Finite Group Databasebeta

← #6365525

Subgroups — #6365525

Order 512.

1739 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61913C212yesmaximal, complemented
2#61582C212yesmaximal, complemented
3#20310C212yesmaximal, complemented
4#20289C212yesmaximal, complemented
5#20266C214yesmaximal, complemented
6#20203C214yesmaximal, complemented
7#20152C212yesmaximal, complemented
8#20084C212yesmaximal, complemented
9#19727C211yesmaximal, characteristic, complemented
10#19522C212yesmaximal, complemented
11#19425C212yesmaximal, complemented
12#18718C211yesmaximal, characteristic, complemented
13#18614C212yesmaximal, complemented
14#18524C211yesmaximal, characteristic, complemented
15#17952C212yesmaximal, complemented
16#3550C2211yescharacteristic, complemented
17#3506C2212yes—
18C2×D42C2211yescharacteristic
19#3401—24no—
20#3399C2214yescomplemented
21#3394C2214yescomplemented
22#3373—24no—
23#3369C2214yescomplemented
24#3362C2214yescomplemented
25#3325—24no—
26#3324C2214yes—
27#3324C2214yes—
28#3274C2214yes—
29#3274C2214yes—
30#3272—24no—
31#3224—24no—
32#3216C2212yescomplemented
33#3208C2212yes—
34#3207C2212yescomplemented
35#3205C2212yes—
36#3194—24no—
37#3184C2211yescharacteristic, complemented
38#3151C2211yescharacteristic
39#3125—24no—
40#3109C2212yes—
41#3101C2212yes—
42#3100C2212yes—
43#3003—24no—
44#2985C2212yescomplemented
45#2975C2212yes—
46#1721C2212yescomplemented
47#1721C2211yescharacteristic, complemented
48#1717C2212yescomplemented
49#1715C2211yescharacteristic, complemented
50#1712C2211yescharacteristic, complemented
51#1708C2211yescharacteristic, complemented
52#1703C2214yescomplemented
53#1702C2214yescomplemented
54#1697C2214yescomplemented
55#1688C2214yescomplemented
56#1687C2214yescomplemented
57#1685C2214yescomplemented
58#1682C2214yescomplemented
59#1668C2214yescomplemented
60#1652C2214yescomplemented
61#1641C2214yescomplemented
62#1629C2212yes—
63#1629C2212yes—
64#1628C2214yescomplemented
65#1623C2212yescomplemented
66#1623C2212yescomplemented
67#1621C2214yescomplemented
68#1620C2212yescomplemented
69#1620C2212yescomplemented
70#1619C2212yes—
71#1619C2212yes—
72#1592C2214yescomplemented
73#1590C2214yescomplemented
74#1588C2214yescomplemented
75#1583C2214yescomplemented
76#1572C2212yescomplemented
77#1555C2211yescharacteristic
78#1553C2212yescomplemented
79#1543C2211yescharacteristic, complemented
80#1490C2212yes—
81#1485C2212yescomplemented
82#1453C2211yescharacteristic, complemented
83C23×D4—22no—
84C2×AGL(1,ℤ/8)—24no—
85C2×AGL(1,ℤ/8)—24no—
86C2×AGL(1,ℤ/8)—41no—
87#563—41no—
88#559—24no—
89#559C2312yescomplemented
90Q82C2312yescomplemented
91#556—22no—
92#556C2314yes—
93#555—22no—
94#553—42no—
95#550C2314yes—
96D4×Q8—22no—
97#547—22no—
98D42—24no—
99D42—41no—
100D42C2312yes—
101D42C2312yes—
102#542—24no—
103#538—42no—
104#536—41no—
105#535—22no—
106#533C2312yes—
107#532C2311yescharacteristic
108#532C2312yes—
109C2×#127C2311yescharacteristic
110#525—24no—
111C2×#121—22no—
112C2×#120—41no—
113#520—22no—
114C2×C4×D4C2312yes—
115#498C2311yescharacteristic, complemented
116#497—22no—
117#496—24no—
118#496—41no—
119#495—24no—
120#493C2312yes—
121#493C2312yes—
122#493C2311yescharacteristic
123#493C2311yescharacteristic
124#493C2312yescomplemented
125#493C2311yescharacteristic, complemented
126#492—22no—
127#492C2311yescharacteristic
128#492C2311yescharacteristic
129#492C2312yescomplemented
130#489—22no—
131#489—24no—
132#487C2312yescomplemented
133#486—22no—
134#486—24no—
135#486C2312yes—
136#486C2312yes—
137#486C2312yescomplemented
138#486C2312yescomplemented
139#483D414yes—
140#482—24no—
141#482—24no—
142#481—22no—
143#480—22no—
144#476—24no—
145#476D414yes—
146#476C2314yes—
147#475—24no—
148#474C2314yes—
149#469—24no—
150#469—24no—
151#469—41no—
152#468—24no—
153#468—24no—
154#464—24no—
155#464—24no—
156#463—24no—
157#463—24no—
158#463—24no—
159#462C2314yescomplemented
160#461D414yes—
161#460—24no—
