Finite Group Databasebeta

← #7624520

Subgroups — #7624520

Order 512.

1489 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62417C211yesmaximal, characteristic, complemented
2D8×#41C211yesmaximal, characteristic, complemented
3#33966C212yesmaximal, complemented
4#33940C212yesmaximal, complemented
5#33899C212yesmaximal, complemented
6#33873C214yesmaximal, complemented
7#33840C212yesmaximal, complemented
8#33830C214yesmaximal, complemented
9#33819C212yesmaximal, complemented
10#33812C211yesmaximal, characteristic, complemented
11#33780C212yesmaximal, complemented
12#33771C211yesmaximal, characteristic, complemented
13#33669C212yesmaximal, complemented
14#33661C211yesmaximal, characteristic, complemented
15#33580C212yesmaximal, complemented
16C2×#2206C211yesmaximal, characteristic, complemented
17#33503C211yesmaximal, characteristic, complemented
18#3508—21no—
19D4×#41—21no—
20C2×#509C2214yescomplemented
21C22×D16C2212yes—
22#3396—21no—
23#3396C2212yescomplemented
24#3391C2211yescharacteristic, complemented
25Q8×D8C2211yescharacteristic, complemented
26#3358—21no—
27#3334C2212yes—
28#3334C2211yescharacteristic, complemented
29#3294—22no—
30#3294—21no—
31#3287—21no—
32#3287C2212yescomplemented
33#3280—22no—
34#3280C2214yescomplemented
35D4×D8C2212yes—
36D4×D8C2211yescharacteristic, complemented
37#3208—21no—
38#3207—22no—
39#3170C2211yescharacteristic, complemented
40#3169C2211yescharacteristic
41#3169C2212yescomplemented
42#3169C2212yescomplemented
43#3147C2212yescomplemented
44C2×#493C2211yescharacteristic, complemented
45#3100—21no—
46#3100—22no—
47C8×#41C2211yescharacteristic
48C2×C4×D8C2211yescharacteristic
49C2×C4×D8C2212yescomplemented
50#2250C2214yescomplemented
51#2246C2212yescomplemented
52#2243C2214yescomplemented
53#2240C2212yescomplemented
54#2235C2212yescomplemented
55#2233C2214yescomplemented
56#2226C2212yescomplemented
57#2226C2211yescharacteristic, complemented
58#2226C2214yescomplemented
59#2226C2211yescharacteristic, complemented
60#2225C2212yescomplemented
61#2218C2214yescomplemented
62#2218C2214yescomplemented
63#2217C2214yescomplemented
64#2210C2214yescomplemented
65#2210C2214yescomplemented
66#2208C2212yescomplemented
67#2208C2211yescharacteristic, complemented
68#2208C2211yescharacteristic, complemented
69#2207C2212yescomplemented
70#2206C2212yes—
71#2206C2212yes—
72#2206C2212yescomplemented
73#2206C2214yescomplemented
74#2206C2212yescomplemented
75#2206C2212yescomplemented
76#2206C2212yescomplemented
77#2187C2212yes—
78#2186C2214yescomplemented
79#2186C2214yescomplemented
80#2184C2214yescomplemented
81#2184C2214yescomplemented
82#2183C2212yescomplemented
83#2177C2214yescomplemented
84#2177C2214yescomplemented
85C4×SD32C2212yescomplemented
86C4×SD32C2212yescomplemented
87#2169C2212yescomplemented
88#2168C2214yescomplemented
89#2163C2212yescomplemented
90#2162C2211yescharacteristic, complemented
91#2150C2212yescomplemented
92C2×C4⋊C16C2211yescharacteristic, complemented
