Finite Group Databasebeta

← #7623516

Subgroups — #7623516

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60673C211yesmaximal, characteristic, complemented
2#60673C211yesmaximal, characteristic, complemented
3#33814C212yesmaximal, complemented
4#33813C212yesmaximal, complemented
5#33806C212yesmaximal, complemented
6#33803C211yesmaximal, characteristic, complemented
7#33801C211yesmaximal, characteristic, complemented
8#33785C212yesmaximal, complemented
9#33782C211yesmaximal, characteristic, complemented
10#33780C211yesmaximal, characteristic, complemented
11#33757C214yesmaximal, complemented
12#33747C212yesmaximal, complemented
13#33746C212yesmaximal, complemented
14#33676C212yesmaximal, complemented
15#33675C212yesmaximal, complemented
16#33671C214yesmaximal, complemented
17C2×#2163C211yesmaximal, characteristic, complemented
18#3444—21no—
19#3444—21no—
20#3170C2211yescharacteristic
21#3170C2211yescharacteristic
22#3170C2211yescharacteristic
23#3170C2211yescharacteristic
24#3169C2211yescharacteristic
25#3169C2211yescharacteristic
26#3169C2211yescharacteristic
27#3169C2211yescharacteristic
28#3168—21no—
29#3168—21no—
30#3168—21no—
31#3168—21no—
32#3168C2212yes—
33#3168C2212yes—
34#3168C2212yes—
35#3168C2212yes—
36#3167—21no—
37#3167—21no—
38#3167—21no—
39#3167—21no—
40#3160C2211yescharacteristic
41#3160C2211yescharacteristic
42#3147C2211yescharacteristic
43#3147C2211yescharacteristic
44#3131—21no—
45#3131—21no—
46#3050—21no—
47#3050—21no—
48#3050C2212yes—
49#3050C2212yes—
50C4×#135—21no—
51C4×#135—21no—
52C4×#135C2212yes—
53C4×#135C2212yes—
54C22×C4×C8C2211yescharacteristic
55#2255C2212yescomplemented
56#2255C2212yescomplemented
57#2251C2212yescomplemented
58#2250C2212yescomplemented
59#2245C2212yescomplemented
60#2244C2212yescomplemented
61#2243C2212yescomplemented
62#2243C2212yescomplemented
63#2237C2214yes—
64#2237C2214yes—
65#2234C2214yes—
66#2234C2214yes—
67#2229C2214yescomplemented
68#2228C2214yescomplemented
69#2227C2212yescomplemented
70#2226C2212yescomplemented
71#2225C2212yescomplemented
72#2224C2212yescomplemented
73#2217C2214yes—
74#2217C2214yes—
75#2216C2214yes—
76#2215C2214yes—
77#2211C2214yescomplemented
78#2210C2214yescomplemented
79#2209C2212yescomplemented
80#2208C2212yescomplemented
81#2207C2212yescomplemented
82#2206C2212yescomplemented
83#2189C2214yes—
84#2189C2214yes—
85#2186C2214yes—
86#2186C2214yes—
87#2177C2214yescomplemented
88#2176C2214yescomplemented
89#2175C2214yescomplemented
90#2175C2214yescomplemented
91#2163C2211yescharacteristic
92#2163C2211yescharacteristic
93#2163C2212yes—
94#2163C2212yes—
95#2163C2212yes—
96C2×C4⋊C16C2212yes—
97C2×C22⋊C16C2212yes—
98C2×#346C2212yes—
99C2×#135—21no—
100C2×#135—21no—
101C2×#135C2312yes—
102C2×#135C2312yes—
103C23×C8D411yescharacteristic
104C23×C8C2311yescharacteristic
105#533—21no—
106#533—21no—
107#533—21no—
108#533—21no—
109#533—21no—
110#533—21no—
111#532—21no—
112#532—21no—
113#532—21no—
114#532—21no—
115#532—21no—
116#532—21no—
117#529—22no—
118#529—22no—
119#529—22no—
