Finite Group Databasebeta

← #7624372

Subgroups — #7624372

Order 512.

1309 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62251C211yesmaximal, characteristic, complemented
2C2×#3341C211yesmaximal, characteristic, complemented
3#33924C212yesmaximal, complemented
4#33914C214yesmaximal, complemented
5#33908C212yesmaximal, complemented
6#33892C212yesmaximal, complemented
7#33886C214yesmaximal, complemented
8#33872C212yesmaximal, complemented
9#33804C211yesmaximal, characteristic, complemented
10#33792C211yesmaximal, characteristic, complemented
11#33760C212yesmaximal, complemented
12#33752C212yesmaximal, complemented
13#33669C211yesmaximal, characteristic, complemented
14#33665C211yesmaximal, characteristic, complemented
15#33648C212yesmaximal, complemented
16#33645C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18#3519—21no—
19C2×#555—21no—
20#3341—21no—
21#3341—22no—
22#3341—21no—
23#3341C2212yes—
24#3341C2211yescharacteristic, complemented
25#3341C2212yescomplemented
26#3341C2214yescomplemented
27#3339C2212yescomplemented
28#3338C2211yescharacteristic, complemented
29#3305—21no—
30#3297—22no—
31#3295—21no—
32#3287C2211yescharacteristic, complemented
33#3280C2212yescomplemented
34D4×D8C2211yescharacteristic, complemented
35#3247—21no—
36#3188—22no—
37#3169C2211yescharacteristic
38#3168C2211yescharacteristic
39C2×#500C2211yescharacteristic, complemented
40C2×#482—22no—
41C2×#476—21no—
42#3050C2212yes—
43C2×#466C2212yes—
44C2×#466C2212yescomplemented
45C2×C4×D8C2211yescharacteristic, complemented
46C2×C8×D4C2211yescharacteristic
47C2×#427C2212yescomplemented
48#2255C2212yescomplemented
49#2252C2212yescomplemented
50#2243C2212yescomplemented
51#2240C2212yescomplemented
52#2237C2214yescomplemented
53#2236C2211yescharacteristic, complemented
54#2235C2212yescomplemented
55#2234C2214yescomplemented
56#2234C2214yescomplemented
57#2234C2214yescomplemented
58#2234C2214yescomplemented
59#2233C2211yescharacteristic
60#2233C2212yescomplemented
61#2232C2212yes—
62#2232C2212yescomplemented
63#2232C2212yescomplemented
64#2232C2214yescomplemented
65#2228C2212yescomplemented
66#2227C2212yescomplemented
67#2226C2212yescomplemented
68#2226C2212yescomplemented
69#2224C2214yescomplemented
70#2220C2214yescomplemented
71#2218C2214yescomplemented
72#2215C2214yescomplemented
73#2215C2214yescomplemented
74#2210C2212yescomplemented
75#2206C2212yescomplemented
76#2187C2211yescharacteristic, complemented
77#2186C2214yescomplemented
78#2184C2211yescharacteristic
79#2184C2214yescomplemented
80#2184C2212yescomplemented
81#2177C2214yescomplemented
82C4×SD32C2212yescomplemented
83C4×D16C2212yescomplemented
84#2169C2212yescomplemented
85#2168C2214yescomplemented
86D4×C16C2212yescomplemented
87#2163C2211yescharacteristic, complemented
88#2162C2211yescharacteristic, complemented
89#2159C2214yescomplemented
90#2158C2214yescomplemented
91#2140C2214yescomplemented
92#2138C2214yescomplemented
93#2112C2212yescomplemented
94C2×C22⋊C16C2212yescomplemented
95C2×#135C2312yes—
96C22×D8C2312yes—
97C22×D8C2311yescharacteristic
98C23×C8C2312yes—
99#555—22no—
100#555—24no—
101#555—22no—
102#555—21no—
103#555—41no—
104#555—42no—
105#555—41no—
106#554—21no—
107#553—22no—
108#553—41no—
109#550—21no—
110#546—22no—
111#546—42no—
112D42—21no—
113D42—41no—
114#540—41no—
115#538—42no—
116#532—21no—
117C2×C4⋊Q8—21no—
118#529—21no—
119C2×#124—42no—
