Finite Group Databasebeta

← #7624239

Subgroups — #7624239

Order 512.

1535 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62252C211yesmaximal, characteristic, complemented
2C2×#3283C211yesmaximal, characteristic, complemented
3#33897C212yesmaximal, complemented
4#33882C214yesmaximal, complemented
5#33869C212yesmaximal, complemented
6#33849C212yesmaximal, complemented
7#33842C212yesmaximal, complemented
8#33835C214yesmaximal, complemented
9#33784C211yesmaximal, characteristic, complemented
10#33775C211yesmaximal, characteristic, complemented
11#33762C212yesmaximal, complemented
12#33755C212yesmaximal, complemented
13#33670C211yesmaximal, characteristic, complemented
14#33665C211yesmaximal, characteristic, complemented
15#33521C212yesmaximal, complemented
16#33516C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18C22×#135C2212yes—
19#3520—21no—
20C2×#547—21no—
21C2×#510C2214yescomplemented
22C2×#508C2214yescomplemented
23#3342C2211yescharacteristic, complemented
24#3336C2212yescomplemented
25#3334C2211yescharacteristic, complemented
26#3302—21no—
27#3300—22no—
28#3298—21no—
29D4×Q16C2211yescharacteristic, complemented
30#3285C2212yescomplemented
31#3283—21no—
32#3283—22no—
33#3283—21no—
34#3283C2214yes—
35#3283C2212yes—
36#3283C2212yes—
37#3283C2211yescharacteristic, complemented
38#3248—21no—
39#3190—22no—
40#3170C2211yescharacteristic
41#3168C2211yescharacteristic
42C2×#494C2211yescharacteristic
43#3050C2212yes—
44C2×#467C2212yes—
45C2×#467C2212yes—
46C2×#461—21no—
47C2×#452—22no—
48C2×C4×D8C2211yescharacteristic
49C2×C8×D4C2211yescharacteristic
50#2251C2212yescomplemented
51#2247C2212yescomplemented
52#2245C2212yescomplemented
53#2241C2212yescomplemented
54#2237C2214yescomplemented
55#2236C2211yescharacteristic
56#2236C2212yes—
57#2236C2214yescomplemented
58#2233C2211yescharacteristic
59#2228C2212yescomplemented
60#2224C2212yescomplemented
61#2220C2214yescomplemented
62#2219C2214yescomplemented
63#2216C2214yescomplemented
64#2216C2214yescomplemented
65#2210C2212yescomplemented
66#2209C2214yescomplemented
67#2209C2212yescomplemented
68#2208C2212yescomplemented
69#2208C2212yescomplemented
70#2189C2214yescomplemented
71#2189C2214yescomplemented
72#2189C2214yescomplemented
73#2189C2214yescomplemented
74#2188C2212yes—
75#2188C2212yes—
76#2188C2212yes—
77#2188C2214yescomplemented
78#2187C2211yescharacteristic
79#2187C2212yes—
80#2186C2214yescomplemented
81#2185C2212yes—
82#2184C2211yescharacteristic
83#2175C2214yescomplemented
84C4×SD32C2212yescomplemented
85C4×D16C2212yescomplemented
86#2169C2212yescomplemented
87#2168C2214yescomplemented
88D4×C16C2212yescomplemented
89#2163C2211yescharacteristic
90#2162C2211yescharacteristic
91#2139C2214yescomplemented
92#2138C2214yescomplemented
93#2112C2212yes—
94C2×C22⋊C16C2212yes—
95C22×#41—22no—
96C2×#135—22no—
97C2×#135—22no—
98C2×#135—48no—
99C2×#135C2314yes—
100C2×#135C2314yes—
101C2×#135C2312yes—
102C22×Q16—21no—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×SD16—22no—
106C22×D8C2311yescharacteristic
107C23×C8C2312yes—
108#556—21no—
109#554—41no—
110#552—22no—
111#551—42no—
112D4×Q8—21no—
113#548—22no—
114#547—21no—
115#547—22no—
116#547—22no—
117#547—24no—
118#547—21no—
119#547—41no—
120#547—42no—
121#547—41no—
122#546—41no—
123#542—41no—
124#539—42no—
125#533—21no—
126C2×C4⋊Q8—21no—
127#529—21no—
128#525—22no—
129C2×#123—42no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×#121—41no—