120#529—22no—
121#529—41no—
122#529—41no—
123#529—41no—
124#529—41no—
125#529—41no—
126#529—41no—
127#529—41no—
128#529—41no—
129#525—22no—
130#525—22no—
131#525—41no—
132#525—41no—
133#525—41no—
134#525—41no—
135C4×#41—22no—
136C4×#41—22no—
137C4×#41—41no—
138C4×#41—41no—
139C4×#41—41no—
140C4×#41—41no—
141C22×C42D411yescharacteristic
142C2×Q32C2314yes—
143C2×SD32C2314yes—
144C2×SD32C2314yes—
145C2×D16C2314yes—
146C22×C16C2314yes—
147#500C2311yescharacteristic
148#500C2311yescharacteristic
149#500C2311yescharacteristic
150#500C2311yescharacteristic
151#499—21no—
152#499—21no—
153#499C2312yes—
154#499C2312yes—
155#498—21no—
156#498—21no—
157#495—21no—
158#495—21no—
159#495C2312yes—
160#495C2312yes—
161#494C2311yescharacteristic
162#494C2311yescharacteristic
163#493C2311yescharacteristic
164#493C2311yescharacteristic
165#492—21no—
166#492—21no—
167#488—22no—
168#488—22no—
169#488—22no—
170#488—22no—
171#487—21no—
172#487—21no—
173#487—21no—
174#487—21no—
175#486—21no—
176#486—21no—
177#486—21no—
178#486—21no—
179#467—21no—
180#467—21no—
181#467—21no—
182#467—21no—
183#467C2312yes—
184#467C2312yes—
185#467C2312yes—
186#467C2312yes—
187#466—21no—
188#466—21no—
189#466—21no—
190#466—21no—
191#466C2312yes—
192#466C2312yes—
193#466C2312yes—
194#466C2312yes—
195#465—22no—
196#465—22no—
197#465—22no—
198#465—22no—
199#465—22no—
200#465—22no—
201#465—22no—
202#465—22no—
203C4×Q16—21no—
204C4×Q16—21no—
205C4×Q16—21no—
206C4×Q16—21no—
207C4×Q16C2312yes—
208C4×Q16C2312yes—
209C4×Q16C2312yes—
210C4×Q16C2312yes—
211C4×SD16—22no—
212C4×SD16—22no—
213C4×SD16—22no—
214C4×SD16—22no—
215C4×SD16—22no—
216C4×SD16—22no—
217C4×SD16—22no—
218C4×SD16—22no—
219C4×D8—21no—
220C4×D8—21no—
221C4×D8—21no—
222C4×D8—21no—
223C4×D8C2312yes—
224C4×D8C2312yes—
225C4×D8C2312yes—
226C4×D8C2312yes—
227#427—21no—
228#427—21no—
229#427C2312yes—
230#427C2312yes—
231#416—22no—
232#416—22no—
233#416—22no—
234#416—22no—
235C2×C4×C8D411yescharacteristic
236C2×C4×C8D411yescharacteristic
237C2×C4×C8D411yescharacteristic
238C2×C4×C8D411yescharacteristic
239C2×C4×C8D411yescharacteristic
240C2×C4×C8D411yescharacteristic
241C2×C4×C8C2311yescharacteristic
242C2×C4×C8C2311yescharacteristic
243C2×C4×C8C2311yescharacteristic
244C2×C4×C8C2311yescharacteristic
245C2×C4×C8C2311yescharacteristic
246C2×C4×C8C2311yescharacteristic
247#367C2314yes—
248#367C2314yes—
249#366C2314yes—
250#366C2314yes—
251C4⋊C16C2312yes—
252C4⋊C16C2312yes—
253C4⋊C16C2314yes—
254#358C2314yes—
255#358C2314yes—
256#358C2314yes—
257#358C2314yes—
258#357C2314yes—
259#357C2314yes—
260#357C2314yes—
261#357C2314yes—
262C22⋊C16C2312yes—
263C22⋊C16C2314yes—
264C22⋊C16C2312yes—
265#346C2314yes—
266#346C2312yes—
267#346C2312yes—
268C2×#41—22no—
269C2×#41—22no—
270C2×#41—41no—
271C2×#41—41no—
272C2×#41—41no—
273C2×#41—41no—
274C23×C4D811yescharacteristic
275C23×C4D811yescharacteristic
276C23×C4C2×D411yescharacteristic
277#135—42no—
278#135—44no—
279#135—42no—
280#135—44no—
281#135—44no—
282C2×Q16—21no—
283C2×Q16—21no—
284C2×Q16C2412yes—
