Finite Group Databasebeta

← #7623127

Subgroups — #7623127

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60708C211yesmaximal, characteristic, complemented
2C2×#2951C211yesmaximal, characteristic, complemented
3#33684C214yesmaximal, complemented
4#33680C214yesmaximal, complemented
5#33677C212yesmaximal, complemented
6#33670C212yesmaximal, complemented
7#33665C212yesmaximal, complemented
8#33664C212yesmaximal, complemented
9#33646C211yesmaximal, characteristic, complemented
10C2×#2236C211yesmaximal, characteristic, complemented
11#33633C212yesmaximal, complemented
12#33590C212yesmaximal, complemented
13#33521C211yesmaximal, characteristic, complemented
14#33516C211yesmaximal, characteristic
15#33459C212yesmaximal, complemented
16#33455C212yesmaximal, complemented
17#33422C211yesmaximal, characteristic, complemented
18C22×#135C2211yescharacteristic
19#3495—21no—
20C2×#520—21no—
21C2×#510C2212yes—
22C2×#508C2212yes—
23#3181—22no—
24#3178—22no—
25#3177C2212yescomplemented
26#3175C2212yescomplemented
27#3170C2212yes—
28#3168C2212yes—
29#3164C2211yescharacteristic, complemented
30#3156C2211yescharacteristic, complemented
31#3137—21no—
32#3056—21no—
33#3050C2211yescharacteristic
34C2×#467C2211yescharacteristic
35#2951—21no—
36#2951—22no—
37#2951—21no—
38#2951C2212yes—
39#2951C2214yes—
40#2951C2212yes—
41#2951C2211yescharacteristic, complemented
42#2950C2211yescharacteristic, complemented
43#2947—21no—
44C2×#440—22no—
45C2×C4×Q16C2212yes—
46C2×C4×D8C2212yes—
47C22×#107C2211yescharacteristic
48C2×#417—21no—
49C2×#405C2211yescharacteristic
50#2236C2212yes—
51#2236C2212yes—
52#2236C2212yes—
53#2236C2214yescomplemented
54#2235C2212yescomplemented
55#2233C2212yes—
56#2232C2212yescomplemented
57#2229C2214yescomplemented
58#2228C2214yescomplemented
59#2227C2214yescomplemented
60#2224C2214yescomplemented
61#2211C2214yescomplemented
62#2210C2214yescomplemented
63#2209C2214yescomplemented
64#2208C2214yescomplemented
65#2189C2212yes—
66#2189C2214yes—
67#2188C2211yescharacteristic
68#2188C2212yes—
69#2187C2212yes—
70#2187C2211yescharacteristic
71#2186C2212yes—
72#2185C2211yescharacteristic
73#2184C2211yescharacteristic
74#2176C2214yescomplemented
75#2175C2214yescomplemented
76#2175C2214yescomplemented
77#2175C2214yescomplemented
78C4×Q32C2214yescomplemented
79C4×Q32C2214yescomplemented
80C4×SD32C2214yescomplemented
81C4×D16C2214yescomplemented
82#2164C2214yescomplemented
83#2163C2212yes—
84#2162C2212yes—
85#2159C2212yescomplemented
86C2×#366C2212yescomplemented
87#2151C2212yescomplemented
88#2139C2212yes—
89#2139C2212yescomplemented
90#2138C2212yes—
91C2×#358C2212yescomplemented
92#2112C2211yescharacteristic
93C2×C22⋊C16C2211yescharacteristic
94#2108C2212yescomplemented
95C22×#41—21no—
96C2×#137—21no—
97C2×#135—28no—
98C2×#135—21no—
99C2×#135—21no—
100C2×#135C2311yescharacteristic
101C2×#135C2312yes—
102C2×#135C2312yes—
103C22×Q16C2311yescharacteristic
104C22×Q16C2311yescharacteristic
105C22×SD16—21no—
106C22×D8C2311yescharacteristic
107C22×M16D411yescharacteristic
108C23×C8C2311yescharacteristic
109#556—41no—
110#554—41no—
111#552—42no—
112#552—41no—
113#552—41no—
114#551—42no—
115#544—21no—
116#542—22no—
117#541—22no—
118#536—21no—
119#533—22no—
120#529—22no—
121#525—21no—
122C2×C22⋊Q8—21no—
123#520—22no—
124#520—22no—
125#520—24no—
126#520—21no—
127#520—42no—
