Finite Group Databasebeta

← #7624521

Subgroups — #7624521

Order 512.

1457 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62418C211yesmaximal, characteristic, complemented
2D8×#41C211yesmaximal, characteristic, complemented
3#33964C212yesmaximal, complemented
4#33934C212yesmaximal, complemented
5#33897C212yesmaximal, complemented
6#33875C214yesmaximal, complemented
7#33838C212yesmaximal, complemented
8#33831C214yesmaximal, complemented
9#33820C212yesmaximal, complemented
10#33811C211yesmaximal, characteristic, complemented
11#33780C212yesmaximal, complemented
12#33772C211yesmaximal, characteristic, complemented
13#33670C212yesmaximal, complemented
14#33660C211yesmaximal, characteristic, complemented
15#33580C212yesmaximal, complemented
16C2×#2208C211yesmaximal, characteristic, complemented
17#33503C211yesmaximal, characteristic, complemented
18#3511—21no—
19D4×#41—21no—
20C2×#509C2214yes—
21C22×SD32C2212yes—
22#3393—21no—
23#3393C2212yescomplemented
24#3391C2211yescharacteristic
25Q8×D8C2211yescharacteristic
26#3358—21no—
27#3334C2212yes—
28#3334C2211yescharacteristic
29#3294—21no—
30#3294—22no—
31#3285—22no—
32#3285C2214yescomplemented
33#3283—21no—
34#3283C2212yescomplemented
35D4×D8C2212yes—
36D4×D8C2211yescharacteristic
37#3210—22no—
38#3206—21no—
39#3170C2212yescomplemented
40#3169C2211yescharacteristic
41#3169C2212yes—
42#3169C2211yescharacteristic, complemented
43#3147C2212yescomplemented
44C2×#494C2211yescharacteristic, complemented
45#3100—21no—
46#3100—22no—
47C8×#41C2211yescharacteristic
48C2×C4×D8C2211yescharacteristic
49C2×C4×D8C2212yes—
50#2250C2214yescomplemented
51#2248C2212yescomplemented
52#2243C2214yescomplemented
53#2241C2212yescomplemented
54#2236C2214yescomplemented
55#2232C2212yescomplemented
56#2227C2214yescomplemented
57#2224C2211yescharacteristic
58#2224C2212yescomplemented
59#2224C2212yescomplemented
60#2224C2211yescharacteristic, complemented
61#2220C2214yescomplemented
62#2220C2214yescomplemented
63#2215C2214yescomplemented
64#2210C2214yes—
65#2210C2214yescomplemented
66#2208C2212yes—
67#2208C2212yes—
68#2208C2212yescomplemented
69#2208C2212yescomplemented
70#2208C2212yescomplemented
71#2208C2212yescomplemented
72#2207C2214yescomplemented
73#2206C2211yescharacteristic
74#2206C2212yescomplemented
75#2206C2212yescomplemented
76#2206C2211yescharacteristic, complemented
77#2187C2214yes—
78#2186C2214yescomplemented
79#2186C2214yescomplemented
80#2185C2214yescomplemented
81#2184C2212yes—
82#2183C2212yescomplemented
83#2175C2214yes—
84#2175C2214yescomplemented
85C4×D16C2212yes—
86C4×D16C2212yescomplemented
87#2169C2212yescomplemented
88#2168C2214yescomplemented
89#2163C2212yescomplemented
90#2162C2211yescharacteristic, complemented
91#2150C2212yescomplemented
92C2×C4⋊C16C2211yescharacteristic, complemented
93#2139C2214yescomplemented
94C2×#358C2212yescomplemented
95C22×#41—41no—
96C2×#135—22no—
97C2×#135C2314yes—
98C22×Q16—21no—
99C22×Q16C2312yes—
100C22×D8C2311yescharacteristic
101C22×D8C2312yes—
102#556—22no—
103#555—41no—
104#554—41no—
105#553—42no—
106#550—21no—
107D4×Q8—21no—
108#548—24no—
109#548—41no—
110#547—21no—
111#547—22no—
112#547—22no—
113#546—42no—
114#546—41no—
115D42—21no—
116D42—22no—
117#542—42no—
118#537—41no—
119#533—22no—
120#532—21no—
121#532—22no—
122#532—21no—
123#532—22no—
124C2×C4⋊Q8—21no—
125#525—41no—
126#525—42no—
127#520—41no—
128C4×#41D411yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131#509—48no—
132C2×SD32—44no—
133C2×SD32C2314yes—
134C2×SD32C2314yes—
135C2×SD32C2314yes—
136C2×SD32C2314yes—
