241C2212yescomplemented
59#2241C2212yescomplemented
60#2240C2212yescomplemented
61#2240C2212yescomplemented
62#2211C2218yescomplemented
63#2210C2218yescomplemented
64#2209C2214yescomplemented
65#2208C2214yescomplemented
66#2207C2214yescomplemented
67#2206C2214yescomplemented
68#2189C2218yescomplemented
69#2188C2214yes—
70#2187C2214yes—
71#2186C2218yescomplemented
72#2185C2214yes—
73#2184C2214yes—
74#2162C2211yescharacteristic
75#2162C2211yescharacteristic
76#2162C2212yes—
77#2162C2212yescomplemented
78#2162C2212yescomplemented
79C2×C4⋊C16C2212yescomplemented
80#2138C2218yescomplemented
81C2×C22⋊C16C2212yes—
82C2×C4×C16C2212yescomplemented
83C22×#41—22no—
84C2×#135—22no—
85C2×#135—22no—
86C2×#135—22no—
87C2×#135—22no—
88C2×#135—22no—
89C2×#135—22no—
90C2×#135—21no—
91C2×#135—48no—
92C2×#135—48no—
93C2×#135C2314yes—
94C2×#135C2312yes—
95C2×#135C2314yes—
96C22×Q16—21no—
97C22×Q16C2312yes—
98C22×SD16—22no—
99C22×SD16—22no—
100C22×D8—21no—
101C22×D8C2312yes—
102C23×C8D411yescharacteristic
103C23×C8C2311yescharacteristic
104#533—21no—
105#533—21no—
106#533—21no—
107#532—21no—
108#532—21no—
109#532—21no—
110#532—22no—
111#532—22no—
112#532—22no—
113#532—22no—
114#532—41no—
115#532—41no—
116#532—41no—
117#532—41no—
118C2×C4⋊Q8—21no—
119C2×#127—21no—
120#529—22no—
121#529—22no—
122#529—41no—
123#529—41no—
124#529—41no—
125#529—41no—
126C2×#124—22no—
127#525—22no—
128#525—41no—
129#525—41no—
130C2×#121—41no—
131C2×#121—41no—
132C4×#41—22no—
133C4×#41—41no—
134C4×#41—41no—
135C2×C4×D4—41no—
136C2×C4×D4—41no—
137C22×C42D411yescharacteristic
138#508—416no—
139C2×Q32C2318yes—
140C2×SD32C2318yes—
141C2×SD32C2318yes—
142C2×D16C2318yes—
143C22×C16C2314yes—
144#500C2311yescharacteristic
145#500C2311yescharacteristic
146#499—21no—
147#499C2312yes—
148#498—21no—
149#495—22no—
150#495—22no—
151#495—22no—
152#495—22no—
153#495—22no—
154#495—22no—
155#495—21no—
156#495C2314yes—
157#495C2314yes—
158#495C2312yes—
159#494—21no—
160#494—21no—
161#494C2312yes—
162#494C2312yes—
163#494C2311yescharacteristic
164#493—21no—
165#493—21no—
166#493C2312yes—
167#493C2312yes—
168#493C2311yescharacteristic
169#492—22no—
170#492—22no—
171#492—22no—
172#492—22no—
173#492—21no—
174#488—22no—
175#488—22no—
176#487—21no—
177#487—21no—
178#486—21no—
179#486—21no—
180#467—21no—
181#467—21no—
182#467C2312yes—
183#467C2312yes—
184#466—21no—
185#466—21no—
186#466C2312yes—
187#466C2312yes—
188#465—22no—
189#465—22no—
190#465—22no—
191#465—22no—
192C4×Q16—21no—
193C4×Q16—21no—
194C4×Q16C2312yes—
195C4×Q16C2312yes—
196C4×SD16—22no—
197C4×SD16—22no—
198C4×SD16—22no—
199C4×SD16—22no—
200C4×D8—21no—
201C4×D8—21no—
202C4×D8C2312yes—
203C4×D8C2312yes—
204#427—21no—
205#427C2312yes—
206#416—22no—
207#416—22no—
208C2×C4×C8D411yescharacteristic
209C2×C4×C8D411yescharacteristic
210C2×C4×C8D411yescharacteristic
211C2×C4×C8D411yescharacteristic
212C2×C4×C8D411yescharacteristic
213C2×C4×C8D411yescharacteristic
214C2×C4×C8C2311yescharacteristic
215C2×C4×C8C2311yescharacteristic
216C2×C4×C8C2311yescharacteristic
217C2×C4×C8C2311yescharacteristic
