Finite Group Databasebeta

← #7623345

Subgroups — #7623345

Order 512.

1697 conjugacy classes of subgroups.

The subgroup lattice has 708 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#33785C214yesmaximal, complemented
4#33784C212yesmaximal, complemented
5#33783C212yesmaximal, complemented
6#33782C212yesmaximal, complemented
7#33781C212yesmaximal, complemented
8#33780C212yesmaximal, complemented
9#33779C212yesmaximal, complemented
10#33716C212yesmaximal, complemented
11#33701C212yesmaximal, complemented
12#33579C214yesmaximal, complemented
13#33516C214yesmaximal, complemented
14C2×#2163C211yesmaximal, characteristic, complemented
15C22×#135—21no—
16C22×#135C2212yes—
17#3444—21no—
18C2×#532—21no—
19C2×#508C2218yescomplemented
20#3170C2211yescharacteristic
21#3170C2211yescharacteristic, complemented
22#3169C2211yescharacteristic
23#3169C2211yescharacteristic, complemented
24#3168—21no—
25#3168—21no—
26#3168C2212yes—
27#3168C2212yescomplemented
28#3167—21no—
29#3167—21no—
30#3160C2211yescharacteristic, complemented
31#3147—21no—
32#3147—21no—
33#3147—21no—
34#3147—21no—
35#3147C2212yes—
36#3147C2212yes—
37#3147C2212yes—
38#3147C2212yes—
39#3147C2211yescharacteristic, complemented
40C2×#495—21no—
41C2×#495C2212yes—
42C2×#494C2211yescharacteristic
43C2×#493C2211yescharacteristic
44C2×#492—21no—
45#3131—21no—
46#3050—21no—
47#3050C2212yes—
48C4×#135—21no—
49C4×#135C2212yescomplemented
50C22×C4×C8C2211yescharacteristic
51#2251C2214yescomplemented
52#2251C2214yescomplemented
53#2250C2214yescomplemented
54#2250C2214yescomplemented
55#2245C2214yescomplemented
56#2244C2214yescomplemented
57#2243C2214yescomplemented
58#2243C2214yescomplemented
59#2211C2218yescomplemented
60#2210C2218yescomplemented
61#2209C2214yescomplemented
62#2208C2214yescomplemented
63#2207C2214yescomplemented
64#2206C2214yescomplemented
65#2189C2218yescomplemented
66#2188C2214yes—
67#2187C2214yes—
68#2186C2218yescomplemented
69#2185C2214yes—
70#2184C2214yes—
71#2163C2212yes—
72#2163C2212yes—
73#2163C2212yescomplemented
74#2163C2212yescomplemented
75C2×C4⋊C16C2212yescomplemented
76#2138C2218yescomplemented
77C2×C22⋊C16C2212yes—
78C2×#346C2212yescomplemented
79C22×#41—22no—
80C2×#135—22no—
81C2×#135—22no—
82C2×#135—22no—
83C2×#135—22no—
84C2×#135—22no—
85C2×#135—22no—
86C2×#135—21no—
87C2×#135—48no—
88C2×#135—48no—
89C2×#135C2314yes—
90C2×#135C2312yes—
91C2×#135C2314yes—
92C22×Q16—21no—
93C22×Q16C2312yes—
94C22×SD16—22no—
95C22×SD16—22no—
96C22×D8—21no—
97C22×D8C2312yes—
98C23×C8D411yescharacteristic
99C23×C8C2311yescharacteristic
100#533—21no—
101#533—21no—
102#533—21no—
103#532—21no—
104#532—21no—
105#532—21no—
106#532—22no—
107#532—22no—
108#532—22no—
109#532—22no—
110#532—41no—
111#532—41no—
112#532—41no—
113#532—41no—
114C2×C4⋊Q8—21no—
115C2×#127—21no—
116#529—22no—
117#529—22no—
118#529—41no—
119#529—41no—
120#529—41no—
121#529—41no—
122C2×#124—22no—
123#525—22no—
124#525—41no—
125#525—41no—
126C2×#121—41no—
127C2×#121—41no—
128C4×#41—22no—
129C4×#41—41no—
130C4×#41—41no—
131C2×C4×D4—41no—
132C2×C4×D4—41no—
133C22×C42D411yescharacteristic
134#508—416no—
135C2×Q32C2318yes—
136C2×SD32C2318yes—
137C2×SD32C2318yes—
138C2×D16C2318yes—