93#2139C2214yescomplemented
94C2×#357C2212yescomplemented
95C22×#41—41no—
96C2×#135—22no—
97C2×#135C2314yes—
98C22×D8—21no—
99C22×D8C2311yescharacteristic
100C22×D8C2312yes—
101C22×D8C2312yes—
102#559—22no—
103#559—41no—
104#550—21no—
105#550—22no—
106#550—41no—
107D4×Q8—21no—
108#548—41no—
109#547—22no—
110#547—21no—
111#546—24no—
112#546—42no—
113#546—41no—
114#546—42no—
115D42—22no—
116D42—21no—
117#540—42no—
118#535—41no—
119#533—21no—
120#532—21no—
121#532—22no—
122#532—22no—
123#532—22no—
124C2×#127—21no—
125#525—41no—
126#525—42no—
127#518—41no—
128C4×#41D411yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131#509—48no—
132C2×SD32C2312yes—
133C2×SD32C2314yes—
134C2×SD32C2312yes—
135C2×D16—44no—
136C2×D16C2314yes—
137C2×D16C2314yes—
138C2×D16C2314yes—
139C2×D16C2314yes—
140C2×M32C2314yes—
141C22×C16C2312yes—
142#500C2312yes—
143#500C2311yescharacteristic
144#500C2311yescharacteristic
145#495—22no—
146#495C2314yes—
147#494C2311yescharacteristic
148#493—21no—
149#493—21no—
150#493C2311yescharacteristic
151#493C2312yes—
152#493C2312yes—
153#493C2312yes—
154#492—22no—
155#487—21no—
156#486—21no—
157#486—22no—
158#486—22no—
159#480—22no—
160#480—22no—
161#480—22no—
162#474—22no—
163#474—21no—
164#474—21no—
165#474—22no—
166#467—21no—
167#467C2312yes—
168#466—21no—
169#466—22no—
170#466—22no—
171#466C2312yes—
172#466C2312yes—
173#466C2312yes—
174#466C2314yes—
175#463—24no—
176#463—24no—
177#460—22no—
178#460—22no—
179#459—22no—
180#459—21no—
181#459—21no—
182#459—22no—
183#450—24no—
184#449—24no—
185#449—24no—
186#447—22no—
187#447—22no—
188#447—22no—
189C8×Q8D411yescharacteristic
190C8×Q8C2311yescharacteristic
191C4×Q16—21no—
192C4×Q16C2312yes—
193C4×D8—21no—
194C4×D8—22no—
195C4×D8—21no—
196C4×D8—22no—
197C4×D8—21no—
198C4×D8—22no—
199C4×D8—22no—
200C4×D8C2312yes—
201C4×D8C2311yescharacteristic
202C4×D8C2311yescharacteristic
203C4×D8C2312yes—
204C4×D8C2314yes—
205C4×D8C2312yes—
206C4×D8C2311yescharacteristic
207C4×D8C2311yescharacteristic
208C8×D4D412yes—
209C8×D4D411yescharacteristic
210C8×D4C2312yes—
211C8×D4C2311yescharacteristic
212#431D412yes—
213#431D411yescharacteristic
214C2×#107C2311yescharacteristic
215C2×#107C2312yes—
216C2×#102—21no—
217C2×#102—22no—
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2312yes—
220#367C2312yes—
221#367C2312yes—
222#367C2314yes—
223C4⋊C16C2312yes—
224C4⋊C16C2312yes—
225C4⋊C16C2311yescharacteristic
226C4⋊C16C2314yes—
227C4⋊C16C2311yescharacteristic
228#358C2312yes—
229#358C2314yes—
230#358C2312yes—
231#357C2314yes—
232#357C2314yes—
233#357C2314yes—
234#357C2312yes—
235#357C2312yes—
236#357C2314yes—
237#357C2314yes—
238C22⋊C16C2312yes—
239C22⋊C16C2314yes—
240#346C2314yes—
241C4×C16C2312yes—
242C2×#41—24no—
243C2×#41—41no—
244C2×#41—42no—
245C2×#41—41no—
246C2×#41—42no—
247C2×#41—41no—
248C2×#41—42no—
249C2×#41—88no—
250C2×#41D811yescharacteristic
251C22×Q8—41no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—22no—