133#520—41no—
134C2×C4×D4—21no—
135C2×C4×D4D411yescharacteristic
136C22×C4⋊C4—41no—
137#510—48no—
138#508—48no—
139C2×Q32C2314yes—
140C2×Q32C2314yes—
141C2×Q32C2314yes—
142C2×SD32C2314yes—
143C2×SD32C2312yes—
144C2×SD32C2314yes—
145C2×D16C2312yes—
146C2×M32C2314yes—
147C22×C16C2314yes—
148#500C2311yescharacteristic
149#499C2311yescharacteristic
150#495C2311yescharacteristic
151#494—21no—
152#494—21no—
153#494C2312yes—
154#494C2312yes—
155#494C2311yescharacteristic
156#488—21no—
157#487—21no—
158#485—22no—
159#485—22no—
160#483—22no—
161#480—22no—
162#479—22no—
163#475—21no—
164#474—21no—
165#467—21no—
166#467—22no—
167#467—22no—
168#467—22no—
169#467—22no—
170#467C2312yes—
171#467C2312yes—
172#467C2312yes—
173#467C2311yescharacteristic
174#467C2312yes—
175#467C2314yes—
176#467C2314yes—
177#466C2312yes—
178#466C2311yescharacteristic
179#465—21no—
180#465—22no—
181#465—22no—
182#464—22no—
183#462—21no—
184#461—21no—
185#461—22no—
186#461—22no—
187#461—24no—
188#452—22no—
189#452—22no—
190#452—24no—
191#452—24no—
192#452—24no—
193#452—24no—
194#451—22no—
195#448—22no—
196C4×Q16—21no—
197C4×Q16C2312yes—
198C4×Q16C2311yescharacteristic
199C4×SD16—21no—
200C4×D8—21no—
201C4×D8—21no—
202C4×D8C2311yescharacteristic
203C4×D8C2312yes—
204C4×D8C2312yes—
205C8×D4D411yescharacteristic
206C8×D4D412yes—
207C8×D4D411yescharacteristic
208C8×D4C2311yescharacteristic
209C8×D4C2312yes—
210C8×D4C2311yescharacteristic
211#427C2312yes—
212C2×#107C2312yes—
213C2×#107C2311yescharacteristic
214C2×C4⋊C8D411yescharacteristic
215#416—22no—
216C2×#103—22no—
217C2×#103—22no—
218C2×#102—21no—
219C2×C22⋊C8D412yes—
220C2×C4×C8C2311yescharacteristic
221#367C2312yes—
222#366C2312yes—
223#366C2314yes—
224C4⋊C16C2312yes—
225C4⋊C16C2312yes—
226#358C2314yes—
227#358C2314yes—
228#358C2314yes—
229#358C2314yes—
230#358C2312yes—
231#358C2314yes—
232#357C2312yes—
233#357C2314yes—
234#357C2312yes—
235#357C2312yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2312yes—
238C22⋊C16C2314yes—
239C22⋊C16C2311yescharacteristic
240C22⋊C16C2314yes—
241#346C2312yes—
242C4×C16C2312yes—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—22no—
246C2×#41—41no—
247C2×#41—42no—
248C2×#41—42no—
249C2×#41—48no—
250C2×#41—84no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×D4—21no—
254C22×D4—42no—
255C22×D4D811yescharacteristic
256C23×C4—41no—
257C23×C4—42no—
258C23×C4C2×D412yes—
259#135—28no—
260#135—28no—
261#135—44no—
262#135—44no—
263#135—44no—
264#135—48no—
265#135—48no—
266C2×Q16—22no—
267C2×Q16—22no—
268C2×Q16—22no—
269C2×Q16—22no—
270C2×Q16—21no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×Q16—82no—
274C2×Q16C2411yescharacteristic
275C2×Q16C2414yes—
276C2×Q16C2412yes—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—22no—
280C2×SD16—21no—
281C2×SD16—48no—
282C2×D8—24no—
283C2×D8—21no—
284C2×D8C2411yescharacteristic
285C2×D8C2412yes—
286C22×C8—44no—
287C22×C8C2×D412yes—
288C22×C8C2×D412yes—
289C22×C8C2×D412yes—
290C22×C8C2×D411yescharacteristic
291C22×C8C2411yescharacteristic
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—21no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—21no—
298C4⋊Q8—41no—
299C4⋊Q8—41no—
300C4⋊Q8—41no—
301#126—42no—
302#126—42no—
303#126—41no—
304#125—21no—
305#125—42no—
306#125—41no—
307#124—21no—
308#124—41no—
309#124—42no—
310#123—42no—
311#123—42no—
312#123—41no—
313#123—44no—
314#123—44no—
315#123—44no—