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2311yescharacteristic
220C4⋊C16C2312yes—
221C4⋊C16C2312yes—
222C4⋊C16C2314yes—
223#358C2318yes—
224#358C2318yes—
225#357C2318yes—
226#357C2318yes—
227C22⋊C16C2312yes—
228C22⋊C16C2312yes—
229C22⋊C16C2314yes—
230C4×C16C2312yes—
231C4×C16C2312yes—
232C4×C16C2314yes—
233C2×#41—22no—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—48no—
237C2×#41—48no—
238C2×#41—41no—
239C2×#41—41no—
240C2×#41—42no—
241C2×#41—42no—
242C22×Q8—22no—
243C22×D4—22no—
244C22×D4—42no—
245C22×D4—41no—
246C22×D4—41no—
247C23×C4—42no—
248C23×C4D811yescharacteristic
249C23×C4D811yescharacteristic
250C23×C4C2×D411yescharacteristic
251#135—216no—
252#135—28no—
253#135—48no—
254#135—48no—
255#135—48no—
256#135—48no—
257#135—48no—
258#135—48no—
259#135—44no—
260#135—48no—
261C2×Q16—22no—
262C2×Q16—22no—
263C2×Q16—22no—
264C2×Q16—21no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16C2414yes—
268C2×Q16C2412yes—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×SD16—48no—
276C2×SD16—48no—
277C2×D8—22no—
278C2×D8—22no—
279C2×D8—22no—
280C2×D8—21no—
281C2×D8—44no—
282C2×D8—44no—
283C2×D8C2414yes—
284C2×D8C2412yes—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D411yescharacteristic
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D411yescharacteristic
294C22×C8C2×D411yescharacteristic
295C22×C8C2×D411yescharacteristic
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298C22×C8C2411yescharacteristic
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—21no—
304C4⋊Q8—41no—
305C4⋊Q8—41no—
306#127—22no—
307#127—22no—
308#127—21no—
309#127—41no—
310#127—41no—
311#126—42no—
312#126—42no—
313#125—22no—
314#125—41no—
315#125—41no—
316#124—24no—
317#124—24no—
318#124—22no—
319#124—42no—
320#124—42no—
321#124—41no—
322#124—41no—
323#123—42no—
324#123—42no—
325#123—42no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—41no—
329C22⋊Q8—41no—
330C22⋊Q8—41no—
331C22⋊Q8—41no—
332#121—22no—
333#121—22no—
334#121—42no—
335#121—42no—
336#121—42no—
337#121—42no—
338#121—42no—
339#121—42no—
340#121—42no—
341#121—42no—
342#121—41no—
343#121—41no—
344#121—41no—
345#121—41no—
346#120—42no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—41no—
350C4×Q8—41no—
351C4×Q8—41no—
352C4×Q8—41no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—42no—
356C4×D4—42no—
357C4×D4—42no—
358C4×D4—42no—
359C4×D4—42no—
360C4×D4—42no—
361C4×D4—42no—
362C4×D4—42no—
363C4×D4—42no—
364C4×D4—42no—
365C4×D4—42no—
366C4×D4—42no—
367C4×D4—42no—
368C4×D4—42no—
369C4×D4—41no—
370C4×D4—41no—
371C4×D4—41no—
372C4×D4—41no—
373#117—22no—
374#117—41no—
375#117—41no—
376#117—42no—
377#117—42no—
378#117—42no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—41no—
381C2×C22⋊C4—42no—
382C2×C22⋊C4—42no—
383C2×C42—42no—
384C2×C42D811yescharacteristic
385C2×C42D811yescharacteristic
386C2×C42D811yescharacteristic
387C2×C42D811yescharacteristic
388C2×C42D811yescharacteristic
389C2×C42D811yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396Q32—48no—
397SD32—48no—
398SD32—48no—
399D16—48no—