162#460C2314yes—
163#460C2314yes—
164#460C2314yescomplemented
165#459—24no—
166#459—24no—
167#459C2314yes—
168#459C2314yes—
169#452—24no—
170#450—24no—
171#450—24no—
172#450—24no—
173#450D414yes—
174#449—24no—
175#449—24no—
176#449—24no—
177#447—22no—
178#447—22no—
179#447—24no—
180#447—24no—
181#447—24no—
182#446—24no—
183#446C2314yescomplemented
184C8×Q8C2314yescomplemented
185#442—24no—
186#442—24no—
187#442—41no—
188#441—24no—
189C4×SD16—24no—
190C4×SD16—24no—
191C4×SD16C2314yes—
192C4×SD16C2314yes—
193C4×D8—24no—
194#436—24no—
195#436C2314yes—
196#436C2314yes—
197#433—24no—
198#432C2312yes—
199#431C2312yes—
200#431C2312yes—
201#418—24no—
202#418—24no—
203#407—24no—
204C4×M16C2311yescharacteristic
205#335C2314yescomplemented
206#334C2314yescomplemented
207#333C2312yescomplemented
208#332C2314yescomplemented
209#331C2312yes—
210#331C2312yes—
211#331C2312yescomplemented
212#331C2314yescomplemented
213Q8⋊C8C2314yescomplemented
214Q8⋊C8C2314yescomplemented
215D4⋊C8C2314yes—
216D4⋊C8C2314yes—
217#322C2312yescomplemented
218#322C2311yescharacteristic, complemented
219C82C2311yescharacteristic, complemented
220C25—41no—
221C2×#41—24no—
222C2×#41—22no—
223C2×#41—21no—
224C22×Q8—41no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—44no—
230C22×D4—42no—
231C22×D4—88no—
232C22×D4C2×D412yes—
233C23×C4—22no—
234AGL(1,ℤ/8)—84no—
235AGL(1,ℤ/8)—88no—
236AGL(1,ℤ/8)—88no—
237C2×Q16—24no—
238C2×SD16—24no—
239C2×SD16—24no—
240C2×SD16—24no—
241C2×SD16—24no—
242C2×SD16—41no—
243C2×SD16—41no—
244C2×SD16C2×D414yes—
245C2×SD16C2×D414yes—
246C2×D8—24no—
247C2×D8—24no—
248C2×D8—24no—
249C2×D8—24no—
250C2×D8—24no—
251C2×D8—41no—
252C2×D8—41no—
253C2×D8—41no—
254C2×M16—21no—
255C2×M16—24no—
256C2×M16—24no—
257C4⋊Q8—24no—
258C4⋊Q8—42no—
259C4⋊Q8C2×D412yes—
260C4⋊Q8C2×D412yes—
261C4⋊Q8C2412yes—
262C4⋊Q8C2411yescharacteristic
263#127—24no—
264#127—22no—
265#127—21no—
266#127C2411yescharacteristic
267#127C2411yescharacteristic
268#127C2412yes—
269#127C2411yescharacteristic
270#126—44no—
271#126—44no—
272#125—22no—
273#124—24no—
274#124—24no—
275#124—22no—
276#124—21no—
277#124—42no—
278#124—44no—
279#123—42no—
280#123—44no—
281#123—42no—
282#123—44no—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285C22⋊Q8—44no—
286C22⋊Q8—42no—
287C22⋊Q8C2×D414yes—
288#121—22no—
289#121—22no—
290#121—22no—
291#121—24no—
292#121—24no—
293#121—24no—
294#121—44no—
295#121—44no—
296#121—42no—
297#121—42no—
298#120—42no—
299#120—42no—
300#120—42no—
301#120—42no—
302#120—44no—
303C4×Q8—22no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8C2414yes—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—44no—
315C4×D4—44no—
316C4×D4C2×D414yes—
317C4×D4C2414yes—
318#117—22no—
319#117—24no—
320#117—42no—
321#117—41no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4C2×D412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—41no—
326C2×C42C2411yescharacteristic
327#107—24no—
328#106—24no—
329#106—41no—
330#106C2×D414yes—
331#106C2×D414yes—
332C4⋊C8—24no—
333C4⋊C8—24no—
334C4⋊C8C2×D414yes—
335C4⋊C8C2414yes—
336C4⋊C8C2414yes—
337#103—24no—
338#103—24no—
339#103C2×D414yes—
340#103C2×D414yes—
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#102—24no—
351#102—42no—
352#102C2×D414yes—
353#102C2×D414yes—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8C2×D414yes—
359#97—24no—
360#97—22no—
361#97—21no—
362C4×C8—24no—
363C4×C8—22no—
364C4×C8C2411yescharacteristic
365C4×C8C2411yescharacteristic
366C4×C8C2412yes—
367C4×C8C2412yes—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—41no—
372C24—88no—
373C24—162no—
374#41—84no—
375#41—82no—
376#41—84no—
377#41—84no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—42no—
382C2×Q8—162no—
383C2×Q8C22×D414yes—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—41no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—82no—
409C2×D4—816no—
410C2×D4—88no—
411C2×D4C22×D414yes—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—84no—