139C22×C16C2314yes—
140#500C2311yescharacteristic
141#500C2311yescharacteristic
142#499—21no—
143#499C2312yes—
144#498—21no—
145#495—22no—
146#495—22no—
147#495—22no—
148#495—22no—
149#495—22no—
150#495—22no—
151#495—21no—
152#495C2314yes—
153#495C2314yes—
154#495C2312yes—
155#494—21no—
156#494—21no—
157#494C2312yes—
158#494C2312yes—
159#494C2311yescharacteristic
160#493—21no—
161#493—21no—
162#493C2312yes—
163#493C2312yes—
164#493C2311yescharacteristic
165#492—22no—
166#492—22no—
167#492—22no—
168#492—22no—
169#492—21no—
170#488—22no—
171#488—22no—
172#487—21no—
173#487—21no—
174#486—21no—
175#486—21no—
176#467—21no—
177#467—21no—
178#467C2312yes—
179#467C2312yes—
180#466—21no—
181#466—21no—
182#466C2312yes—
183#466C2312yes—
184#465—22no—
185#465—22no—
186#465—22no—
187#465—22no—
188C4×Q16—21no—
189C4×Q16—21no—
190C4×Q16C2312yes—
191C4×Q16C2312yes—
192C4×SD16—22no—
193C4×SD16—22no—
194C4×SD16—22no—
195C4×SD16—22no—
196C4×D8—21no—
197C4×D8—21no—
198C4×D8C2312yes—
199C4×D8C2312yes—
200#427—21no—
201#427C2312yes—
202#416—22no—
203#416—22no—
204C2×C4×C8D411yescharacteristic
205C2×C4×C8D411yescharacteristic
206C2×C4×C8D411yescharacteristic
207C2×C4×C8D411yescharacteristic
208C2×C4×C8D411yescharacteristic
209C2×C4×C8D411yescharacteristic
210C2×C4×C8C2311yescharacteristic
211C2×C4×C8C2311yescharacteristic
212C2×C4×C8C2311yescharacteristic
213C2×C4×C8C2311yescharacteristic
214C2×C4×C8C2311yescharacteristic
215C2×C4×C8C2311yescharacteristic
216C4⋊C16C2314yes—
217C4⋊C16C2314yes—
218#358C2318yes—
219#358C2318yes—
220#357C2318yes—
221#357C2318yes—
222C22⋊C16C2314yes—
223C22⋊C16C2314yes—
224#346C2314yes—
225#346C2314yes—
226C2×#41—22no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—48no—
230C2×#41—48no—
231C2×#41—41no—
232C2×#41—41no—
233C2×#41—42no—
234C2×#41—42no—
235C22×Q8—22no—
236C22×D4—22no—
237C22×D4—42no—
238C22×D4—41no—
239C22×D4—41no—
240C23×C4—42no—
241C23×C4D811yescharacteristic
242C23×C4D811yescharacteristic
243C23×C4C2×D411yescharacteristic
244#135—216no—
245#135—28no—
246#135—48no—
247#135—48no—
248#135—48no—
249#135—48no—
250#135—48no—
251#135—48no—
252#135—44no—
253#135—48no—
254C2×Q16—22no—
255C2×Q16—22no—
256C2×Q16—22no—
257C2×Q16—21no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16C2414yes—
261C2×Q16C2412yes—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—22no—
267C2×SD16—22no—
268C2×SD16—48no—
269C2×SD16—48no—
270C2×D8—22no—
271C2×D8—22no—
272C2×D8—22no—
273C2×D8—21no—
274C2×D8—44no—
275C2×D8—44no—
276C2×D8C2414yes—
277C2×D8C2412yes—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8C2×D411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2411yescharacteristic
290C22×C8C2411yescharacteristic
291C22×C8C2411yescharacteristic
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—21no—
297C4⋊Q8—41no—
298C4⋊Q8—41no—
299#127—22no—
300#127—22no—
301#127—21no—
302#127—41no—
303#127—41no—
304#126—42no—
305#126—42no—
306#125—22no—
307#125—41no—
308#125—41no—
309#124—24no—
310#124—24no—
311#124—22no—
312#124—42no—
313#124—42no—
314#124—41no—
315#124—41no—
316#123—42no—
317#123—42no—
318#123—42no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—41no—
322C22⋊Q8—41no—
323C22⋊Q8—41no—
