Finite Group Databasebeta

← #7623360

Subgroups — #7623360

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3168C211yesmaximal, characteristic, complemented
2C2×#3147C211yesmaximal, characteristic, complemented
3#33785C214yesmaximal, complemented
4#33784C212yesmaximal, complemented
5#33783C212yesmaximal, complemented
6#33778C214yesmaximal, complemented
7#33777C214yesmaximal, complemented
8#33719C212yesmaximal, complemented
9#33708C212yesmaximal, complemented
10#33581C212yesmaximal, complemented
11#33580C212yesmaximal, complemented
12#33521C212yesmaximal, complemented
13#33520C212yesmaximal, complemented
14#33418C211yesmaximal, characteristic, complemented
15C22×#135—21no—
16C22×#135C2212yes—
17C2×#532—21no—
18C2×#529—21no—
19C2×#510C2214yescomplemented
20C2×#509C2214yescomplemented
21#3168—21no—
22#3168—21no—
23#3168—21no—
24#3168—21no—
25#3168C2212yes—
26#3168C2212yes—
27#3168C2212yescomplemented
28#3168C2212yescomplemented
29C2×#499C2211yescharacteristic, complemented
30#3147—21no—
31#3147—21no—
32#3147—21no—
33#3147—21no—
34#3147C2212yes—
35#3147C2212yes—
36#3147C2212yescomplemented
37#3147C2212yescomplemented
38C2×#495—21no—
39C2×#495C2211yescharacteristic, complemented
40C2×#495C2212yescomplemented
41C2×#494C2211yescharacteristic, complemented
42C2×#493C2211yescharacteristic, complemented
43C2×#492—21no—
44C2×#488—21no—
45C2×#467C2211yescharacteristic
46C2×#466C2211yescharacteristic
47C2×#465—21no—
48C2×C4×Q16C2211yescharacteristic, complemented
49C2×C4×SD16—21no—
50C2×C4×D8C2211yescharacteristic, complemented
51C22×C4×C8C2211yescharacteristic
52#2251C2214yescomplemented
53#2250C2214yescomplemented
54#2249C2218yescomplemented
55#2245C2214yescomplemented
56#2244C2214yescomplemented
57#2242C2218yescomplemented
58#2211C2218yescomplemented
59#2210C2218yescomplemented
60#2209C2214yescomplemented
61#2208C2214yescomplemented
62#2207C2214yescomplemented
63#2206C2214yescomplemented
64#2189C2218yescomplemented
65#2188C2214yescomplemented
66#2187C2214yescomplemented
67#2186C2218yescomplemented
68#2185C2214yes—
69#2184C2214yes—
70#2163C2212yescomplemented
71#2163C2212yescomplemented
72#2162C2212yescomplemented
73#2162C2212yescomplemented
74#2150C2212yescomplemented
75#2139C2214yescomplemented
76#2139C2214yescomplemented
77#2112C2212yes—
78C4×M32C2212yescomplemented
79C22×#41—22no—
80C2×#135—22no—
81C2×#135—22no—
82C2×#135—22no—
83C2×#135—22no—
84C2×#135—22no—
85C2×#135—22no—
86C2×#135—48no—
87C2×#135—48no—
88C2×#135C2314yes—
89C2×#135C2314yes—
90C22×Q16—21no—
91C22×Q16C2311yescharacteristic
92C22×Q16C2312yes—
93C22×SD16—22no—
94C22×SD16—22no—
95C22×SD16—21no—
96C22×D8—21no—
97C22×D8C2311yescharacteristic
98C22×D8C2312yes—
99C23×C8D411yescharacteristic
100C23×C8C2311yescharacteristic
101#532—22no—
102#532—22no—
103#532—22no—
104#532—22no—
105#532—41no—
106#532—41no—
107#532—41no—
108#532—41no—
109C2×C4⋊Q8—21no—
110C2×#127—21no—
111#529—22no—
112#529—22no—
113#529—22no—
114#529—22no—
115#529—41no—
116#529—41no—
117#529—41no—
118#529—41no—
119C2×#126—41no—
120C2×#125—21no—
121C2×#124—21no—
122C2×#124—22no—
123C2×#123—41no—
124C2×C22⋊Q8—21no—
125C2×#121—21no—
126C2×#121—41no—
127C2×#121—41no—
128C2×C4×Q8—21no—
129C2×C4×D4—21no—
130C2×C4×D4—41no—
131C2×C4×D4—41no—
132C2×C4×D4—41no—
133C2×#117—41no—
134C22×C42D411yescharacteristic
135#510—48no—
136#509—48no—
137C2×Q32C2318yes—
138C2×SD32C2318yes—
139C2×SD32C2318yes—
140C2×D16C2318yes—
141C2×M32C2314yes—
142#499—21no—
143#499—21no—
144#499C2312yes—
145#499C2312yes—
146#495—22no—
147#495—22no—
148#495—22no—
149#495—22no—
150#495—22no—
151#495—22no—
152#495—21no—
153#495—21no—
154#495C2314yes—
155#495C2314yes—
156#495C2312yes—
157#495C2312yes—
158#494—21no—
159#494—21no—
160#494C2312yes—
161#494C2312yes—
162#493—21no—
163#493—21no—
164#493C2312yes—
165#493C2312yes—
166#492—22no—
167#492—22no—
168#492—22no—
169#492—22no—
170#488—22no—
171#488—22no—
172#488—22no—
173#488—22no—
174#467—21no—
175#467—21no—
176#467C2312yes—
177#467C2312yes—
178#466—21no—
179#466—21no—
180#466C2312yes—