128#520—41no—
129#520—41no—
130#519—21no—
131C2×C4×Q8—22no—
132C2×C4×D4—22no—
133C2×C4×D4—42no—
134C2×#117—41no—
135C2×#117D411yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—41no—
138#510—28no—
139#508—28no—
140C2×Q32C2312yes—
141C2×Q32C2314yes—
142C2×SD32C2312yes—
143C2×SD32C2312yes—
144C2×SD32C2314yes—
145C2×D16C2312yes—
146C2×M32C2312yes—
147C22×C16C2312yes—
148#501—22no—
149#500C2312yes—
150#499C2312yes—
151#497—22no—
152#495C2312yes—
153#494C2312yes—
154#491—22no—
155#489—22no—
156#488—22no—
157#487—22no—
158#470—22no—
159#468—22no—
160#467—22no—
161#467—21no—
162#467—21no—
163#467C2312yes—
164#467C2311yescharacteristic
165#467C2312yes—
166#467C2311yescharacteristic
167#466C2311yescharacteristic
168#466C2311yescharacteristic
169#465—22no—
170#465—21no—
171#465—21no—
172#441—22no—
173#441—22no—
174#440—22no—
175#440—22no—
176#440—24no—
177#440—24no—
178#440—24no—
179#440—24no—
180C4×Q16—22no—
181C4×Q16—22no—
182C4×Q16—22no—
183C4×Q16C2312yes—
184C4×Q16C2312yes—
185C4×Q16C2312yes—
186C4×Q16C2314yes—
187C4×Q16C2314yes—
188C4×SD16—22no—
189C4×D8—22no—
190C4×D8—22no—
191C4×D8C2312yes—
192C4×D8C2314yes—
193C4×D8C2314yes—
194#428—21no—
195#427C2311yescharacteristic
196C2×#107—21no—
197C2×#107—21no—
198C2×#107—21no—
199C2×#107C2312yes—
200C2×#107C2311yescharacteristic
201C2×#107C2312yes—
202C2×#107C2312yes—
203#419—21no—
204#417—21no—
205#417—22no—
206#417—22no—
207#417—24no—
208#416—21no—
209C2×#103—21no—
210C2×#103—22no—
211C2×#102—22no—
212#405D411yescharacteristic
213#405D412yes—
214#405D411yescharacteristic
215#405C2311yescharacteristic
216#405C2311yescharacteristic
217#405C2312yes—
218C2×#97D412yes—
219C2×C4×C8C2312yes—
220#367C2314yes—
221#366C2314yes—
222#366C2314yes—
223#366C2314yes—
224C4⋊C16C2314yes—
225C4⋊C16C2314yes—
226#358C2314yes—
227#358C2314yes—
228#358C2312yes—
229#358C2312yes—
230#358C2314yes—
231#358C2314yes—
232#357C2312yes—
233#357C2312yes—
234#357C2314yes—
235#357C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2311yescharacteristic
238C22⋊C16C2312yes—
239C22⋊C16C2312yes—
240C22⋊C16C2312yes—
241#346C2314yes—
242C4×C16C2314yes—
243C2×#41—28no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—21no—
247C2×#41—44no—
248C2×#41—41no—
249C2×#41—41no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×D4—21no—
253C22×D4—41no—
254C23×C4—41no—
255C23×C4—41no—
256C23×C4—42no—
257C23×C4C2×D411yescharacteristic
258#137—44no—
259#135—28no—
260#135—28no—
261#135—24no—
262#135—24no—
263#135—24no—
264#135—24no—
265#135#4118yes—
266C2×Q16—24no—
267C2×Q16—21no—
268C2×Q16—21no—
269C2×Q16—22no—
270C2×Q16—24no—
271C2×Q16C2412yes—
272C2×Q16C2411yescharacteristic
273C2×Q16C2412yes—
274C2×SD16—28no—
275C2×SD16—22no—
276C2×SD16—22no—
277C2×SD16—21no—
278C2×SD16—22no—
279C2×D8—24no—
280C2×D8—21no—
281C2×D8C2411yescharacteristic
282C2×D8C2412yes—
283C2×M16—42no—
284C2×M16C2×D411yescharacteristic
285C2×M16C2×D411yescharacteristic
286C2×M16C2×D412yes—
287C22×C8—24no—
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D412yes—
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—41no—
298#126—42no—
299#126—42no—
300#126—42no—
301#126—42no—
302#125—22no—
303#125—42no—
304#125—42no—
305#125—41no—
306#125—42no—
307#124—22no—
308#124—42no—
309#123—41no—
310#123—42no—
311#123—42no—
312#123—42no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8—41no—
