Finite Group Databasebeta

← #6357470

Subgroups — #6357470

Order 512.

2276 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60968C211yesmaximal, characteristic, complemented
2#60798C211yesmaximal, characteristic, complemented
3#60782C212yesmaximal, complemented
4C2×#3205C212yesmaximal, complemented
5#19729C214yesmaximal, complemented
6#19727C214yesmaximal, complemented
7#19701C214yesmaximal, complemented
8#19698C214yesmaximal, complemented
9#19661C214yesmaximal, complemented
10#19656C214yesmaximal, complemented
11#17325C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13C2×#532C2212yes—
14C22×#127C2211yescharacteristic
15#3253—22no—
16#3227C2214yescomplemented
17#3217—24no—
18#3212—24no—
19#3211—24no—
20#3210C2214yescomplemented
21#3209C2212yescomplemented
22#3208C2214yescomplemented
23#3208C2212yescomplemented
24#3207C2214yescomplemented
25#3206C2212yescomplemented
26#3205—24no—
27#3205—24no—
28#3205C2214yes—
29#3205C2214yes—
30#3205C2218yescomplemented
31#3205C2214yescomplemented
32#3205C2214yescomplemented
33#3205C2212yescomplemented
34#3192—22no—
35#3184C2214yescomplemented
36#3151C2212yescomplemented
37#3147C2214yescomplemented
38C2×#493C2214yescomplemented
39#3138C2212yescomplemented
40#3131C2214yescomplemented
41C2×#486C2214yescomplemented
42#3109—24no—
43#3038—22no—
44#3035—24no—
45C2×#459—24no—
46#3003—22no—
47#3002—24no—
48C2×#447—24no—
49#2920C2211yescharacteristic
50C2×#431C2212yes—
51#1721C2218yescomplemented
52#1721C2218yescomplemented
53#1721C2214yescomplemented
54#1720C2218yescomplemented
55#1717C2218yescomplemented
56#1717C2218yescomplemented
57#1715C2214yescomplemented
58#1712C2214yescomplemented
59#1709C2218yescomplemented
60#1708C2214yescomplemented
61#1453C2214yescomplemented
62#1452C2214yescomplemented
63#1450C2214yescomplemented
64C22×#41—22no—
65C23×D4—22no—
66C23×D4—21no—
67C2×AGL(1,ℤ/8)—24no—
68C2×#135—28no—
69C22×D8—28no—
70#533C2312yes—
71#533C2311yescharacteristic
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532—22no—
78#532C2314yes—
79#532C2312yes—
80#532C2311yescharacteristic
81#532C2314yes—
82C2×C4⋊Q8C2312yes—
83C2×#127—22no—
84C2×#127—22no—
85C2×#127—22no—
86C2×#127—22no—
87C2×#127—21no—
88C2×#127C2314yes—
89C2×#127C2312yes—
90C2×#127C2312yes—
91#529—22no—
92#529—22no—
93#529—22no—
94C2×#124—22no—
95#525—22no—
96#525—21no—
97C2×#121—22no—
98C2×#121—22no—
99C4×#41—22no—
100C4×#41—21no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C22×C42C2311yescharacteristic
104#498C2314yescomplemented
105#496—24no—
106#495—28no—
107#494C2314yescomplemented
108#493—28no—
109#493C2318yes—
110#493C2318yes—
111#493C2314yes—
112#493C2314yescomplemented
113#493C2314yescomplemented
114#492C2314yes—
115#492C2314yescomplemented
116#492C2314yescomplemented
117#490—24no—
118#488—28no—
119#487C2314yescomplemented
120#487C2314yescomplemented
121#486—28no—
122#486C2318yes—
123#486C2318yes—
124#486C2318yes—
125#486C2314yescomplemented
126#486C2314yescomplemented
127#486C2318yescomplemented
128#481—28no—
129#476—28no—
130#469—28no—
131#464—28no—
132#463—28no—
133#462—24no—
134#461—24no—
135#460—24no—
136#460—28no—
137#459—28no—
138#459—28no—
139#459—28no—
140#459—216no—
141#459—24no—
142#459—28no—
143#452—28no—
144#451—24no—
145#450—24no—
146#450—28no—
147#449—28no—
148#448—24no—
149#447—28no—
150#447—28no—
151#447—28no—
152#447—216no—
153#447—24no—