316#123—44no—
317#123—42no—
318#123—42no—
319C22⋊Q8—21no—
320C22⋊Q8—22no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—42no—
327C22⋊Q8—42no—
328C22⋊Q8—42no—
329C22⋊Q8—42no—
330C22⋊Q8—41no—
331C22⋊Q8—42no—
332C22⋊Q8—41no—
333C22⋊Q8—42no—
334#121—21no—
335#121—22no—
336#121—42no—
337#121—42no—
338#121—44no—
339#121—42no—
340#121—41no—
341#120—42no—
342C4×Q8—21no—
343C4×Q8—22no—
344C4×Q8—41no—
345C4×Q8—41no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—21no—
349C4×D4—41no—
350C4×D4—42no—
351C4×D4—42no—
352C4×D4—41no—
353C4×D4—42no—
354C4×D4—41no—
355C4×D4—41no—
356C4×D4D811yescharacteristic
357C4×D4D812yes—
358C4×D4D811yescharacteristic
359C4×D4C2×D411yescharacteristic
360C4×D4C2×D411yescharacteristic
361C4×D4C2×D412yes—
362#117—22no—
363#117—41no—
364#117—42no—
365#117—41no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4—44no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4D811yescharacteristic
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4D812yes—
381C2×C42C2×D411yescharacteristic
382Q32—44no—
383Q32—44no—
384Q32—44no—
385SD32—28no—
386SD32—24no—
387SD32—44no—
388D16—24no—
389M32—44no—
390C2×C16—44no—
391C2×C16C2412yes—
392C2×C16C2412yes—
393C2×C16C2414yes—
394#107—21no—
395#107—22no—
396#107—21no—
397#107C2411yescharacteristic
398#107C2412yes—
399#107C2411yescharacteristic
400#107C2412yes—
401#106—22no—
402#106—21no—
403C4⋊C8C2×D411yescharacteristic
404C4⋊C8C2×D411yescharacteristic
405C4⋊C8C2×D412yes—
406#103—21no—
407#103—22no—
408#103—22no—
409#103—24no—
410#103—24no—
411#103—22no—
412#103—22no—
413#103—22no—
414#103—21no—
415#102—22no—
416#102—22no—
417#102—21no—
418#102—22no—
419#102—21no—
420C22⋊C8C2×D412yes—
421C22⋊C8C2×D412yes—
422C22⋊C8C2×D412yes—
423C22⋊C8C2×D412yes—
424C4×C8C2×D411yescharacteristic
425C4×C8C2×D411yescharacteristic
426C4×C8C2411yescharacteristic
427C4×C8C2411yescharacteristic
428C24—41no—
429C24C2×D812yes—
430#41—48no—
431#41—48no—
432#41—84no—
433#41—84no—
434#41—82no—
435#41—82no—
436#41—88no—
437#41—88no—
438#41—84no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—41no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—88no—
461C2×D4C2×D811yescharacteristic
462C2×D4C2×D811yescharacteristic
463C2×D4C2×D812yes—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—41no—
475C22×C4—42no—
476C22×C4—84no—
477C22×C4—88no—
478C22×C4C2×D811yescharacteristic
479C22×C4C2×D811yescharacteristic
480C22×C4C2×D812yes—
481C22×C4C2×D812yes—
482C22×C4C2×D812yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486Q16—22no—
487Q16—24no—
488Q16—24no—
489Q16—24no—
490Q16—44no—
491Q16—81no—
492Q16—82no—
493Q16—82no—
494Q16—82no—
495SD16—48no—
496SD16—48no—
497SD16—81no—
498SD16—82no—
499D8—24no—
500D8—22no—
501D8—22no—
502D8H(5,𝔽2)14yes—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—42no—
507C2×C8—42no—
508C2×C8—42no—
509C2×C8—41no—
510C2×C8C22×D411yescharacteristic
511C2×C8C22×D411yescharacteristic
512C2×C8C22×D411yescharacteristic
513C2×C8C22×D412yes—
514C2×C8C22×D412yes—
515C2×C8C2511yescharacteristic
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—41no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—41no—
527C4⋊C4—42no—
528C4⋊C4—41no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—42no—
537C4⋊C4—41no—
538C4⋊C4—42no—
539C4⋊C4—41no—
540C4⋊C4C2×D811yescharacteristic
541C4⋊C4C2×D812yes—
542C4⋊C4C2×D811yescharacteristic
543C22⋊C4—41no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—42no—