285C2×Q16C2412yes—
286C2×SD16—22no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—22no—
290C2×D8—21no—
291C2×D8—21no—
292C2×D8C2412yes—
293C2×D8C2412yes—
294C22×C8—42no—
295C22×C8—42no—
296C22×C8C2×D411yescharacteristic
297C22×C8C2×D411yescharacteristic
298C22×C8C2×D411yescharacteristic
299C22×C8C2×D411yescharacteristic
300C22×C8C2×D411yescharacteristic
301C22×C8C2×D411yescharacteristic
302C22×C8C2×D411yescharacteristic
303C22×C8C2×D411yescharacteristic
304C22×C8C2×D411yescharacteristic
305C22×C8C2411yescharacteristic
306C22×C8C2411yescharacteristic
307C22×C8C2411yescharacteristic
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314#127—21no—
315#127—21no—
316#126—42no—
317#126—42no—
318#126—42no—
319#126—42no—
320#125—22no—
321#125—22no—
322#125—41no—
323#125—41no—
324#125—41no—
325#125—41no—
326#124—22no—
327#124—22no—
328#124—41no—
329#124—41no—
330#124—41no—
331#124—41no—
332#123—42no—
333#123—42no—
334#123—42no—
335#123—42no—
336#123—42no—
337#123—42no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—41no—
343C22⋊Q8—41no—
344C22⋊Q8—41no—
345C22⋊Q8—41no—
346C22⋊Q8—41no—
347C22⋊Q8—41no—
348C22⋊Q8—41no—
349C22⋊Q8—41no—
350#121—22no—
351#121—22no—
352#121—22no—
353#121—22no—
354#121—41no—
355#121—41no—
356#121—41no—
357#121—41no—
358#121—41no—
359#121—41no—
360#121—41no—
361#121—41no—
362#120—42no—
363#120—42no—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—41no—
369C4×Q8—41no—
370C4×Q8—41no—
371C4×Q8—41no—
372C4×Q8—41no—
373C4×Q8—41no—
374C4×Q8—41no—
375C4×Q8—41no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—42no—
381C4×D4—42no—
382C4×D4—42no—
383C4×D4—42no—
384C4×D4—42no—
385C4×D4—42no—
386C4×D4—42no—
387C4×D4—42no—
388C4×D4—42no—
389C4×D4—42no—
390C4×D4—42no—
391C4×D4—42no—
392C4×D4—41no—
393C4×D4—41no—
394C4×D4—41no—
395C4×D4—41no—
396C4×D4—41no—
397C4×D4—41no—
398C4×D4—41no—
399C4×D4—41no—
400#117—22no—
401#117—22no—
402#117—41no—
403#117—41no—
404#117—41no—
405#117—41no—
406#117—42no—
407#117—42no—
408#117—42no—
409#117—42no—
410#117—42no—
411#117—42no—
412C2×C42—42no—
413C2×C42—42no—
414C2×C42D811yescharacteristic
415C2×C42D811yescharacteristic
416C2×C42D811yescharacteristic
417C2×C42D811yescharacteristic
418C2×C42D811yescharacteristic
419C2×C42D811yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2×D411yescharacteristic
422C2×C42C2×D411yescharacteristic
423C2×C42C2×D411yescharacteristic
424C2×C42C2×D411yescharacteristic
425C2×C42C2×D411yescharacteristic
426Q32—44no—
427SD32—44no—
428SD32—44no—
429D16—44no—
430C2×C16—44no—
431C2×C16C2414yes—
432C2×C16C2414yes—
433#107—21no—
434#107—21no—
435#107—21no—
436#107—21no—
437#107C2412yes—
438#107C2412yes—
439#107C2412yes—
440#107C2412yes—
441#106—22no—
442#106—22no—
443#106—22no—
444#106—22no—
445#103—22no—
446#103—22no—
447#103—22no—
448#103—22no—
449#103—22no—
450#103—22no—
451#103—22no—
452#103—22no—
453#102—22no—
454#102—22no—
455#102—22no—
456#102—22no—
457#102—22no—
458#102—22no—
459#102—22no—
460#102—22no—
461C4×C8C2×D411yescharacteristic
462C4×C8C2×D411yescharacteristic