154#447—28no—
155#442—28no—
156#432D414yes—
157#431D414yes—
158#431D414yes—
159#431C2312yes—
160#431C2312yes—
161#431C2314yes—
162#431C2314yes—
163#423D412yes—
164C2×C4⋊C8D414yes—
165#418—24no—
166#416—28no—
167C2×#102—28no—
168#407D412yes—
169C2×C22⋊C8D414yes—
170C4×M16C2312yes—
171C2×C4×C8C2314yes—
172#322C2318yescomplemented
173#322C2314yescomplemented
174C82C2314yescomplemented
175C25—41no—
176C2×#41—22no—
177C2×#41—21no—
178C2×#41—24no—
179C2×#41—24no—
180C2×#41—42no—
181C2×#41—42no—
182C2×#41—88no—
183C22×Q8—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—41no—
198C22×D4—88no—
199C22×D4—84no—
200C23×C4—42no—
201C23×C4C2×D412yes—
202C23×C4C2×D411yescharacteristic
203AGL(1,ℤ/8)—88no—
204#135—816no—
205C2×Q16—28no—
206C2×SD16—28no—
207C2×SD16—28no—
208C2×SD16—28no—
209C2×D8—216no—
210C2×D8—216no—
211C2×D8—28no—
212C2×D8—28no—
213C2×D8—28no—
214C2×D8—816no—
215C2×M16C2×D414yes—
216C22×C8C2×D418yes—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8C2414yes—
221C4⋊Q8C2412yes—
222C4⋊Q8C2411yescharacteristic
223#127—24no—
224#127—24no—
225#127—24no—
226#127—24no—
227#127—22no—
228#127—22no—
229#127—22no—
230#127—22no—
231#127—41no—
232#127—42no—
233#127—41no—
234#127—42no—
235#127C2414yes—
236#127C2414yes—
237#127C2411yescharacteristic
238#126—42no—
239#125—22no—
240#125—21no—
241#124—24no—
242#124—24no—
243#124—22no—
244#124—21no—
245#124—42no—
246#124—42no—
247#123—42no—
248#123—41no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252#121—24no—
253#121—24no—
254#121—24no—
255#121—24no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—42no—
260#121—42no—
261#121—42no—
262#121—42no—
263#120—41no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—42no—
279C4×D4—42no—
280C4×D4—42no—
281#117—22no—
282#117—21no—
283#117—42no—
284#117—41no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C42—41no—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2412yes—
296C2×C42C2411yescharacteristic
297#106—28no—
298C4⋊C8C2×D414yes—
299C4⋊C8C2×D414yes—
300C4⋊C8C2×D418yes—
301C4⋊C8C2×D418yes—
302#103—28no—
303#103—28no—
304#102—216no—
305#102—216no—
306#102—28no—
307#102—28no—
308#102—216no—
309#102—216no—
310C22⋊C8C2×D414yes—
311C22⋊C8C2×D414yes—
312C22⋊C8C2×D418yes—
313C22⋊C8C2×D418yes—
314#97C2×D414yes—
315C4×C8C2×D418yes—
316C4×C8C2414yes—
317C4×C8C2414yes—
318C4×C8C2414yes—
319C24—42no—
320C24—42no—
321C24—44no—
322C24—88no—
323C24—84no—
324C24#12711yescharacteristic
325#41—44no—
326#41—48no—
327#41—416no—
328#41—88no—
329#41—88no—
330#41—88no—
331C2×Q8—24no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—42no—
335C2×Q8—84no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—28no—
339C2×D4—24no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—48no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—416no—
358C2×D4—84no—
359C2×D4—88no—
360C2×D4—88no—
361C2×D4—88no—
362C2×D4—84no—
363C2×D4—88no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—88no—
367C2×D4—84no—
368C2×D4—88no—
369C22×C4—28no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—88no—
377C22×C4#12712yes—
378C22×C4#12712yes—
379C22×C4#12712yes—
380C22×C4#12712yes—
381C22×C4#12711yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387Q16—88no—
388SD16—88no—
389SD16—88no—
390SD16—88no—
391D8—88no—
392D8—816no—
393D8—816no—
394D8—88no—
395D8—88no—
396M16—44no—
397C2×C8—216no—
398C2×C8C22×D418yes—