255C22×D4—42no—
256C22×D4—41no—
257C22×D4—41no—
258C22×D4—41no—
259C23×C4—42no—
260C23×C4—41no—
261#135—44no—
262#135—44no—
263#135—84no—
264C2×Q16—21no—
265C2×Q16C2412yes—
266C2×SD16—24no—
267C2×SD16—24no—
268C2×D8—22no—
269C2×D8—24no—
270C2×D8—22no—
271C2×D8—22no—
272C2×D8—21no—
273C2×D8—22no—
274C2×D8—22no—
275C2×D8—22no—
276C2×D8—21no—
277C2×D8—41no—
278C2×D8—42no—
279C2×D8—42no—
280C2×D8—42no—
281C2×D8—82no—
282C2×D8C2414yes—
283C2×D8C2411yescharacteristic
284C2×D8C2412yes—
285C2×D8C2411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D412yes—
288C22×C8C2411yescharacteristic
289C22×C8C2412yes—
290C4⋊Q8—21no—
291C4⋊Q8—22no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294#127—22no—
295#127—22no—
296#127—22no—
297#127—21no—
298#127—42no—
299#127—41no—
300#127—41no—
301#125—41no—
302#124—24no—
303#124—42no—
304#124—42no—
305#124—41no—
306#124—42no—
307#123—44no—
308#123—44no—
309#123—42no—
310#123—42no—
311#123—42no—
312C22⋊Q8—22no—
313C22⋊Q8—41no—
314C22⋊Q8—41no—
315C22⋊Q8—42no—
316C22⋊Q8—41no—
317C22⋊Q8—42no—
318#121—22no—
319#121—22no—
320#121—22no—
321#121—24no—
322#121—42no—
323#121—42no—
324#121—42no—
325#121—41no—
326#121—42no—
327#121—41no—
328#121—42no—
329#121—41no—
330#121—42no—
331#121—44no—
332#121—44no—
333#120—44no—
334#120—42no—
335#120—42no—
336#120—42no—
337C4×Q8—22no—
338C4×Q8—41no—
339C4×Q8D811yescharacteristic
340C4×Q8C2×D411yescharacteristic
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—22no—
347C4×D4—21no—
348C4×D4—24no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—42no—
352C4×D4—41no—
353C4×D4—42no—
354C4×D4—41no—
355C4×D4—42no—
356C4×D4—42no—
357C4×D4—42no—
358C4×D4—41no—
359C4×D4—42no—
360C4×D4D812yes—
361C4×D4D811yescharacteristic
362C4×D4C2×D412yes—
363C4×D4C2×D411yescharacteristic
364#117—41no—
365#117D811yescharacteristic
366#117D812yes—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—44no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D412yes—
374SD32—24no—
375SD32—42no—
376SD32—44no—
377D16—28no—
378D16—44no—
379D16—44no—
380D16—44no—
381M32—44no—
382C2×C16—42no—
383C2×C16C2412yes—
384C2×C16C2412yes—
385C2×C16C2414yes—
386#107—22no—
387#107—21no—
388#107—21no—
389#107—22no—
390#107—21no—
391#107C2411yescharacteristic
392#107C2411yescharacteristic
393#107C2412yes—
394#107C2412yes—
395C4⋊C8C2×D411yescharacteristic
396C4⋊C8C2×D412yes—
397C4⋊C8C2×D411yescharacteristic
398C4⋊C8C2×D412yes—
399#103—22no—
400#103—22no—
401#102—21no—
402#102—22no—
403#102—21no—
404#102—22no—
405#102—21no—
406#102—21no—
407#102—22no—
408#102—22no—
409#102—22no—
410#102—22no—
411#102—24no—
412#102—24no—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C22⋊C8C2×D412yes—
416C4×C8C2×D412yes—
417C4×C8C2×D411yescharacteristic
418C4×C8C2×D412yes—
419C4×C8C2×D411yescharacteristic
420C4×C8C2411yescharacteristic
421C4×C8C2412yes—
422C4×C8C2411yescharacteristic
423C24—42no—
424C24—41no—
425C24—42no—
426#41—24no—
427#41—88no—
428#41—84no—