181#466C2312yes—
182#465—22no—
183#465—22no—
184#465—22no—
185#465—22no—
186C4×Q16—21no—
187C4×Q16—21no—
188C4×Q16C2312yes—
189C4×Q16C2312yes—
190C4×SD16—22no—
191C4×SD16—22no—
192C4×SD16—22no—
193C4×SD16—22no—
194C4×D8—21no—
195C4×D8—21no—
196C4×D8C2312yes—
197C4×D8C2312yes—
198C2×#107C2311yescharacteristic
199C2×#107C2311yescharacteristic
200C2×#106—21no—
201C2×#103—21no—
202C2×#103—21no—
203C2×#102—21no—
204C2×#102—21no—
205C2×C4×C8D411yescharacteristic
206C2×C4×C8D411yescharacteristic
207C2×C4×C8D411yescharacteristic
208C2×C4×C8D411yescharacteristic
209C2×C4×C8D411yescharacteristic
210C2×C4×C8D411yescharacteristic
211C2×C4×C8C2311yescharacteristic
212C2×C4×C8C2311yescharacteristic
213C2×C4×C8C2311yescharacteristic
214C2×C4×C8C2311yescharacteristic
215C2×C4×C8C2311yescharacteristic
216C2×C4×C8C2311yescharacteristic
217C4⋊C16C2314yes—
218C4⋊C16C2314yes—
219#358C2318yes—
220#358C2318yes—
221#357C2318yes—
222#357C2318yes—
223C22⋊C16C2314yes—
224C22⋊C16C2314yes—
225#346C2314yes—
226C4×C16C2314yes—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—48no—
230C2×#41—48no—
231C2×#41—42no—
232C2×#41—42no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—41no—
238C22×D4—42no—
239C22×D4—41no—
240C22×D4—41no—
241C23×C4—41no—
242C23×C4—42no—
243C23×C4D811yescharacteristic
244C23×C4D811yescharacteristic
245C23×C4C2×D411yescharacteristic
246#135—28no—
247#135—28no—
248#135—44no—
249#135—44no—
250#135—48no—
251#135—48no—
252#135—48no—
253#135—48no—
254#135—48no—
255#135—48no—
256C2×Q16—22no—
257C2×Q16—22no—
258C2×Q16—22no—
259C2×Q16—21no—
260C2×Q16—24no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×Q16C2414yes—
264C2×Q16C2412yes—
265C2×SD16—24no—
266C2×SD16—24no—
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16—22no—
270C2×SD16—22no—
271C2×SD16—48no—
272C2×SD16—48no—
273C2×SD16—44no—
274C2×D8—22no—
275C2×D8—22no—
276C2×D8—22no—
277C2×D8—21no—
278C2×D8—24no—
279C2×D8—44no—
280C2×D8—44no—
281C2×D8C2414yes—
282C2×D8C2412yes—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D411yescharacteristic
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—41no—
300C4⋊Q8—41no—
301#127—22no—
302#127—22no—
303#127—41no—
304#127—41no—
305#126—42no—
306#126—42no—
307#126—42no—
308#126—42no—
309#125—22no—
310#125—22no—
311#125—41no—
312#125—41no—
313#124—22no—
314#124—22no—
315#124—24no—
316#124—24no—
317#124—41no—
318#124—41no—
319#124—42no—
320#124—42no—
321#123—42no—
322#123—42no—
323#123—42no—
324#123—42no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—41no—
328C22⋊Q8—41no—
329#121—22no—
330#121—22no—
331#121—41no—
332#121—41no—
333#121—42no—
334#121—42no—
335#121—42no—
336#121—42no—
337#121—42no—
338#121—42no—
339#121—42no—
340#121—42no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—41no—
344C4×Q8—41no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—42no—
348C4×D4—42no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—41no—
352C4×D4—41no—
353C4×D4—42no—
354C4×D4—42no—
355C4×D4—42no—
356C4×D4—42no—
357C4×D4—42no—
358C4×D4—42no—
359C4×D4—42no—
360C4×D4—42no—
361#117—42no—
362#117—42no—
363#117—42no—
364#117—42no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—41no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—41no—
380C2×C42—41no—
381C2×C42D811yescharacteristic
382C2×C42D811yescharacteristic
383C2×C42D811yescharacteristic
384C2×C42D811yescharacteristic
385C2×C42D811yescharacteristic
386C2×C42D811yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393Q32—48no—
394SD32—48no—
395SD32—48no—
396D16—48no—
397M32—28no—
398C2×C16C2418yes—
399#107—21no—
400#107—21no—
401#107C2412yes—
402#107C2412yes—
403#106—22no—
404#106—22no—
405#103—22no—
406#103—22no—
407#103—22no—
408#103—22no—
409#102—22no—
410#102—22no—
411#102—22no—
412#102—22no—
413C4×C8C2×D411yescharacteristic
414C4×C8C2×D411yescharacteristic
415C4×C8C2×D411yescharacteristic
416C4×C8C2×D411yescharacteristic
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×C8C2411yescharacteristic
426C4×C8C2411yescharacteristic
427C4×C8C2411yescharacteristic