399C2×C8C22×D418yes—
400C2×C8C22×D418yes—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C22⋊C4—41no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C42—41no—
421C42—42no—
422C42#12711yescharacteristic
423C42#12712yes—
424C42#12711yescharacteristic
425C42#12712yes—
426C42C22×D412yes—
427C42C22×D411yescharacteristic
428C42C22×D412yes—
429C42C22×D412yes—
430C42C22×D412yes—
431C42C2511yescharacteristic
432C23—24no—
433C23—41no—
434C23—41no—
435C23—44no—
436C23—44no—
437C23—84no—
438C23—84no—
439C23—88no—
440C23—88no—
441C23—88no—
442C23—84no—
443C23—88no—
444C23—84no—
445C23—1616no—
446C23C2×#12711yescharacteristic
447C23C2×#12712yes—
448Q8—42no—
449Q8—44no—
450Q8—48no—
451Q8—84no—
452D4—42no—
453D4—48no—
454D4—48no—
455D4—48no—
456D4—44no—
457D4—416no—
458D4—48no—
459D4—84no—
460D4—84no—
461D4—88no—
462D4—88no—
463D4—88no—
464D4—84no—
465D4—88no—
466D4—82no—
467D4—88no—
468D4—84no—
469D4—84no—
470D4—88no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—88no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—88no—
487C2×C4—84no—
488C2×C4C2×#12711yescharacteristic
489C2×C4C2×#12712yes—
490C2×C4C2×#12712yes—
491C2×C4C2×#12712yes—
492C2×C4C2×#12712yes—
493C2×C4#57618yes—
494C2×C4C23×D411yescharacteristic
495C2×C4C23×D412yes—
496C8—28no—
497C8—28no—
498C8—28no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—21no—
503C22—44no—
504C22—84no—
505C22—82no—
506C22—88no—
507C22—84no—
508C22—81no—
509C22—84no—
510C22—84no—
511C22—168no—
512C22—1616no—
513C22—168no—
514C22#315514yes—
515C22C22×#12711yescharacteristic
516C4—22no—
517C4—22no—
518C4—21no—
519C4—22no—
520C4—84no—
521C4—82no—
522C4—81no—
523C4#315514yes—
524C4C2×#57614yes—
525C4#358512yes—
526C2—24no—
527C2—41no—
528C2—161no—
529C2—164no—
530C2—164no—
531C2C2×#315512yes—
532C2#6064411yescharacteristic
subgroupG/H
center, socleC22C22×#127
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60968C2semidirect
2. #60798C2semidirect
3. #60782C2semidirect
4. C2×#3205C2semidirect
5. #19729C2semidirect
6. #19727C2semidirect
7. #19701C2semidirect
8. #19698C2semidirect
9. #19661C2semidirect
10. #19656C2semidirect
11. #17325C2semidirect
16. #3227C22semidirect
20. #3210C22semidirect
21. #3209C22semidirect
22. #3208C22semidirect
23. #3208C22semidirect
24. #3207C22semidirect
25. #3206C22semidirect
30. #3205C22semidirect
31. #3205C22semidirect
32. #3205C22semidirect
33. #3205C22semidirect
35. #3184C22semidirect
36. #3151C22semidirect
37. #3147C22semidirect
38. C2×#493C22semidirect
39. #3138C22semidirect
40. #3131C22semidirect
41. C2×#486C22semidirect
51. #1721C22semidirect
52. #1721C22semidirect
53. #1721C22semidirect
54. #1720C22semidirect
55. #1717C22semidirect
56. #1717C22semidirect
57. #1715C22semidirect
58. #1712C22semidirect
59. #1709C22semidirect
60. #1708C22semidirect
61. #1453C22semidirect
62. #1452C22semidirect
63. #1450C22semidirect
104. #498C23semidirect
107. #494C23semidirect
112. #493C23semidirect
113. #493C23semidirect
115. #492C23semidirect
116. #492C23semidirect
119. #487C23semidirect
120. #487C23semidirect
125. #486C23semidirect
126. #486C23semidirect
127. #486C23semidirect
172. #322C23semidirect
173. #322C23semidirect
174. C82C23semidirect

Maximal subgroups

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

subgroupindex
#609682
#607982
#607822
C2×#32052
#197292
#197272
#197012
#196982
#196612
#196562
#173252