Finite Group Databasebeta

← #6365547

Subgroups — #6365547

Order 512.

1757 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61802C212yesmaximal, complemented
2#61582C212yesmaximal, complemented
3#20314C212yesmaximal, complemented
4#20302C212yesmaximal, complemented
5#20243C214yesmaximal, complemented
6#20159C214yesmaximal, complemented
7#20131C212yesmaximal, complemented
8#20023C212yesmaximal, complemented
9#19713C211yesmaximal, characteristic, complemented
10#19504C212yesmaximal, complemented
11#19430C212yesmaximal, complemented
12#18714C211yesmaximal, characteristic, complemented
13#18617C212yesmaximal, complemented
14#18524C211yesmaximal, characteristic, complemented
15#17950C212yesmaximal, complemented
16#3549C2211yescharacteristic, complemented
17#3506C2212yes—
18C2×D42C2211yescharacteristic, complemented
19#3354C2214yescomplemented
20#3347—24no—
21#3341C2214yescomplemented
22#3325C2214yescomplemented
23#3325C2214yescomplemented
24#3325C2214yescomplemented
25#3324—24no—
26#3318—24no—
27#3294C2214yescomplemented
28#3274—24no—
29#3272C2214yescomplemented
30#3272C2214yescomplemented
31#3235C2212yescomplemented
32#3228C2212yescomplemented
33#3208C2212yescomplemented
34#3205C2212yes—
35#3199—24no—
36#3194—24no—
37#3185C2211yescharacteristic, complemented
38#3151C2211yescharacteristic
39#3109C2212yes—
40#3101C2212yescomplemented
41#3100C2212yes—
42#3096—24no—
43#3003—24no—
44#2981C2212yescomplemented
45#2975C2212yescomplemented
46#1722C2211yescharacteristic, complemented
47#1717C2212yescomplemented
48#1715C2211yescharacteristic, complemented
49#1714C2212yescomplemented
50#1709C2211yescharacteristic, complemented
51#1705C2211yescharacteristic, complemented
52#1701C2214yescomplemented
53#1697C2214yescomplemented
54#1697C2214yescomplemented
55#1695C2214yescomplemented
56#1691C2214yescomplemented
57#1685C2214yescomplemented
58#1677C2214yescomplemented
59#1665C2214yescomplemented
60#1654C2214yescomplemented
61#1646C2214yescomplemented
62#1635C2214yescomplemented
63#1634C2214yescomplemented
64#1629C2212yescomplemented
65#1629C2212yescomplemented
66#1624C2212yescomplemented
67#1624C2212yescomplemented
68#1619C2212yes—
69#1619C2212yes—
70#1619C2212yescomplemented
71#1619C2212yescomplemented
72#1594C2214yescomplemented
73#1589C2214yescomplemented
74#1586C2214yescomplemented
75#1581C2214yescomplemented
76#1572C2212yescomplemented
77#1555C2211yescharacteristic, complemented
78#1554C2212yescomplemented
79#1543C2211yescharacteristic, complemented
80#1490C2212yescomplemented
81#1484C2212yescomplemented
82#1453C2211yescharacteristic, complemented
83C23×D4—22no—
84C2×AGL(1,ℤ/8)—24no—
85C2×AGL(1,ℤ/8)—24no—
86C2×AGL(1,ℤ/8)—41no—
87H(3,𝔽4)—41no—
88#559—42no—
89#556—22no—
90#555—22no—
91#555C2314yes—
92#555C2312yescomplemented
93#553—24no—
94#550—22no—
95#547C2314yes—
96D42—22no—
97D42—24no—
98D42—41no—
99D42C2312yes—
100D42C2312yes—
101D42C2312yescomplemented
102#538—42no—
103#535—22no—
104#535—24no—
105#535—41no—
106#533C2312yes—
107#532C2311yescharacteristic
108#532C2312yes—
109C2×#127C2311yescharacteristic
110#525—24no—