557C22⋊C4—41no—
558C22⋊C4C2×D812yes—
559C22⋊C4C2×D812yes—
560C22⋊C4C2×D812yes—
561C22⋊C4C2×D812yes—
562C42—42no—
563C42—42no—
564C42—41no—
565C42C2×D811yescharacteristic
566C42C2×D811yescharacteristic
567C42C22×D411yescharacteristic
568C42C22×D411yescharacteristic
569C16—22no—
570C16—24no—
571C16—22no—
572C23—24no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23—41no—
578C23—82no—
579C23—82no—
580C23C22×D812yes—
581C23C22×D812yes—
582C23C22×D811yescharacteristic
583Q8—41no—
584Q8—44no—
585Q8—42no—
586Q8—44no—
587Q8—42no—
588Q8—41no—
589Q8—84no—
590Q8—82no—
591Q8—162no—
592D4—22no—
593D4—22no—
594D4—24no—
595D4—42no—
596D4—44no—
597D4—42no—
598D4—41no—
599D4—41no—
600D4—88no—
601D4—88no—
602D4—84no—
603D4—82no—
604D4—162no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4—82no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—82no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4C22×D811yescharacteristic
634C2×C4C22×D812yes—
635C2×C4C22×D811yescharacteristic
636C2×C4C22×D811yescharacteristic
637C2×C4C22×D811yescharacteristic
638C2×C4C22×D812yes—
639C2×C4C22×D811yescharacteristic
640C2×C4C23×D411yescharacteristic
641C8—21no—
642C8—21no—
643C8—21no—
644C8—22no—
645C8—22no—
646C8C2×H(5,𝔽2)12yes—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—44no—
656C22—84no—
657C22—88no—
658C22—82no—
659C22—82no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#341712yes—
664C22C23×D811yescharacteristic
665C4—21no—
666C4—22no—
667C4—22no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—84no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4#341412yes—
679C4#348411yescharacteristic
680C4#358511yescharacteristic
681C2—22no—
682C2—42no—
683C2—42no—
684C2—84no—
685C2—161no—
686C2#6155911yescharacteristic
687C2C2×#341411yescharacteristic
688C2C2×#341711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62252C2semidirect
2. C2×#3283C2semidirect
3. #33897C2semidirect
4. #33882C2semidirect
5. #33869C2semidirect
6. #33849C2semidirect
7. #33842C2semidirect
8. #33835C2semidirect
9. #33784C2semidirect
10. #33775C2semidirect
11. #33762C2semidirect
12. #33755C2semidirect
13. #33670C2semidirect
14. #33665C2semidirect
15. #33521C2semidirect
16. #33516C2semidirect
17. #33488C2semidirect
21. C2×#510C22semidirect
22. C2×#508C22semidirect
23. #3342C22semidirect
24. #3336C22semidirect
25. #3334C22semidirect
29. D4×Q16C22semidirect
30. #3285C22semidirect
37. #3283C22semidirect
50. #2251C22semidirect
51. #2247C22semidirect
52. #2245C22semidirect
53. #2241C22semidirect
54. #2237C22semidirect
57. #2236C22semidirect
59. #2228C22semidirect
60. #2224C22semidirect
61. #2220C22semidirect
62. #2219C22semidirect
63. #2216C22semidirect
64. #2216C22semidirect
65. #2210C22semidirect
66. #2209C22semidirect
67. #2209C22semidirect
68. #2208C22semidirect
69. #2208C22semidirect
70. #2189C22semidirect
71. #2189C22semidirect
72. #2189C22semidirect
73. #2189C22semidirect
77. #2188C22semidirect
80. #2186C22semidirect
83. #2175C22semidirect
84. C4×SD32C22semidirect
85. C4×D16C22semidirect
86. #2169C22semidirect
87. #2168C22semidirect
88. D4×C16C22semidirect
91. #2139C22semidirect
92. #2138C22semidirect

Maximal subgroups

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

subgroupindex
#622522
C2×#32832
#338972
#338822
#338692
#338492
#338422
#338352
#337842
#337752
#337622
#337552
#336702
#336652
#335212
#335162
#334882