324C22⋊Q8—41no—
325#121—22no—
326#121—22no—
327#121—42no—
328#121—42no—
329#121—42no—
330#121—42no—
331#121—42no—
332#121—42no—
333#121—42no—
334#121—42no—
335#121—41no—
336#121—41no—
337#121—41no—
338#121—41no—
339#120—42no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—41no—
343C4×Q8—41no—
344C4×Q8—41no—
345C4×Q8—41no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—42no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—42no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4—42no—
355C4×D4—42no—
356C4×D4—42no—
357C4×D4—42no—
358C4×D4—42no—
359C4×D4—42no—
360C4×D4—42no—
361C4×D4—42no—
362C4×D4—41no—
363C4×D4—41no—
364C4×D4—41no—
365C4×D4—41no—
366#117—22no—
367#117—41no—
368#117—41no—
369#117—42no—
370#117—42no—
371#117—42no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—41no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4—42no—
376C2×C42—42no—
377C2×C42D811yescharacteristic
378C2×C42D811yescharacteristic
379C2×C42D811yescharacteristic
380C2×C42D811yescharacteristic
381C2×C42D811yescharacteristic
382C2×C42D811yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389Q32—48no—
390SD32—48no—
391SD32—48no—
392D16—48no—
393C2×C16—28no—
394C2×C16C2418yes—
395#107—21no—
396#107—21no—
397#107C2412yes—
398#107C2412yes—
399#106—22no—
400#106—22no—
401#103—22no—
402#103—22no—
403#103—22no—
404#103—22no—
405#102—22no—
406#102—22no—
407#102—22no—
408#102—22no—
409C4×C8C2×D411yescharacteristic
410C4×C8C2×D411yescharacteristic
411C4×C8C2×D411yescharacteristic
412C4×C8C2×D411yescharacteristic
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×C8C2411yescharacteristic
422C4×C8C2411yescharacteristic
423C4×C8C2411yescharacteristic
424C4×C8C2411yescharacteristic
425C24—42no—
426C24C2×D811yescharacteristic
427#41—48no—
428#41—48no—
429#41—48no—
430#41—44no—
431#41—48no—
432#41—44no—
433#41—84no—
434#41—84no—
435#41—88no—
436#41—88no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—84no—
461C2×D4—84no—
462C2×D4—88no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—88no—
474C22×C4C2×D811yescharacteristic
475C22×C4C2×D811yescharacteristic
476C22×C4C2×D811yescharacteristic
477C22×C4C2×D811yescharacteristic
478C22×C4C2×D811yescharacteristic
479C22×C4C2×D811yescharacteristic
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×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492Q16—24no—
493Q16—28no—
494Q16—44no—
495Q16—44no—
496Q16—44no—
497Q16—42no—
498SD16—48no—
499SD16—48no—
500SD16—48no—
501SD16—44no—
502SD16—48no—
503SD16—44no—
504D8—24no—
505D8—28no—
506D8—44no—
507D8—44no—
508D8—44no—
509D8—42no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8AGL(1,ℤ/8)14yes—
521C2×C8C22×D411yescharacteristic
522C2×C8C22×D411yescharacteristic
523C2×C8C22×D411yescharacteristic
524C2×C8C22×D411yescharacteristic
525C2×C8C22×D411yescharacteristic
526C2×C8C22×D411yescharacteristic
527C2×C8C22×D411yescharacteristic
528C2×C8H(5,𝔽2)14yes—
529C2×C8C2511yescharacteristic
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42C2×D811yescharacteristic
563C42C2×D811yescharacteristic
564C42C2×D811yescharacteristic
565C42C2×D811yescharacteristic
566C42C2×D811yescharacteristic
567C42C2×D811yescharacteristic