111C2×#121—22no—
112C2×#120—41no—
113#520—22no—
114C2×C4×D4C2312yes—
115#501—22no—
116#500C2312yescomplemented
117#498C2312yescomplemented
118#498C2311yescharacteristic, complemented
119#496—41no—
120#494C2311yescharacteristic, complemented
121#493—22no—
122#493C2311yescharacteristic
123#493C2311yescharacteristic
124#492C2312yes—
125#492C2312yes—
126#492C2311yescharacteristic
127#492C2311yescharacteristic
128#490—22no—
129#487C2312yescomplemented
130#486—22no—
131#486C2312yes—
132#486C2312yes—
133#486C2312yescomplemented
134#486C2312yescomplemented
135#483—22no—
136#482—24no—
137#482—24no—
138#482—24no—
139#482—24no—
140#482—24no—
141#482—24no—
142#482—24no—
143#482—24no—
144#481—22no—
145#481—24no—
146#480—24no—
147#480D414yes—
148#478—24no—
149#476C2314yes—
150#476C2314yes—
151#476C2314yescomplemented
152#474D414yes—
153#474C2314yescomplemented
154#470—41no—
155#469—24no—
156#469—24no—
157#469—24no—
158#469—24no—
159#468—24no—
160#468—24no—
161#466—24no—
162#466—24no—
163#461C2314yes—
164#460—24no—
165#459D414yes—
166#459C2314yes—
167#459C2314yes—
168#459C2314yes—
169#450—22no—
170#450—24no—
171#450—24no—
172#450—24no—
173#449—24no—
174#449—24no—
175#447—22no—
176#447—24no—
177#447—24no—
178#447—24no—
179#447—24no—
180#447—24no—
181#447D414yes—
182#442—24no—
183#441—41no—
184C4×SD16—24no—
185C4×D8C2314yes—
186C4×D8C2314yes—
187#436—24no—
188#436C2314yes—
189#436C2314yes—
190#436C2314yescomplemented
191#435—24no—
192C8×D4C2314yescomplemented
193#432C2312yes—
194#431C2312yes—
195#431C2312yes—
196#428—24no—
197#427—24no—
198#418—24no—
199#418—24no—
200#418—24no—
201C2×#102—24no—
202#408—24no—
203#407—24no—
204C4×M16C2311yescharacteristic
205#335C2314yescomplemented
206#334C2314yescomplemented
207#333C2312yescomplemented
208#333C2314yescomplemented
209#332C2314yescomplemented
210#331C2312yes—
211#331C2312yes—
212#331C2312yescomplemented
213D4⋊C8C2314yes—
214D4⋊C8C2314yes—
215D4⋊C8C2314yescomplemented
216D4⋊C8C2314yescomplemented
217#322C2312yescomplemented
218#322C2311yescharacteristic, complemented
219C82C2311yescharacteristic, complemented
220C25—41no—
221C2×#41—22no—
222C2×#41—24no—
223C2×#41—21no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—44no—
230C22×D4—42no—
231C22×D4—41no—
232C22×D4—88no—
233C22×D4C2×D412yes—
234C23×C4—22no—
235AGL(1,ℤ/8)—84no—
236AGL(1,ℤ/8)—88no—
237AGL(1,ℤ/8)—88no—
238C2×Q16—41no—
239C2×SD16—24no—
240C2×SD16—24no—
241C2×SD16—24no—
242C2×SD16—41no—
243C2×SD16—41no—
244C2×D8—24no—
245C2×D8—24no—
246C2×D8—24no—
247C2×D8—41no—
248C2×D8—41no—
249C2×D8C2×D414yes—
250C2×D8C2×D414yes—
251C2×M16—21no—
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—24no—
255C22×C8—24no—
256C4⋊Q8C2×D412yes—
257C4⋊Q8C2411yescharacteristic
258C4⋊Q8C2412yes—
259C4⋊Q8C2412yes—
260#127—22no—
261#127—21no—
262#127—42no—
263#127—42no—
264#127C2×D412yes—
265#127C2411yescharacteristic
266#127C2411yescharacteristic
267#127C2411yescharacteristic
268#126—44no—
269#125—22no—
270#124—22no—
271#124—21no—
272#124—42no—
273#124—44no—
274#124—44no—
275#124—42no—
276#123—44no—
277#123—44no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—24no—