137C2×D16C2312yes—
138C2×D16C2314yes—
139C2×D16C2312yes—
140C2×M32C2314yes—
141C22×C16C2312yes—
142#500C2311yescharacteristic
143#500C2312yes—
144#500C2311yescharacteristic
145#495—22no—
146#495C2314yes—
147#494—21no—
148#494—21no—
149#494C2312yes—
150#494C2312yes—
151#493C2311yescharacteristic
152#493C2312yes—
153#493C2311yescharacteristic
154#492—22no—
155#487—22no—
156#486—21no—
157#486—22no—
158#486—21no—
159#480—22no—
160#480—22no—
161#480—22no—
162#474—21no—
163#474—22no—
164#474—21no—
165#474—22no—
166#467—22no—
167#467C2314yes—
168#466—21no—
169#466—22no—
170#466—21no—
171#466C2312yes—
172#466C2312yes—
173#466C2312yes—
174#466C2312yes—
175#464—24no—
176#464—24no—
177#461—22no—
178#461—22no—
179#459—21no—
180#459—22no—
181#459—21no—
182#459—22no—
183#452—24no—
184#452—24no—
185#448—24no—
186#447—22no—
187#447—22no—
188#447—22no—
189C8×Q8D411yescharacteristic
190C8×Q8C2311yescharacteristic
191C4×Q16—22no—
192C4×Q16C2314yes—
193C4×D8—21no—
194C4×D8—22no—
195C4×D8—21no—
196C4×D8—22no—
197C4×D8—21no—
198C4×D8—22no—
199C4×D8—21no—
200C4×D8C2311yescharacteristic
201C4×D8C2312yes—
202C4×D8C2311yescharacteristic
203C4×D8C2312yes—
204C4×D8C2312yes—
205C4×D8C2312yes—
206C4×D8C2311yescharacteristic
207C4×D8C2311yescharacteristic
208C8×D4D411yescharacteristic
209C8×D4D412yes—
210C8×D4C2312yes—
211C8×D4C2311yescharacteristic
212#431D411yescharacteristic
213#431D412yes—
214C2×#107C2311yescharacteristic
215C2×#107C2312yes—
216C2×#102—21no—
217C2×#102—22no—
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2312yes—
220#366C2312yes—
221#366C2312yes—
222#366C2314yes—
223C4⋊C16C2312yes—
224C4⋊C16C2312yes—
225C4⋊C16C2311yescharacteristic
226C4⋊C16C2314yes—
227C4⋊C16C2311yescharacteristic
228#358C2314yes—
229#358C2314yes—
230#358C2314yes—
231#358C2314yes—
232#357C2312yes—
233#357C2314yes—
234#357C2312yes—
235#357C2312yes—
236#357C2312yes—
237#357C2314yes—
238C22⋊C16C2312yes—
239C22⋊C16C2314yes—
240#346C2314yes—
241C4×C16C2312yes—
242C2×#41—24no—
243C2×#41—41no—
244C2×#41—41no—
245C2×#41—42no—
246C2×#41—41no—
247C2×#41—42no—
248C2×#41—41no—
249C2×#41—88no—
250C2×#41D811yescharacteristic
251C22×Q8—22no—
252C22×Q8—41no—
253C22×D4—21no—
254C22×D4—22no—
255C22×D4—42no—
256C22×D4—41no—
257C23×C4—42no—
258C23×C4—41no—
259#135—44no—
260#135—44no—
261#135—84no—
262C2×Q16—22no—
263C2×Q16—22no—
264C2×Q16—22no—
265C2×Q16—42no—
266C2×Q16—42no—
267C2×Q16—82no—
268C2×Q16C2414yes—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×D8—22no—
272C2×D8—24no—
273C2×D8—21no—
274C2×D8—22no—
275C2×D8—21no—
276C2×D8—21no—
277C2×D8—22no—
278C2×D8—41no—
279C2×D8—42no—
280C2×D8C2412yes—
281C2×D8C2411yescharacteristic
282C2×D8C2412yes—
283C2×D8C2411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D412yes—
286C22×C8C2411yescharacteristic
287C22×C8C2412yes—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—41no—
294C4⋊Q8—41no—
295#127—21no—
296#127—21no—
297#127—22no—
298#126—44no—
299#125—42no—
300#124—24no—
301#124—44no—
302#124—41no—
303#124—42no—
304#124—41no—
305#123—42no—
306#123—42no—
307#123—42no—
308#123—44no—
309C22⋊Q8—24no—
310C22⋊Q8—42no—
311C22⋊Q8—42no—
312C22⋊Q8—41no—
313C22⋊Q8—42no—
314C22⋊Q8—41no—
315C22⋊Q8—42no—
316C22⋊Q8—44no—
317#121—22no—
318#121—22no—
319#121—22no—
320#121—22no—
321#121—42no—
322#121—44no—
323#121—42no—
324#121—41no—
325#121—42no—
326#121—41no—
327#121—42no—
328#121—41no—
329#121—42no—
330#121—41no—
331#121—42no—
332#120—42no—
333#120—42no—
334#120—42no—
335C4×Q8—24no—
336C4×Q8—42no—
337C4×Q8D811yescharacteristic
338C4×Q8C2×D411yescharacteristic