400C2×C16—28no—
401C2×C16C2414yes—
402C2×C16C2414yes—
403#107—21no—
404#107—21no—
405#107C2412yes—
406#107C2412yes—
407#106—22no—
408#106—22no—
409#103—22no—
410#103—22no—
411#103—22no—
412#103—22no—
413#102—22no—
414#102—22no—
415#102—22no—
416#102—22no—
417C4×C8C2×D411yescharacteristic
418C4×C8C2×D411yescharacteristic
419C4×C8C2×D411yescharacteristic
420C4×C8C2×D411yescharacteristic
421C4×C8C2×D411yescharacteristic
422C4×C8C2×D411yescharacteristic
423C4×C8C2×D411yescharacteristic
424C4×C8C2×D411yescharacteristic
425C4×C8C2×D411yescharacteristic
426C4×C8C2×D411yescharacteristic
427C4×C8C2×D411yescharacteristic
428C4×C8C2×D411yescharacteristic
429C4×C8C2411yescharacteristic
430C4×C8C2411yescharacteristic
431C4×C8C2411yescharacteristic
432C4×C8C2411yescharacteristic
433C24—42no—
434C24C2×D811yescharacteristic
435#41—48no—
436#41—48no—
437#41—48no—
438#41—44no—
439#41—48no—
440#41—44no—
441#41—84no—
442#41—88no—
443#41—88no—
444#41—84no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—42no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—84no—
469C2×D4—88no—
470C2×D4—84no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—88no—
482C22×C4C2×D811yescharacteristic
483C22×C4C2×D811yescharacteristic
484C22×C4C2×D811yescharacteristic
485C22×C4C2×D811yescharacteristic
486C22×C4C2×D811yescharacteristic
487C22×C4C2×D811yescharacteristic
488C22×C4C2×D811yescharacteristic
489C22×C4C2×D811yescharacteristic
490C22×C4C2×D811yescharacteristic
491C22×C4C2×D811yescharacteristic
492C22×C4C2×D811yescharacteristic
493C22×C4C2×D811yescharacteristic
494C22×C4C2×D811yescharacteristic
495C22×C4C2×D811yescharacteristic
496C22×C4C2×D811yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500Q16—24no—
501Q16—28no—
502Q16—44no—
503Q16—44no—
504Q16—44no—
505Q16—42no—
506SD16—48no—
507SD16—48no—
508SD16—48no—
509SD16—44no—
510SD16—48no—
511SD16—44no—
512D8—24no—
513D8—28no—
514D8—44no—
515D8—44no—
516D8—44no—
517D8—42no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8AGL(1,ℤ/8)14yes—
529C2×C8C22×D411yescharacteristic
530C2×C8C22×D411yescharacteristic
531C2×C8C22×D411yescharacteristic
532C2×C8C22×D411yescharacteristic
533C2×C8C22×D411yescharacteristic
534C2×C8C22×D411yescharacteristic
535C2×C8C22×D411yescharacteristic
536C2×C8H(5,𝔽2)14yes—
537C2×C8C2511yescharacteristic
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42C2×D811yescharacteristic
571C42C2×D811yescharacteristic
572C42C2×D811yescharacteristic
573C42C2×D811yescharacteristic
574C42C2×D811yescharacteristic
575C42C2×D811yescharacteristic
576C42C2×D811yescharacteristic
577C42C2×D811yescharacteristic
578C42C2×D811yescharacteristic
579C42C2×D811yescharacteristic
580C42C2×D811yescharacteristic
581C42C2×D811yescharacteristic
582C42C22×D411yescharacteristic
583C42C22×D411yescharacteristic
584C42C22×D411yescharacteristic
585C42C22×D411yescharacteristic
586C16—24no—
587C16—24no—
588C23—22no—
589C23—22no—
590C23—41no—
591C23—42no—
592C23—44no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23C22×D811yescharacteristic
597C23C22×D811yescharacteristic