120#525—22no—
121C2×#121—22no—
122C2×#121—22no—
123C2×#121—41no—
124#518—41no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×#117—22no—
128C2×#117—41no—
129C2×SD32C2312yes—
130C2×D16C2312yes—
131C2×D16C2314yes—
132C2×M32C2314yes—
133C22×C16C2314yes—
134#500—21no—
135#500—21no—
136#500C2311yescharacteristic
137#500C2312yes—
138#500C2312yes—
139#499C2311yescharacteristic
140#495C2311yescharacteristic
141#493C2311yescharacteristic
142#488—21no—
143#486—21no—
144#484—22no—
145#482—22no—
146#482—24no—
147#482—24no—
148#482—24no—
149#482—24no—
150#482—22no—
151#481—22no—
152#478—22no—
153#477—21no—
154#476—22no—
155#476—22no—
156#476—24no—
157#476—21no—
158#467C2312yes—
159#467C2311yescharacteristic
160#466—22no—
161#466—22no—
162#466—21no—
163#466—22no—
164#466—22no—
165#466C2312yes—
166#466C2312yes—
167#466C2314yes—
168#466C2314yes—
169#466C2312yes—
170#466C2311yescharacteristic
171#466C2312yes—
172#465—21no—
173#465—22no—
174#465—22no—
175#463—22no—
176#460—21no—
177#459—21no—
178#450—22no—
179#449—22no—
180#449—22no—
181#447—22no—
182C4×Q16C2311yescharacteristic
183C4×SD16—21no—
184C4×D8—21no—
185C4×D8—21no—
186C4×D8—21no—
187C4×D8C2312yes—
188C4×D8C2312yes—
189C4×D8C2312yes—
190C4×D8C2311yescharacteristic
191C8×D4D411yescharacteristic
192C8×D4D412yes—
193C8×D4D411yescharacteristic
194C8×D4C2311yescharacteristic
195C8×D4C2312yes—
196C8×D4C2311yescharacteristic
197#427—22no—
198#427—22no—
199#427C2312yes—
200#427C2314yes—
201#427C2314yes—
202C2×#107—21no—
203C2×#107C2312yes—
204C2×#107C2312yes—
205C2×#107C2311yescharacteristic
206C2×#106—22no—
207C2×C4⋊C8D411yescharacteristic
208#416—22no—
209C2×#102—21no—
210C2×#102—22no—
211C2×#102—22no—
212C2×C22⋊C8D412yes—
213C2×C4×C8C2311yescharacteristic
214#367C2314yes—
215#367C2314yes—
216#367C2312yes—
217#366C2312yes—
218#366C2314yes—
219#366C2314yes—
220#366C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223#358C2312yes—
224#358C2314yes—
225#357C2314yes—
226#357C2314yes—
227#357C2312yes—
228#357C2314yes—
229#357C2314yes—
230#357C2312yes—
231#357C2312yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2314yes—
238#346C2312yes—
239C4×C16C2312yes—
240C2×#41—22no—
241C2×#41—41no—
242C22×Q8—41no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4—42no—
247C22×D4—41no—
248C22×D4D811yescharacteristic
249C23×C4—41no—
250C23×C4C2×D412yes—
251#135—44no—
252C2×Q16C2411yescharacteristic
253C2×SD16—22no—
254C2×SD16—21no—
255C2×D8—21no—
256C2×D8—22no—
257C2×D8—21no—
258C2×D8—42no—
259C2×D8—44no—
260C2×D8C2411yescharacteristic
261C2×D8C2412yes—
262C2×D8C2412yes—
263C22×C8—44no—
264C22×C8C2×D412yes—
265C22×C8C2×D412yes—
266C22×C8C2×D412yes—
267C22×C8C2×D411yescharacteristic
268C22×C8C2411yescharacteristic
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—21no—
274C4⋊Q8—41no—
275C4⋊Q8—41no—
276#127—21no—
277#127—41no—
278#126—42no—
279#126—41no—
280#125—21no—
281#125—41no—
282#124—21no—
283#124—44no—
284#124—44no—
285#124—44no—
286#124—44no—
287#124—42no—
288#124—41no—
289#123—42no—
290#123—41no—
291#123—42no—
292#123—42no—
293C22⋊Q8—21no—
294C22⋊Q8—22no—
295C22⋊Q8—42no—
296#121—21no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—24no—
301#121—24no—
302#121—22no—
303#121—42no—
304#121—42no—
305#121—44no—
306#121—42no—
307#121—41no—
308#121—42no—
309#121—42no—
310#121—42no—
311#121—42no—
312#121—41no—
313#121—41no—
314#121—42no—
315#121—42no—