281#121—22no—
282#121—24no—
283#121—24no—
284#121—24no—
285#121—24no—
286#121—22no—
287#121—24no—
288#121—24no—
289#121—24no—
290#121—44no—
291#121—44no—
292#121—42no—
293#121C2×D414yes—
294#120—42no—
295#120—42no—
296#120—42no—
297#120—42no—
298#120—42no—
299#120—44no—
300#120—42no—
301#120—44no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—42no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—44no—
310C4×D4—42no—
311C4×D4—44no—
312C4×D4C2×D414yes—
313C4×D4C2414yes—
314C4×D4C2414yes—
315#117—22no—
316#117—24no—
317#117—24no—
318#117—24no—
319#117—41no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4C2×D412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—41no—
324C2×C42C2411yescharacteristic
325#107—24no—
326#107—24no—
327#107C2×D414yes—
328#107C2×D414yes—
329#106—24no—
330#106—24no—
331#106—24no—
332#106—24no—
333#106—41no—
334C4⋊C8C2×D414yes—
335C4⋊C8C2414yes—
336C4⋊C8C2414yes—
337#103—24no—
338#103—42no—
339#102—24no—
340#102—24no—
341#102—24no—
342#102—24no—
343#102—24no—
344#102—24no—
345#102—24no—
346#102—24no—
347#102—24no—
348#102—24no—
349#102—24no—
350#102C2×D414yes—
351#102C2×D414yes—
352#102C2×D414yes—
353#102C2×D414yes—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8C2×D414yes—
361#97—22no—
362#97—21no—
363C4×C8—22no—
364C4×C8C2411yescharacteristic
365C4×C8C2411yescharacteristic
366C4×C8C2412yes—
367C4×C8C2412yes—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—41no—
373C24—88no—
374C24—162no—
375#41—84no—
376#41—82no—
377#41—84no—
378#41—84no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—41no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—82no—
411C2×D4—816no—
412C2×D4—88no—
413C2×D4—162no—
414C2×D4C22×D414yes—
415C2×D4C22×D414yes—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—41no—
422C22×C4—84no—
423C22×C4C22×D412yes—
424C22×C4H(5,𝔽2)11yescharacteristic
425Q16—82no—
426SD16—48no—
427SD16—84no—
428SD16—84no—
429SD16—82no—
430SD16—82no—
431D8—44no—
432D8—44no—
433D8—82no—
434D8—84no—
435D8—84no—
436D8—84no—
437D8—82no—
438M16—42no—
439M16—84no—
440M16—84no—
441M16—84no—
442C2×C8—21no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—21no—
446C2×C8—84no—
447C2×C8C22×D414yes—
448C2×C8C22×D414yes—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4C22×D414yes—
460C4⋊C4C22×D414yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C42—42no—
475C42—44no—
476C42—42no—
477C42C22×D412yes—
478C42H(5,𝔽2)11yescharacteristic
479C42C2511yescharacteristic
480C23—24no—
481C23—24no—
482C23—44no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—82no—
487C23—82no—
488C23—84no—
489C23—82no—
490C23—88no—
491C23—88no—
492C23—162no—
493C23—162no—
494C23—164no—
495C23—162no—
496C23—1616no—
497C23D4211yescharacteristic
498Q8—44no—
499Q8—82no—
500Q8—81no—
501Q8—82no—
502D4—24no—
503D4—24no—
504D4—24no—
505D4—24no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—44no—
510D4—82no—
511D4—82no—
512D4—81no—
513D4—81no—
514D4—81no—
515D4—84no—
516D4—84no—
517D4—88no—
518D4—88no—
519D4—161no—
520D4—164no—
521D4—161no—
522D4—164no—
523D4—162no—