318C22⋊Q8—41no—
319C22⋊Q8—42no—
320C22⋊Q8—41no—
321C22⋊Q8—41no—
322C22⋊Q8—42no—
323C22⋊Q8—42no—
324#121—21no—
325#121—21no—
326#120—41no—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8—41no—
333C4×Q8—41no—
334C4×Q8—42no—
335C4×Q8—42no—
336C4×Q8—42no—
337C4×D4—24no—
338C4×D4—24no—
339C4×D4—22no—
340C4×D4—41no—
341C4×D4—41no—
342C4×D4—42no—
343C4×D4—44no—
344C4×D4—44no—
345C4×D4—44no—
346C4×D4—44no—
347C4×D4—42no—
348C4×D4—42no—
349C4×D4—42no—
350#117—21no—
351#117—42no—
352#117—42no—
353#117—42no—
354#117—44no—
355#117—41no—
356#117—42no—
357#117—42no—
358#117D811yescharacteristic
359#117D812yes—
360#117D811yescharacteristic
361#117C2×D411yescharacteristic
362#117C2×D411yescharacteristic
363#117C2×D412yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—42no—
374C2×C4⋊C4—44no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—41no—
377C2×C4⋊C4D811yescharacteristic
378C2×C4⋊C4D811yescharacteristic
379C2×C22⋊C4—41no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4D812yes—
382C2×C42—42no—
383C2×C42C2×D412yes—
384Q32—24no—
385Q32—28no—
386SD32—24no—
387SD32—24no—
388SD32—28no—
389D16—24no—
390M32—24no—
391C2×C16—24no—
392C2×C16C2412yes—
393C2×C16C2414yes—
394C2×C16C2412yes—
395#107—22no—
396#107—22no—
397#107—22no—
398#107—21no—
399#107—22no—
400#107C2412yes—
401#107C2411yescharacteristic
402#107C2411yescharacteristic
403#107C2414yes—
404#106—21no—
405#106—22no—
406#103—24no—
407#103—24no—
408#103—22no—
409#103—22no—
410#103—21no—
411#103—21no—
412#103—22no—
413#103—22no—
414#103—22no—
415#102—24no—
416#102—24no—
417#102—21no—
418#102—21no—
419#102—22no—
420#97C2×D412yes—
421#97C2×D412yes—
422#97C2×D412yes—
423#97C2×D412yes—
424C4×C8C2×D412yes—
425C4×C8C2×D412yes—
426C4×C8C2412yes—
427C4×C8C2412yes—
428C24—41no—
429C24C2×D811yescharacteristic
430#41—28no—
431#41—28no—
432#41—48no—
433#41—48no—
434#41—44no—
435#41—44no—
436#41—42no—
437#41—42no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—42no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—48no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—48no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—41no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—164no—
475C22×C4—162no—
476C22×C4C2×D812yes—
477C22×C4C2×D811yescharacteristic
478C22×C4C2×D811yescharacteristic
479C22×C4C2×D811yescharacteristic
480C22×C4C2×D812yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484Q16—24no—
485Q16—22no—
486Q16—24no—
487Q16—22no—
488Q16—44no—
489Q16—42no—
490Q16C2×#4114yes—
491SD16—28no—
492SD16—28no—
493SD16—44no—
494SD16—42no—
495D8—24no—
496D8—22no—
497D8—22no—
498D8C2×#4114yes—
499M16—22no—
500M16—22no—
501M16—42no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—42no—
508C2×C8C22×D411yescharacteristic
509C2×C8C22×D412yes—
510C2×C8C22×D412yes—
511C2×C8C22×D411yescharacteristic
512C2×C8C22×D411yescharacteristic
513C2×C8C2×#4114yes—
514C2×C8C2511yescharacteristic
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—41no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4C2×D811yescharacteristic
538C4⋊C4C2×D812yes—
539C4⋊C4C2×D811yescharacteristic
540C4⋊C4C2×D811yescharacteristic
541C4⋊C4C2×D811yescharacteristic
542C4⋊C4C2×D812yes—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—41no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—41no—