316#120—42no—
317#120—42no—
318#120—42no—
319C4×Q8—21no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—21no—
323C4×D4—22no—
324C4×D4—41no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4—41no—
328C4×D4—41no—
329C4×D4—42no—
330C4×D4—41no—
331C4×D4—41no—
332C4×D4—41no—
333C4×D4D811yescharacteristic
334C4×D4D812yes—
335C4×D4D811yescharacteristic
336C4×D4C2×D411yescharacteristic
337C4×D4C2×D412yes—
338C4×D4C2×D411yescharacteristic
339#117—24no—
340#117—24no—
341#117—22no—
342#117—42no—
343#117—42no—
344#117—42no—
345#117—42no—
346#117—44no—
347#117—41no—
348#117—41no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4D811yescharacteristic
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4D812yes—
360C2×C42—42no—
361C2×C42C2×D411yescharacteristic
362SD32—42no—
363D16—42no—
364D16—44no—
365M32—44no—
366C2×C16—44no—
367C2×C16C2412yes—
368C2×C16C2412yes—
369C2×C16C2414yes—
370#107—22no—
371#107—22no—
372#107—21no—
373#107—22no—
374#107—22no—
375#107—21no—
376#107C2414yes—
377#107C2412yes—
378#107C2411yescharacteristic
379#107C2412yes—
380#107C2411yescharacteristic
381#106—24no—
382#106—24no—
383#106—22no—
384#106—21no—
385C4⋊C8C2×D411yescharacteristic
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D412yes—
388#103—21no—
389#103—22no—
390#103—21no—
391#102—21no—
392#102—22no—
393#102—22no—
394#102—22no—
395#102—22no—
396#102—22no—
397#102—24no—
398#102—24no—
399#102—22no—
400#102—21no—
401#102—22no—
402C22⋊C8C2×D412yes—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C22⋊C8C2×D412yes—
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2411yescharacteristic
409C4×C8C2411yescharacteristic
410C24—41no—
411C24—42no—
412C24C2×D812yes—
413#41—82no—
414#41—82no—
415#41—162no—
416C2×Q8—21no—
417C2×Q8—41no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—162no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—42no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—84no—
442C2×D4—164no—
443C2×D4—164no—
444C2×D4C2×D811yescharacteristic
445C2×D4C2×D811yescharacteristic
446C2×D4C2×D812yes—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—84no—
458C22×C4C2×D811yescharacteristic
459C22×C4C2×D811yescharacteristic
460C22×C4C2×D812yes—
461C22×C4C2×D812yes—
462C22×C4C2×D812yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466Q16—22no—
467SD16—81no—
468SD16—82no—
469D8—22no—
470D8—22no—
471D8—22no—
472D8—24no—
473D8—42no—
474D8—81no—
475D8—82no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—41no—
483C2×C8C22×D411yescharacteristic
484C2×C8C22×D411yescharacteristic
485C2×C8C22×D411yescharacteristic
486C2×C8C22×D412yes—
487C2×C8C22×D412yes—
488C2×C8C2511yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4C2×D811yescharacteristic
506C4⋊C4C2×D811yescharacteristic
507C4⋊C4C2×D812yes—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—41no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—41no—
525C22⋊C4—42no—
526C22⋊C4C2×D812yes—
527C22⋊C4C2×D812yes—
528C22⋊C4C2×D812yes—
529C22⋊C4C2×D812yes—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—41no—
534C42C2×D811yescharacteristic
535C42C2×D811yescharacteristic
536C42C22×D411yescharacteristic
537C42C22×D411yescharacteristic
538C16—22no—
539C16—24no—
540C16—22no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—82no—
548C23—82no—
549C23—84no—
550C23—82no—
551C23—162no—
552C23—162no—
553C23C22×D812yes—
554C23C22×D812yes—