429#41—84no—
430#41—88no—
431#41—84no—
432#41—84no—
433#41—84no—
434#41—164no—
435C2×Q8—22no—
436C2×Q8—41no—
437C2×Q8—42no—
438C2×Q8—41no—
439C2×Q8—84no—
440C2×Q8C2×D811yescharacteristic
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—41no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—41no—
459C2×D4—42no—
460C2×D4—41no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—82no—
465C2×D4—82no—
466C2×D4—82no—
467C2×D4—88no—
468C2×D4—84no—
469C2×D4—84no—
470C2×D4—162no—
471C2×D4C2×D812yes—
472C2×D4C2×D811yescharacteristic
473C22×C4—42no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—41no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—88no—
485C22×C4—84no—
486C22×C4C2×D812yes—
487C22×C4C2×D812yes—
488C22×C4C2×D811yescharacteristic
489C22×C4C2×D811yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492Q16—22no—
493Q16—22no—
494Q16—42no—
495SD16—84no—
496SD16—84no—
497D8—24no—
498D8—21no—
499D8—22no—
500D8—21no—
501D8—22no—
502D8—24no—
503D8—21no—
504D8—42no—
505D8—44no—
506D8—44no—
507D8—81no—
508D8—82no—
509D8—82no—
510D8—82no—
511D8H(5,𝔽2)12yes—
512C2×C8—21no—
513C2×C8—22no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—41no—
517C2×C8—42no—
518C2×C8C22×D411yescharacteristic
519C2×C8C22×D412yes—
520C2×C8C22×D411yescharacteristic
521C2×C8C22×D412yes—
522C2×C8C22×D411yescharacteristic
523C2×C8C2511yescharacteristic
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—42no—
534C4⋊C4—41no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4C2×D811yescharacteristic
538C4⋊C4C2×D811yescharacteristic
539C4⋊C4C2×D812yes—
540C4⋊C4C2×D812yes—
541C22⋊C4—42no—
542C22⋊C4—41no—
543C22⋊C4—42no—
544C22⋊C4—41no—
545C22⋊C4—42no—
546C22⋊C4—41no—
547C22⋊C4—42no—
548C22⋊C4—41no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4C2×D812yes—
557C22⋊C4C2×D812yes—
558C22⋊C4C2×D812yes—
559C42—42no—
560C42C2×D812yes—
561C42C2×D811yescharacteristic
562C42C2×D812yes—
563C42C2×D811yescharacteristic
564C42C22×D411yescharacteristic
565C42C22×D412yes—
566C42C22×D411yescharacteristic
567C16—24no—
568C16—22no—
569C16—22no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23—44no—
574C23—82no—
575C23—84no—
576C23—84no—
577C23—82no—
578C23—84no—
579C23—161no—
580C23—162no—
581C23—162no—
582C23—162no—
583C23C22×D812yes—
584C23C22×D811yescharacteristic
585Q8—22no—
586Q8—42no—
587Q8—42no—
588Q8—84no—
589Q8—82no—
590Q8—82no—
591D4—24no—
592D4—22no—
593D4—22no—
594D4—41no—
595D4—44no—
596D4—42no—
597D4—41no—
598D4—41no—
599D4—42no—
600D4—44no—
601D4—84no—
602D4—82no—
603D4—84no—
604D4—84no—
605D4—84no—
606D4—84no—
607D4—84no—
608D4—84no—
609D4—84no—
610D4—82no—
611D4—162no—
612D4—161no—
613D4—162no—
614D4—162no—
615D4—164no—
616D4—164no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—84no—
637C2×C4—82no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4#50914yescomplemented
641C2×C4C22×D811yescharacteristic
642C2×C4C22×D812yes—
643C2×C4C22×D811yescharacteristic
644C2×C4C22×D812yes—