598C23C22×D811yescharacteristic
599Q8—44no—
600Q8—42no—
601Q8—44no—
602Q8—42no—
603Q8—84no—
604Q8—82no—
605Q8—82no—
606D4—44no—
607D4—42no—
608D4—44no—
609D4—42no—
610D4—84no—
611D4—84no—
612D4—88no—
613D4—84no—
614D4—84no—
615D4—88no—
616D4—84no—
617D4—84no—
618D4—84no—
619D4—82no—
620D4—84no—
621D4—82no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—88no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4#50814yescomplemented
651C2×C4#53512yes—
652C2×C4C22×D811yescharacteristic
653C2×C4C22×D811yescharacteristic
654C2×C4C22×D811yescharacteristic
655C2×C4C22×D811yescharacteristic
656C2×C4C22×D811yescharacteristic
657C2×C4C22×D811yescharacteristic
658C2×C4C22×D811yescharacteristic
659C2×C4C22×D811yescharacteristic
660C2×C4C22×D811yescharacteristic
661C2×C4C22×D811yescharacteristic
662C2×C4C22×D811yescharacteristic
663C2×C4C2×AGL(1,ℤ/8)12yes—
664C2×C4C23×D411yescharacteristic
665C8—21no—
666C8—21no—
667C8—21no—
668C8—21no—
669C8#53512yes—
670C8C2×AGL(1,ℤ/8)12yes—
671C8C2×AGL(1,ℤ/8)12yes—
672C8C2×H(5,𝔽2)12yes—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—44no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—84no—
685C22—82no—
686C22—84no—
687C22—168no—
688C22#320512yes—
689C22C2×#50812yes—
690C22C23×D811yescharacteristic
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#320512yes—
703C4C2×#50812yes—
704C4C2×#50812yes—
705C4#341412yes—
706C4C2×#53511yescharacteristic
707C4C22×AGL(1,ℤ/8)11yescharacteristic
708C2—22no—
709C2—22no—
710C2—41no—
711C2—162no—
712C2—164no—
713C2C2×#320511yescharacteristic
714C2C22×#50811yescharacteristic
715C2C2×#341411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutator, Frattini, cosocleC2×C8C25
socleC22C23×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60673C2semidirect
2. C2×#3147C2semidirect
3. #33778C2semidirect
4. #33777C2semidirect
5. #33776C2semidirect
6. #33775C2semidirect
7. #33774C2semidirect
8. #33773C2semidirect
9. #33772C2semidirect
10. #33771C2semidirect
11. #33713C2semidirect
12. #33697C2semidirect
13. #33579C2semidirect
14. #33516C2semidirect
15. C2×#2162C2semidirect
20. C2×#508C22semidirect
22. #3170C22semidirect
24. #3169C22semidirect
28. #3168C22semidirect
31. #3160C22semidirect
40. #3147C22semidirect
50. C4×#135C22semidirect
52. #2249C22semidirect
53. #2248C22semidirect
54. #2248C22semidirect
55. #2247C22semidirect
56. #2246C22semidirect
57. #2242C22semidirect
58. #2241C22semidirect
59. #2241C22semidirect
60. #2240C22semidirect
61. #2240C22semidirect
62. #2211C22semidirect
63. #2210C22semidirect
64. #2209C22semidirect
65. #2208C22semidirect
66. #2207C22semidirect
67. #2206C22semidirect
68. #2189C22semidirect
71. #2186C22semidirect
77. #2162C22semidirect
78. #2162C22semidirect
79. C2×C4⋊C16C22semidirect
80. #2138C22semidirect
82. C2×C4×C16C22semidirect
650. C2×C4#508semidirect

Maximal subgroups

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

subgroupindex
#606732
C2×#31472
#337782
#337772
#337762
#337752
#337742
#337732
#337722
#337712
#337132
#336972
#335792
#335162
C2×#21622