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—22no—
342C4×D4—21no—
343C4×D4—22no—
344C4×D4—21no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—44no—
348C4×D4—42no—
349C4×D4—41no—
350C4×D4—42no—
351C4×D4—41no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4—41no—
355C4×D4—42no—
356C4×D4—41no—
357C4×D4D812yes—
358C4×D4D811yescharacteristic
359C4×D4C2×D411yescharacteristic
360C4×D4C2×D412yes—
361#117—42no—
362#117—42no—
363#117D811yescharacteristic
364#117D812yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—42no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—42no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D412yes—
374SD32—28no—
375SD32—44no—
376SD32—44no—
377SD32—44no—
378D16—24no—
379D16—42no—
380D16—44no—
381M32—44no—
382C2×C16—42no—
383C2×C16C2412yes—
384C2×C16C2412yes—
385C2×C16C2414yes—
386#107—21no—
387#107—22no—
388#107—21no—
389#107—21no—
390#107—22no—
391#107C2412yes—
392#107C2411yescharacteristic
393#107C2412yes—
394#107C2411yescharacteristic
395C4⋊C8C2×D411yescharacteristic
396C4⋊C8C2×D412yes—
397C4⋊C8C2×D411yescharacteristic
398C4⋊C8C2×D412yes—
399#103—24no—
400#103—24no—
401#102—21no—
402#102—21no—
403#102—22no—
404#102—21no—
405#102—22no—
406#102—21no—
407#102—22no—
408#102—22no—
409#102—22no—
410#102—22no—
411#102—22no—
412#102—22no—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C22⋊C8C2×D412yes—
416C4×C8C2×D411yescharacteristic
417C4×C8C2×D412yes—
418C4×C8C2×D411yescharacteristic
419C4×C8C2×D412yes—
420C4×C8C2411yescharacteristic
421C4×C8C2412yes—
422C4×C8C2411yescharacteristic
423C24—42no—
424C24—41no—
425#41—24no—
426#41—88no—
427#41—84no—
428#41—84no—
429#41—88no—
430#41—84no—
431#41—84no—
432#41—84no—
433#41—84no—
434C2×Q8—24no—
435C2×Q8—41no—
436C2×Q8—42no—
437C2×Q8—41no—
438C2×Q8—42no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8—84no—
442C2×Q8C2×D811yescharacteristic
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—42no—
457C2×D4—41no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—42no—
461C2×D4—41no—
462C2×D4—81no—
463C2×D4—82no—
464C2×D4—88no—
465C2×D4—84no—
466C2×D4C2×D812yes—
467C2×D4C2×D811yescharacteristic
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—41no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—41no—
479C22×C4—44no—
480C22×C4—88no—
481C22×C4—84no—
482C22×C4C2×D812yes—
483C22×C4C2×D811yescharacteristic
484C22×C4C2×D812yes—
485C22×C4C2×D811yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488Q16—24no—
489Q16—24no—
490Q16—44no—
491Q16—82no—
492Q16—82no—
493SD16—84no—
494SD16—84no—
495D8—22no—
496D8—22no—
497D8—21no—
498D8—21no—
499D8—21no—
500D8—22no—
501D8—22no—
502D8—44no—
503D8—42no—
504D8—42no—
505D8—82no—
506D8—81no—
507D8H(5,𝔽2)12yes—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—41no—
513C2×C8—42no—
514C2×C8C22×D411yescharacteristic
515C2×C8C22×D411yescharacteristic
516C2×C8C22×D412yes—
517C2×C8C22×D411yescharacteristic
518C2×C8C22×D412yes—
519C2×C8C2511yescharacteristic
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—41no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—41no—
532C4⋊C4—42no—
533C4⋊C4—41no—
534C4⋊C4—44no—
535C4⋊C4C2×D811yescharacteristic
536C4⋊C4C2×D811yescharacteristic
537C4⋊C4C2×D812yes—
538C4⋊C4C2×D812yes—
539C22⋊C4—44no—
540C22⋊C4—41no—
541C22⋊C4—42no—
542C22⋊C4—41no—
543C22⋊C4—42no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—44no—
554C22⋊C4C2×D812yes—
555C22⋊C4C2×D812yes—
556C22⋊C4C2×D812yes—
557C42—44no—
558C42C2×D812yes—
559C42C2×D811yescharacteristic
560C42C2×D812yes—
561C42C2×D811yescharacteristic
562C42C22×D411yescharacteristic