555C23C22×D811yescharacteristic
556Q8—41no—
557Q8—41no—
558Q8—162no—
559Q8—162no—
560Q8—161no—
561D4—22no—
562D4—22no—
563D4—42no—
564D4—42no—
565D4—41no—
566D4—42no—
567D4—42no—
568D4—41no—
569D4—42no—
570D4—82no—
571D4—84no—
572D4—82no—
573D4—84no—
574D4—82no—
575D4—162no—
576D4—161no—
577D4—164no—
578D4—164no—
579D4—162no—
580D4—162no—
581D4—162no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—82no—
604C2×C4—162no—
605C2×C4—162no—
606C2×C4—162no—
607C2×C4C22×D812yes—
608C2×C4C22×D812yes—
609C2×C4C22×D811yescharacteristic
610C2×C4C22×D811yescharacteristic
611C2×C4C22×D811yescharacteristic
612C2×C4C22×D811yescharacteristic
613C2×C4C22×D811yescharacteristic
614C2×C4C23×D411yescharacteristic
615C8—21no—
616C8—21no—
617C8—21no—
618C8—22no—
619C8—22no—
620C8#58512yes—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—84no—
630C22—82no—
631C22—81no—
632C22—82no—
633C22—82no—
634C22—81no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—81no—
639C22—164no—
640C22—324no—
641C22#341512yes—
642C22C23×D811yescharacteristic
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—22no—
648C4—22no—
649C4—82no—
650C4—81no—
651C4—84no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4#341412yes—
656C4#348511yescharacteristic
657C4#358611yescharacteristic
658C2—22no—
659C2—42no—
660C2—42no—
661C2—162no—
662C2—162no—
663C2—161no—
664C2#6169411yescharacteristic
665C2C2×#341411yescharacteristic
666C2C2×#341511yescharacteristic
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. #62251C2semidirect
2. C2×#3341C2semidirect
3. #33924C2semidirect
4. #33914C2semidirect
5. #33908C2semidirect
6. #33892C2semidirect
7. #33886C2semidirect
8. #33872C2semidirect
9. #33804C2semidirect
10. #33792C2semidirect
11. #33760C2semidirect
12. #33752C2semidirect
13. #33669C2semidirect
14. #33665C2semidirect
15. #33648C2semidirect
16. #33645C2semidirect
17. #33488C2semidirect
24. #3341C22semidirect
25. #3341C22semidirect
26. #3341C22semidirect
27. #3339C22semidirect
28. #3338C22semidirect
32. #3287C22semidirect
33. #3280C22semidirect
34. D4×D8C22semidirect
39. C2×#500C22semidirect
44. C2×#466C22semidirect
45. C2×C4×D8C22semidirect
47. C2×#427C22semidirect
48. #2255C22semidirect
49. #2252C22semidirect
50. #2243C22semidirect
51. #2240C22semidirect
52. #2237C22semidirect
53. #2236C22semidirect
54. #2235C22semidirect
55. #2234C22semidirect
56. #2234C22semidirect
57. #2234C22semidirect
58. #2234C22semidirect
60. #2233C22semidirect
62. #2232C22semidirect
63. #2232C22semidirect
64. #2232C22semidirect
65. #2228C22semidirect
66. #2227C22semidirect
67. #2226C22semidirect
68. #2226C22semidirect
69. #2224C22semidirect
70. #2220C22semidirect
71. #2218C22semidirect
72. #2215C22semidirect
73. #2215C22semidirect
74. #2210C22semidirect
75. #2206C22semidirect
76. #2187C22semidirect
77. #2186C22semidirect
79. #2184C22semidirect
80. #2184C22semidirect
81. #2177C22semidirect
82. C4×SD32C22semidirect
83. C4×D16C22semidirect
84. #2169C22semidirect
85. #2168C22semidirect
86. D4×C16C22semidirect
87. #2163C22semidirect
88. #2162C22semidirect
89. #2159C22semidirect
90. #2158C22semidirect
91. #2140C22semidirect
92. #2138C22semidirect
93. #2112C22semidirect
94. C2×C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#622512
C2×#33412
#339242
#339142
#339082
#338922
#338862
#338722
#338042
#337922
#337602
#337522
#336692
#336652
#336482
#336452
#334882