645C2×C4C22×D812yes—
646C2×C4C22×D811yescharacteristic
647C2×C4C22×D811yescharacteristic
648C2×C4C22×D811yescharacteristic
649C2×C4C23×D411yescharacteristic
650C8—21no—
651C8—21no—
652C8—22no—
653C8—21no—
654C8—22no—
655C8C2×H(5,𝔽2)11yescharacteristic
656C8#58511yescharacteristic
657C22—21no—
658C22—21no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—82no—
666C22—81no—
667C22—84no—
668C22—84no—
669C22—81no—
670C22—82no—
671C22—84no—
672C22—81no—
673C22—82no—
674C22—164no—
675C22—168no—
676C22—324no—
677C22C2×#50912yes—
678C22C23×D811yescharacteristic
679C4—22no—
680C4—21no—
681C4—22no—
682C4—21no—
683C4—22no—
684C4—21no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4—82no—
690C4C2×#50912yes—
691C4C2×#50912yes—
692C4#348811yescharacteristic
693C4#358511yescharacteristic
694C2—22no—
695C2—42no—
696C2—82no—
697C2—161no—
698C2—162no—
699C2—164no—
700C2#6194711yescharacteristic
701C2C22×#50911yescharacteristic
702C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62417C2semidirect
2. D8×#41C2semidirect
3. #33966C2semidirect
4. #33940C2semidirect
5. #33899C2semidirect
6. #33873C2semidirect
7. #33840C2semidirect
8. #33830C2semidirect
9. #33819C2semidirect
10. #33812C2semidirect
11. #33780C2semidirect
12. #33771C2semidirect
13. #33669C2semidirect
14. #33661C2semidirect
15. #33580C2semidirect
16. C2×#2206C2semidirect
17. #33503C2semidirect
20. C2×#509C22semidirect
23. #3396C22semidirect
24. #3391C22semidirect
25. Q8×D8C22semidirect
28. #3334C22semidirect
32. #3287C22semidirect
34. #3280C22semidirect
36. D4×D8C22semidirect
39. #3170C22semidirect
41. #3169C22semidirect
42. #3169C22semidirect
43. #3147C22semidirect
44. C2×#493C22semidirect
49. C2×C4×D8C22semidirect
50. #2250C22semidirect
51. #2246C22semidirect
52. #2243C22semidirect
53. #2240C22semidirect
54. #2235C22semidirect
55. #2233C22semidirect
56. #2226C22semidirect
57. #2226C22semidirect
58. #2226C22semidirect
59. #2226C22semidirect
60. #2225C22semidirect
61. #2218C22semidirect
62. #2218C22semidirect
63. #2217C22semidirect
64. #2210C22semidirect
65. #2210C22semidirect
66. #2208C22semidirect
67. #2208C22semidirect
68. #2208C22semidirect
69. #2207C22semidirect
72. #2206C22semidirect
73. #2206C22semidirect
74. #2206C22semidirect
75. #2206C22semidirect
76. #2206C22semidirect
78. #2186C22semidirect
79. #2186C22semidirect
80. #2184C22semidirect
81. #2184C22semidirect
82. #2183C22semidirect
83. #2177C22semidirect
84. #2177C22semidirect
85. C4×SD32C22semidirect
86. C4×SD32C22semidirect
87. #2169C22semidirect
88. #2168C22semidirect
89. #2163C22semidirect
90. #2162C22semidirect
91. #2150C22semidirect
92. C2×C4⋊C16C22semidirect
93. #2139C22semidirect
94. C2×#357C22semidirect
640. C2×C4#509semidirect

Maximal subgroups

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

subgroupindex
#624172
D8×#412
#339662
#339402
#338992
#338732
#338402
#338302
#338192
#338122
#337802
#337712
#336692
#336612
#335802
C2×#22062
#335032