428C4×C8C2411yescharacteristic
429C24—41no—
430C24—42no—
431C24C2×D811yescharacteristic
432#41—48no—
433#41—48no—
434#41—48no—
435#41—48no—
436#41—88no—
437#41—88no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—41no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—84no—
463C2×D4—84no—
464C2×D4—84no—
465C2×D4—88no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—84no—
479C22×C4—88no—
480C22×C4C2×D811yescharacteristic
481C22×C4C2×D811yescharacteristic
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×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498Q16—24no—
499Q16—24no—
500Q16—24no—
501Q16—44no—
502Q16—44no—
503Q16—42no—
504Q16—44no—
505SD16—48no—
506SD16—48no—
507SD16—48no—
508SD16—44no—
509SD16—44no—
510SD16—48no—
511D8—24no—
512D8—24no—
513D8—24no—
514D8—44no—
515D8—44no—
516D8—42no—
517D8—44no—
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—41no—
541C4⋊C4—41no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—42no—
567C22⋊C4—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—28no—
587C23—22no—
588C23—22no—
589C23—41no—
590C23—42no—
591C23—44no—
592C23—84no—
593C23—82no—
594C23—84no—
595C23—82no—
596C23—84no—
597C23—82no—
598C23C22×D811yescharacteristic
599C23C22×D811yescharacteristic
600C23C22×D811yescharacteristic
601Q8—44no—
602Q8—42no—
603Q8—44no—
604Q8—42no—
605Q8—84no—
606Q8—82no—
607D4—44no—
608D4—42no—
609D4—44no—
610D4—42no—
611D4—84no—
612D4—84no—
613D4—84no—
614D4—84no—
615D4—84no—
616D4—88no—
617D4—84no—
618D4—88no—
619D4—84no—
620D4—82no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—84no—
645C2×C4—88no—
646C2×C4—82no—
647C2×C4—84no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—84no—
651C2×C4—88no—
652C2×C4#53512yes—
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×C4C22×D811yescharacteristic
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C23×D411yescharacteristic
666C8—21no—
667C8—21no—
668C8—21no—
669C8—21no—
670C8#53512yes—
671C8C2×AGL(1,ℤ/8)12yes—
672C8C2×AGL(1,ℤ/8)12yes—
673C8C2×H(5,𝔽2)12yes—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—44no—
682C22—84no—
683C22—82no—
684C22—82no—
685C22—84no—
686C22—82no—
687C22—84no—
688C22—168no—
689C22—164no—
690C22#320512yes—
691C22#341412yes—
692C22C23×D811yescharacteristic
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—84no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4#320512yes—
707C4#341412yes—
708C4C2×#53511yescharacteristic
709C4C22×AGL(1,ℤ/8)11yescharacteristic
710C2—22no—
711C2—22no—
712C2—41no—
713C2—164no—
714C2—162no—
715C2C2×#320511yescharacteristic
716C2C2×#341411yescharacteristic
717C2C2×#341411yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. C2×#3168C2semidirect
2. C2×#3147C2semidirect
3. #33785C2semidirect
4. #33784C2semidirect
5. #33783C2semidirect
6. #33778C2semidirect
7. #33777C2semidirect
8. #33719C2semidirect
9. #33708C2semidirect
10. #33581C2semidirect
11. #33580C2semidirect
12. #33521C2semidirect
13. #33520C2semidirect
14. #33418C2semidirect
19. C2×#510C22semidirect
20. C2×#509C22semidirect
27. #3168C22semidirect
28. #3168C22semidirect
29. C2×#499C22semidirect
36. #3147C22semidirect
37. #3147C22semidirect
39. C2×#495C22semidirect
40. C2×#495C22semidirect
41. C2×#494C22semidirect
42. C2×#493C22semidirect
48. C2×C4×Q16C22semidirect
50. C2×C4×D8C22semidirect
52. #2251C22semidirect
53. #2250C22semidirect
54. #2249C22semidirect
55. #2245C22semidirect
56. #2244C22semidirect
57. #2242C22semidirect
58. #2211C22semidirect
59. #2210C22semidirect
60. #2209C22semidirect
61. #2208C22semidirect
62. #2207C22semidirect
63. #2206C22semidirect
64. #2189C22semidirect
65. #2188C22semidirect
66. #2187C22semidirect
67. #2186C22semidirect
70. #2163C22semidirect
71. #2163C22semidirect
72. #2162C22semidirect
73. #2162C22semidirect
74. #2150C22semidirect
75. #2139C22semidirect
76. #2139C22semidirect
78. C4×M32C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31682
C2×#31472
#337852
#337842
#337832
#337782
#337772
#337192
#337082
#335812
#335802
#335212
#335202
#334182