554C22⋊C4—42no—
555C22⋊C4C2×D812yes—
556C22⋊C4C2×D812yes—
557C22⋊C4C2×D812yes—
558C22⋊C4C2×D812yes—
559C42—44no—
560C42—44no—
561C42—41no—
562C42—42no—
563C42—42no—
564C42—42no—
565C42C2×D812yes—
566C42C2×D812yes—
567C42C22×D412yes—
568C42C22×D412yes—
569C16—22no—
570C16—22no—
571C16—24no—
572C23—22no—
573C23—22no—
574C23—41no—
575C23—44no—
576C23—42no—
577C23—41no—
578C23—82no—
579C23—82no—
580C23C22×D811yescharacteristic
581C23C22×D811yescharacteristic
582C23C22×D811yescharacteristic
583Q8—24no—
584Q8—44no—
585Q8—42no—
586Q8—41no—
587Q8—42no—
588Q8—42no—
589Q8—41no—
590Q8—82no—
591Q8—162no—
592D4—24no—
593D4—44no—
594D4—48no—
595D4—48no—
596D4—41no—
597D4—42no—
598D4—41no—
599D4—82no—
600D4—82no—
601D4—161no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—42no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—84no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—162no—
625C2×C4—164no—
626C2×C4—164no—
627C2×C4—161no—
628C2×C4—162no—
629C2×C4—162no—
630C2×C4#52512yes—
631C2×C4C22×D812yes—
632C2×C4C22×D812yes—
633C2×C4C22×D812yes—
634C2×C4C22×D811yescharacteristic
635C2×C4C22×D812yes—
636C2×C4C22×D811yescharacteristic
637C2×C4C22×D811yescharacteristic
638C2×C4#57512yes—
639C2×C4C23×D411yescharacteristic
640C8—21no—
641C8—21no—
642C8—22no—
643C8—22no—
644C8#52512yes—
645C8C22×#4112yes—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—44no—
654C22—44no—
655C22—88no—
656C22—82no—
657C22—82no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22#310012yes—
662C22#341712yes—
663C22C23×D811yescharacteristic
664C4—21no—
665C4—21no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—84no—
675C4—82no—
676C4—81no—
677C4#310012yes—
678C4C2×#52511yescharacteristic
679C4C2×#57511yescharacteristic
680C2—22no—
681C2—22no—
682C2—41no—
683C2—84no—
684C2—161no—
685C2C2×#310011yescharacteristic
686C2C2×#341711yescharacteristic
687C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#33516C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60708C2semidirect
2. C2×#2951C2semidirect
3. #33684C2semidirect
4. #33680C2semidirect
5. #33677C2semidirect
6. #33670C2semidirect
7. #33665C2semidirect
8. #33664C2semidirect
9. #33646C2semidirect
10. C2×#2236C2semidirect
11. #33633C2semidirect
12. #33590C2semidirect
13. #33521C2semidirect
15. #33459C2semidirect
16. #33455C2semidirect
17. #33422C2semidirect
25. #3177C22semidirect
26. #3175C22semidirect
29. #3164C22semidirect
30. #3156C22semidirect
41. #2951C22semidirect
42. #2950C22semidirect
53. #2236C22semidirect
54. #2235C22semidirect
56. #2232C22semidirect
57. #2229C22semidirect
58. #2228C22semidirect
59. #2227C22semidirect
60. #2224C22semidirect
61. #2211C22semidirect
62. #2210C22semidirect
63. #2209C22semidirect
64. #2208C22semidirect
74. #2176C22semidirect
75. #2175C22semidirect
76. #2175C22semidirect
77. #2175C22semidirect
78. C4×Q32C22semidirect
79. C4×Q32C22semidirect
80. C4×SD32C22semidirect
81. C4×D16C22semidirect
82. #2164C22semidirect
85. #2159C22semidirect
86. C2×#366C22semidirect
87. #2151C22semidirect
89. #2139C22semidirect
91. C2×#358C22semidirect
94. #2108C22semidirect

Maximal subgroups

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

subgroupindex
#607082
C2×#29512
#336842
#336802
#336772
#336702
#336652
#336642
#336462
C2×#22362
#336332
#335902
#335212
#335162
#334592
#334552
#334222