463C4×C8C2×D411yescharacteristic
464C4×C8C2×D411yescharacteristic
465C4×C8C2×D411yescharacteristic
466C4×C8C2×D411yescharacteristic
467C4×C8C2×D411yescharacteristic
468C4×C8C2×D411yescharacteristic
469C4×C8C2×D411yescharacteristic
470C4×C8C2×D411yescharacteristic
471C4×C8C2×D411yescharacteristic
472C4×C8C2×D411yescharacteristic
473C4×C8C2411yescharacteristic
474C4×C8C2411yescharacteristic
475C4×C8C2411yescharacteristic
476C4×C8C2411yescharacteristic
477C24C2×D811yescharacteristic
478#41—44no—
479#41—44no—
480#41—44no—
481#41—44no—
482#41—84no—
483#41—84no—
484#41—84no—
485#41—84no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—41no—
489C2×Q8—41no—
490C2×Q8—41no—
491C2×Q8—41no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—41no—
501C2×D4—41no—
502C2×D4—41no—
503C2×D4—41no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4C2×D811yescharacteristic
515C22×C4C2×D811yescharacteristic
516C22×C4C2×D811yescharacteristic
517C22×C4C2×D811yescharacteristic
518C22×C4C2×D811yescharacteristic
519C22×C4C2×D811yescharacteristic
520C22×C4C2×D811yescharacteristic
521C22×C4C2×D811yescharacteristic
522C22×C4C2×D811yescharacteristic
523C22×C4C2×D811yescharacteristic
524C22×C4C2×D811yescharacteristic
525C22×C4C2×D811yescharacteristic
526C22×C4C2×D811yescharacteristic
527C22×C4C2×D811yescharacteristic
528C22×C4C2×D811yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532Q16—22no—
533Q16—24no—
534Q16—22no—
535Q16—42no—
536Q16—42no—
537SD16—44no—
538SD16—44no—
539SD16—44no—
540SD16—44no—
541D8—22no—
542D8—24no—
543D8—22no—
544D8—42no—
545D8—42no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—41no—
553C2×C8—41no—
554C2×C8—41no—
555C2×C8—41no—
556C2×C8—41no—
557C2×C8—41no—
558C2×C8C22×D411yescharacteristic
559C2×C8C22×D411yescharacteristic
560C2×C8C22×D411yescharacteristic
561C2×C8C22×D411yescharacteristic
562C2×C8C22×D411yescharacteristic
563C2×C8C22×D411yescharacteristic
564C2×C8C22×D411yescharacteristic
565C2×C8C2511yescharacteristic
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—41no—
575C4⋊C4—41no—
576C4⋊C4—41no—
577C4⋊C4—41no—
578C4⋊C4—41no—
579C4⋊C4—41no—
580C4⋊C4—41no—
581C4⋊C4—41no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—42no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C22⋊C4—42no—
591C22⋊C4—42no—
592C22⋊C4—42no—
593C22⋊C4—42no—
594C22⋊C4—42no—
595C22⋊C4—42no—
596C22⋊C4—42no—
597C22⋊C4—42no—
598C22⋊C4—42no—
599C22⋊C4—42no—
600C22⋊C4—42no—
601C22⋊C4—42no—
602C22⋊C4—42no—
603C22⋊C4—42no—
604C22⋊C4—42no—
605C22⋊C4—42no—
606C22⋊C4—42no—
607C22⋊C4—42no—
608C42—42no—
609C42—42no—
610C42—42no—
611C42—42no—
612C42—42no—
613C42—42no—
614C42C2×D811yescharacteristic
615C42C2×D811yescharacteristic
616C42C2×D811yescharacteristic
617C42C2×D811yescharacteristic
618C42C2×D811yescharacteristic
619C42C2×D811yescharacteristic
620C42C2×D811yescharacteristic
621C42C2×D811yescharacteristic
622C42C2×D811yescharacteristic
623C42C2×D811yescharacteristic
624C42C2×D811yescharacteristic
625C42C2×D811yescharacteristic
626C42C22×D411yescharacteristic