524D4—164no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—164no—
540C2×C4D4212yes—
541C2×C4D4211yescharacteristic
542C2×C4C23×D412yes—
543C2×C4C2×H(5,𝔽2)11yescharacteristic
544C8—22no—
545C8—24no—
546C8—24no—
547C8—41no—
548C8—44no—
549C8—44no—
550C22—22no—
551C22—21no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—88no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—164no—
566C22—168no—
567C22—168no—
568C22—324no—
569C22C2×D4211yescharacteristic
570C4—22no—
571C4—21no—
572C4—21no—
573C4—44no—
574C4—44no—
575C4—82no—
576C4—81no—
577C4—82no—
578C4#358514yes—
579C2—41no—
580C2—84no—
581C2—84no—
582C2—161no—
583C2—161no—
584C2—161no—
585C2#6105812yes—
586C2#6111011yescharacteristic
subgroupG/H
center, socleC22C2×D42
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61802C2semidirect
2. #61582C2semidirect
3. #20314C2semidirect
4. #20302C2semidirect
5. #20243C2semidirect
6. #20159C2semidirect
7. #20131C2semidirect
8. #20023C2semidirect
9. #19713C2semidirect
10. #19504C2semidirect
11. #19430C2semidirect
12. #18714C2semidirect
13. #18617C2semidirect
14. #18524C2semidirect
15. #17950C2semidirect
16. #3549C22semidirect
18. C2×D42C22semidirect
19. #3354C22semidirect
21. #3341C22semidirect
22. #3325C22semidirect
23. #3325C22semidirect
24. #3325C22semidirect
27. #3294C22semidirect
29. #3272C22semidirect
30. #3272C22semidirect
31. #3235C22semidirect
32. #3228C22semidirect
33. #3208C22semidirect
37. #3185C22semidirect
40. #3101C22semidirect
44. #2981C22semidirect
45. #2975C22semidirect
46. #1722C22semidirect
47. #1717C22semidirect
48. #1715C22semidirect
49. #1714C22semidirect
50. #1709C22semidirect
51. #1705C22semidirect
52. #1701C22semidirect
53. #1697C22semidirect
54. #1697C22semidirect
55. #1695C22semidirect
56. #1691C22semidirect
57. #1685C22semidirect
58. #1677C22semidirect
59. #1665C22semidirect
60. #1654C22semidirect
61. #1646C22semidirect
62. #1635C22semidirect
63. #1634C22semidirect
64. #1629C22semidirect
65. #1629C22semidirect
66. #1624C22semidirect
67. #1624C22semidirect
70. #1619C22semidirect
71. #1619C22semidirect
72. #1594C22semidirect
73. #1589C22semidirect
74. #1586C22semidirect
75. #1581C22semidirect
76. #1572C22semidirect
77. #1555C22semidirect
78. #1554C22semidirect
79. #1543C22semidirect
80. #1490C22semidirect
81. #1484C22semidirect
82. #1453C22semidirect
92. #555C23semidirect
101. D42C23semidirect
116. #500C23semidirect
117. #498C23semidirect
118. #498C23semidirect
120. #494C23semidirect
129. #487C23semidirect
133. #486C23semidirect
134. #486C23semidirect
151. #476C23semidirect
153. #474C23semidirect
190. #436C23semidirect
192. C8×D4C23semidirect
205. #335C23semidirect
206. #334C23semidirect
207. #333C23semidirect
208. #333C23semidirect
209. #332C23semidirect
212. #331C23semidirect
215. D4⋊C8C23semidirect
216. D4⋊C8C23semidirect
217. #322C23semidirect
218. #322C23semidirect
219. C82C23semidirect

Maximal subgroups

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

subgroupindex
#618022
#615822
#203142
#203022
#202432
#201592
#201312
#200232
#197132
#195042
#194302
#187142
#186172
#185242
#179502