563C42C22×D411yescharacteristic
564C42C22×D412yes—
565C16—24no—
566C16—22no—
567C16—22no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—82no—
573C23—84no—
574C23—84no—
575C23—82no—
576C23—161no—
577C23—162no—
578C23C22×D812yes—
579C23C22×D811yescharacteristic
580Q8—22no—
581Q8—44no—
582Q8—44no—
583Q8—84no—
584Q8—82no—
585Q8—82no—
586Q8—162no—
587D4—24no—
588D4—22no—
589D4—22no—
590D4—41no—
591D4—42no—
592D4—41no—
593D4—42no—
594D4—42no—
595D4—41no—
596D4—42no—
597D4—84no—
598D4—82no—
599D4—82no—
600D4—84no—
601D4—84no—
602D4—84no—
603D4—84no—
604D4—84no—
605D4—82no—
606D4—82no—
607D4—162no—
608D4—161no—
609D4—164no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—84no—
630C2×C4—82no—
631C2×C4—84no—
632C2×C4—164no—
633C2×C4—162no—
634C2×C4#50914yescomplemented
635C2×C4C22×D812yes—
636C2×C4C22×D811yescharacteristic
637C2×C4C22×D811yescharacteristic
638C2×C4C22×D812yes—
639C2×C4C22×D811yescharacteristic
640C2×C4C22×D812yes—
641C2×C4C22×D811yescharacteristic
642C2×C4C22×D811yescharacteristic
643C2×C4C23×D411yescharacteristic
644C8—21no—
645C8—22no—
646C8—21no—
647C8—22no—
648C8—21no—
649C8C2×H(5,𝔽2)11yescharacteristic
650C8#58511yescharacteristic
651C22—21no—
652C22—21no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—82no—
660C22—81no—
661C22—82no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—82no—
666C22—81no—
667C22—82no—
668C22—164no—
669C22—168no—
670C22C2×#50912yes—
671C22C23×D811yescharacteristic
672C4—22no—
673C4—21no—
674C4—22no—
675C4—21no—
676C4—22no—
677C4—21no—
678C4—81no—
679C4—82no—
680C4—81no—
681C4—82no—
682C4—84no—
683C4C2×#50912yes—
684C4C2×#50912yes—
685C4#348811yescharacteristic
686C4#358511yescharacteristic
687C2—22no—
688C2—42no—
689C2—82no—
690C2—162no—
691C2—161no—
692C2—162no—
693C2#6194711yescharacteristic
694C2C22×#50911yescharacteristic
695C2#6263711yescharacteristic
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. #62418C2semidirect
2. D8×#41C2semidirect
3. #33964C2semidirect
4. #33934C2semidirect
5. #33897C2semidirect
6. #33875C2semidirect
7. #33838C2semidirect
8. #33831C2semidirect
9. #33820C2semidirect
10. #33811C2semidirect
11. #33780C2semidirect
12. #33772C2semidirect
13. #33670C2semidirect
14. #33660C2semidirect
15. #33580C2semidirect
16. C2×#2208C2semidirect
17. #33503C2semidirect
23. #3393C22semidirect
32. #3285C22semidirect
34. #3283C22semidirect
39. #3170C22semidirect
42. #3169C22semidirect
43. #3147C22semidirect
44. C2×#494C22semidirect
50. #2250C22semidirect
51. #2248C22semidirect
52. #2243C22semidirect
53. #2241C22semidirect
54. #2236C22semidirect
55. #2232C22semidirect
56. #2227C22semidirect
58. #2224C22semidirect
59. #2224C22semidirect
60. #2224C22semidirect
61. #2220C22semidirect
62. #2220C22semidirect
63. #2215C22semidirect
65. #2210C22semidirect
68. #2208C22semidirect
69. #2208C22semidirect
70. #2208C22semidirect
71. #2208C22semidirect
72. #2207C22semidirect
74. #2206C22semidirect
75. #2206C22semidirect
76. #2206C22semidirect
78. #2186C22semidirect
79. #2186C22semidirect
80. #2185C22semidirect
82. #2183C22semidirect
84. #2175C22semidirect
86. C4×D16C22semidirect
87. #2169C22semidirect
88. #2168C22semidirect
89. #2163C22semidirect
90. #2162C22semidirect
91. #2150C22semidirect
92. C2×C4⋊C16C22semidirect
93. #2139C22semidirect
94. C2×#358C22semidirect
634. C2×C4#509semidirect

Maximal subgroups

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

subgroupindex
#624182
D8×#412
#339642
#339342
#338972
#338752
#338382
#338312
#338202
#338112
#337802
#337722
#336702
#336602
#335802
C2×#22082
#335032