627C42C22×D411yescharacteristic
628C42C22×D411yescharacteristic
629C42C22×D411yescharacteristic
630C16—24no—
631C16—24no—
632C23—22no—
633C23—41no—
634C23—41no—
635C23—42no—
636C23—42no—
637C23C22×D811yescharacteristic
638C23C22×D811yescharacteristic
639C23C22×D811yescharacteristic
640Q8—42no—
641Q8—42no—
642Q8—42no—
643Q8—42no—
644Q8—82no—
645Q8—82no—
646Q8—82no—
647Q8—82no—
648D4—42no—
649D4—42no—
650D4—42no—
651D4—42no—
652D4—82no—
653D4—82no—
654D4—82no—
655D4—84no—
656D4—84no—
657D4—84no—
658D4—84no—
659D4—84no—
660D4—84no—
661D4—82no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—84no—
690C2×C4—84no—
691C2×C4—84no—
692C2×C4—84no—
693C2×C4—84no—
694C2×C4—84no—
695C2×C4C22×D811yescharacteristic
696C2×C4C22×D811yescharacteristic
697C2×C4C22×D811yescharacteristic
698C2×C4C22×D811yescharacteristic
699C2×C4C22×D811yescharacteristic
700C2×C4C22×D811yescharacteristic
701C2×C4C22×D811yescharacteristic
702C2×C4C22×D811yescharacteristic
703C2×C4C22×D811yescharacteristic
704C2×C4C22×D811yescharacteristic
705C2×C4C22×D811yescharacteristic
706C2×C4C23×D411yescharacteristic
707C8—21no—
708C8—21no—
709C8—21no—
710C8—21no—
711C8—21no—
712C8—21no—
713C8#57512yes—
714C8#58512yes—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—21no—
719C22—21no—
720C22—21no—
721C22—21no—
722C22—21no—
723C22—44no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22#341412yes—
731C22C23×D811yescharacteristic
732C4—21no—
733C4—21no—
734C4—21no—
735C4—21no—
736C4—21no—
737C4—21no—
738C4—21no—
739C4—21no—
740C4—21no—
741C4—21no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#341412yes—
749C4#344911yescharacteristic
750C4C2×#57511yescharacteristic
751C2—22no—
752C2—41no—
753C2—41no—
754C2—162no—
755C2—162no—
756C2#6094011yescharacteristic
757C2C2×#341411yescharacteristic
758C2C2×#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. #60673C2semidirect
2. #60673C2semidirect
3. #33814C2semidirect
4. #33813C2semidirect
5. #33806C2semidirect
6. #33803C2semidirect
7. #33801C2semidirect
8. #33785C2semidirect
9. #33782C2semidirect
10. #33780C2semidirect
11. #33757C2semidirect
12. #33747C2semidirect
13. #33746C2semidirect
14. #33676C2semidirect
15. #33675C2semidirect
16. #33671C2semidirect
17. C2×#2163C2semidirect
55. #2255C22semidirect
56. #2255C22semidirect
57. #2251C22semidirect
58. #2250C22semidirect
59. #2245C22semidirect
60. #2244C22semidirect
61. #2243C22semidirect
62. #2243C22semidirect
67. #2229C22semidirect
68. #2228C22semidirect
69. #2227C22semidirect
70. #2226C22semidirect
71. #2225C22semidirect
72. #2224C22semidirect
77. #2211C22semidirect
78. #2210C22semidirect
79. #2209C22semidirect
80. #2208C22semidirect
81. #2207C22semidirect
82. #2206C22semidirect
87. #2177C22semidirect
88. #2176C22semidirect
89. #2175C22semidirect
90. #2175C22semidirect

Maximal subgroups

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

subgroupindex
#606732
#338142
#338132
#338062
#338032
#338012
#337852
#337822
#337802
#337572
#337472
#337462
#336762
#336752
#336712
C2×#21632