420C22×C4C22×D412yes—
421C22×C4H(5,𝔽2)11yescharacteristic
422Q16—84no—
423SD16—44no—
424SD16—44no—
425SD16—84no—
426SD16—82no—
427SD16—84no—
428SD16—84no—
429SD16—84no—
430SD16—82no—
431D8—48no—
432D8—84no—
433D8—84no—
434D8—84no—
435D8—82no—
436D8—82no—
437D8—84no—
438D8—82no—
439M16—42no—
440M16—84no—
441M16—84no—
442C2×C8—21no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—21no—
446C2×C8C22×D414yes—
447C2×C8C22×D414yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4C22×D414yes—
460C4⋊C4C22×D414yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C42—24no—
475C42—44no—
476C42—42no—
477C42C22×D412yes—
478C42H(5,𝔽2)11yescharacteristic
479C42C2511yescharacteristic
480C23—24no—
481C23—44no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—82no—
487C23—82no—
488C23—84no—
489C23—82no—
490C23—88no—
491C23—88no—
492C23—162no—
493C23—162no—
494C23—1616no—
495C23D4211yescharacteristic
496Q8—24no—
497Q8—24no—
498Q8—44no—
499Q8—82no—
500Q8—81no—
501Q8—162no—
502D4—24no—
503D4—24no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—44no—
510D4—81no—
511D4—81no—
512D4—81no—
513D4—82no—
514D4—84no—
515D4—88no—
516D4—88no—
517D4—84no—
518D4—161no—
519D4—164no—
520D4—164no—
521D4—161no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—162no—
536C2×C4—164no—
537C2×C4D4211yescharacteristic
538C2×C4D4212yes—
539C2×C4C23×D412yes—
540C2×C4C2×H(5,𝔽2)11yescharacteristic
541C8—22no—
542C8—24no—
543C8—24no—
544C8—41no—
545C8—44no—
546C8—44no—
547C22—22no—
548C22—21no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—88no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—81no—
559C22—82no—
560C22—81no—
561C22—81no—
562C22—164no—
563C22—168no—
564C22—168no—
565C22C2×D4211yescharacteristic
566C4—22no—
567C4—21no—
568C4—21no—
569C4—44no—
570C4—44no—
571C4—44no—
572C4—82no—
573C4—81no—
574C4#358514yes—
575C2—41no—
576C2—84no—
577C2—162no—
578C2—161no—
579C2—161no—
580C2—161no—
581C2#6105912yes—
582C2#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. #61913C2semidirect
2. #61582C2semidirect
3. #20310C2semidirect
4. #20289C2semidirect
5. #20266C2semidirect
6. #20203C2semidirect
7. #20152C2semidirect
8. #20084C2semidirect
9. #19727C2semidirect
10. #19522C2semidirect
11. #19425C2semidirect
12. #18718C2semidirect
13. #18614C2semidirect
14. #18524C2semidirect
15. #17952C2semidirect
16. #3550C22semidirect
20. #3399C22semidirect
21. #3394C22semidirect
23. #3369C22semidirect
24. #3362C22semidirect
32. #3216C22semidirect
34. #3207C22semidirect
37. #3184C22semidirect
44. #2985C22semidirect
46. #1721C22semidirect
47. #1721C22semidirect
48. #1717C22semidirect
49. #1715C22semidirect
50. #1712C22semidirect
51. #1708C22semidirect
52. #1703C22semidirect
53. #1702C22semidirect
54. #1697C22semidirect
55. #1688C22semidirect
56. #1687C22semidirect
57. #1685C22semidirect
58. #1682C22semidirect
59. #1668C22semidirect
60. #1652C22semidirect
61. #1641C22semidirect
64. #1628C22semidirect
65. #1623C22semidirect
66. #1623C22semidirect
67. #1621C22semidirect
68. #1620C22semidirect
69. #1620C22semidirect
72. #1592C22semidirect
73. #1590C22semidirect
74. #1588C22semidirect
75. #1583C22semidirect
76. #1572C22semidirect
78. #1553C22semidirect
79. #1543C22semidirect
81. #1485C22semidirect
82. #1453C22semidirect
89. #559C23semidirect
90. Q82C23semidirect
115. #498C23semidirect
124. #493C23semidirect
125. #493C23semidirect
129. #492C23semidirect
132. #487C23semidirect
137. #486C23semidirect
138. #486C23semidirect
159. #462C23semidirect
164. #460C23semidirect
183. #446C23semidirect
184. C8×Q8C23semidirect
205. #335C23semidirect
206. #334C23semidirect
207. #333C23semidirect
208. #332C23semidirect
211. #331C23semidirect
212. #331C23semidirect
213. Q8⋊C8C23semidirect
214. Q8⋊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
#619132
#615822
#203102
#202892
#202662
#202032
#201522
#200842
#197272
#195222
#194252
#187182
#186142
#185242
#179522