568C42C2×D811yescharacteristic
569C42C2×D811yescharacteristic
570C42C2×D811yescharacteristic
571C42C2×D811yescharacteristic
572C42C2×D811yescharacteristic
573C42C2×D811yescharacteristic
574C42C22×D411yescharacteristic
575C42C22×D411yescharacteristic
576C42C22×D411yescharacteristic
577C42C22×D411yescharacteristic
578C16—28no—
579C23—22no—
580C23—22no—
581C23—41no—
582C23—42no—
583C23—44no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23C22×D811yescharacteristic
588C23C22×D811yescharacteristic
589C23C22×D811yescharacteristic
590Q8—44no—
591Q8—42no—
592Q8—44no—
593Q8—42no—
594Q8—84no—
595Q8—82no—
596Q8—82no—
597D4—44no—
598D4—42no—
599D4—44no—
600D4—42no—
601D4—84no—
602D4—84no—
603D4—84no—
604D4—84no—
605D4—82no—
606D4—82no—
607D4—88no—
608D4—84no—
609D4—84no—
610D4—88no—
611D4—84no—
612D4—84no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—88no—
641C2×C4—84no—
642C2×C4#53512yes—
643C2×C4C22×D811yescharacteristic
644C2×C4C22×D811yescharacteristic
645C2×C4C22×D811yescharacteristic
646C2×C4C22×D811yescharacteristic
647C2×C4C22×D811yescharacteristic
648C2×C4C22×D811yescharacteristic
649C2×C4C22×D811yescharacteristic
650C2×C4C22×D811yescharacteristic
651C2×C4C22×D811yescharacteristic
652C2×C4C22×D811yescharacteristic
653C2×C4C22×D811yescharacteristic
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4C23×D411yescharacteristic
656C8—21no—
657C8—21no—
658C8—21no—
659C8—21no—
660C8#53512yes—
661C8C2×AGL(1,ℤ/8)12yes—
662C8C2×AGL(1,ℤ/8)12yes—
663C8C2×H(5,𝔽2)12yes—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—44no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22—84no—
678C22—168no—
679C22#320512yes—
680C22#341412yes—
681C22C23×D811yescharacteristic
682C4—21no—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#320512yes—
696C4#341412yes—
697C4C2×#53511yescharacteristic
698C4C22×AGL(1,ℤ/8)11yescharacteristic
699C2—22no—
700C2—22no—
701C2—41no—
702C2—162no—
703C2—164no—
704C2C2×#320511yescharacteristic
705C2C2×#341411yescharacteristic
706C2C2×#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. #33785C2semidirect
4. #33784C2semidirect
5. #33783C2semidirect
6. #33782C2semidirect
7. #33781C2semidirect
8. #33780C2semidirect
9. #33779C2semidirect
10. #33716C2semidirect
11. #33701C2semidirect
12. #33579C2semidirect
13. #33516C2semidirect
14. C2×#2163C2semidirect
19. C2×#508C22semidirect
21. #3170C22semidirect
23. #3169C22semidirect
27. #3168C22semidirect
30. #3160C22semidirect
39. #3147C22semidirect
49. C4×#135C22semidirect
51. #2251C22semidirect
52. #2251C22semidirect
53. #2250C22semidirect
54. #2250C22semidirect
55. #2245C22semidirect
56. #2244C22semidirect
57. #2243C22semidirect
58. #2243C22semidirect
59. #2211C22semidirect
60. #2210C22semidirect
61. #2209C22semidirect
62. #2208C22semidirect
63. #2207C22semidirect
64. #2206C22semidirect
65. #2189C22semidirect
68. #2186C22semidirect
73. #2163C22semidirect
74. #2163C22semidirect
75. C2×C4⋊C16C22semidirect
76. #2138C22semidirect
78. C2×#346C22semidirect

Maximal subgroups

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

subgroupindex
#606732
C2×#31472
#337852
#337842
#337832
#337822
#337812
#337802
#337792
#337162
#337012
#335792
#335162
C2×#21632