Finite Group Databasebeta

← #6357953

Subgroups — #6357953

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60968C211yesmaximal, characteristic, complemented
2#60965C212yesmaximal, complemented
3#60862C211yesmaximal, characteristic, complemented
4#60804C212yesmaximal, complemented
5#19750C214yesmaximal, complemented
6#19748C214yesmaximal, complemented
7#19741C214yesmaximal, complemented
8#19739C214yesmaximal, complemented
9#19737C214yesmaximal, complemented
10#19735C214yesmaximal, complemented
11#17339C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C2×#532C2212yes—
15#3253—22no—
16#3251—24no—
17#3240—24no—
18#3231C2214yescomplemented
19#3229C2212yescomplemented
20#3228C2214yes—
21#3228C2214yescomplemented
22#3227C2214yes—
23#3226C2212yescomplemented
24#3216—24no—
25#3215—24no—
26#3210C2214yes—
27#3210C2214yes—
28#3209C2214yescomplemented
29#3208C2214yescomplemented
30#3208C2212yescomplemented
31#3207C2214yes—
32#3205C2214yes—
33#3205C2212yescomplemented
34#3194—24no—
35#3192—22no—
36#3184C2214yes—
37#3184C2214yescomplemented
38#3182C2214yescomplemented
39#3165C2212yescomplemented
40#3152C2214yescomplemented
41#3141C2214yescomplemented
42#3141C2212yescomplemented
43#3115—24no—
44#3114—24no—
45#3109—24no—
46#3093—22no—
47#3080—22no—
48#3037—24no—
49#3007—24no—
50#2920C2212yes—
51#2920C2211yescharacteristic
52#1723C2214yescomplemented
53#1722C2214yescomplemented
54#1721C2214yescomplemented
55#1720C2214yescomplemented
56#1720C2214yescomplemented
57#1720C2214yescomplemented
58#1719C2214yescomplemented
59#1717C2214yescomplemented
60#1717C2214yescomplemented
61#1716C2214yescomplemented
62#1715C2214yescomplemented
63#1714C2214yescomplemented
64#1709C2214yescomplemented
65#1708C2214yescomplemented
66#1707C2214yescomplemented
67#1706C2214yescomplemented
68#1455C2214yescomplemented
69#1454C2214yescomplemented
70#1454C2214yescomplemented
71C22×#41—22no—
72C2×#137—28no—
73#533—22no—
74#533—22no—
75#533—22no—
76#533C2311yescharacteristic
77#533C2314yes—
78#533C2312yes—
79#532—22no—
80#532—22no—
81#532—22no—
82#532—22no—
83#532—22no—
84#532—22no—
85#532C2311yescharacteristic
86#532C2314yes—
87#532C2314yes—
88#532C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2312yes—
92#529—22no—
93#529—22no—
94#529—22no—
95C2×#125—21no—
96C2×#124—22no—
97#525—21no—
98#525—22no—
99C2×C22⋊Q8—22no—
100C2×#121—22no—
101C2×#121—22no—
102C4×#41—21no—
103C4×#41—22no—
104C2×C4×Q8—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×#117—21no—
108C22×C42C2311yescharacteristic
109#501—24no—
110#500C2314yes—
111#500C2314yes—
112#498C2314yes—
113#498C2314yes—
114#498C2314yes—
115#497—28no—
116#494C2314yes—
117#494C2314yes—
118#493C2314yes—
119#493C2314yes—
120#492C2314yes—
121#492C2314yes—
122#492C2314yes—
123#489—28no—
124#489—24no—
125#487C2314yes—
126#487C2314yes—
127#487C2314yes—
128#487C2314yes—
129#486C2314yes—
130#486C2314yes—
131#486C2314yes—
132#486C2314yes—
133#486C2314yes—
134#486C2314yes—
135#486C2314yes—
136#486C2314yes—
137#485—28no—
138#483—24no—
139#482—28no—
140#482—28no—
141#481—28no—
142#480—24no—
143#479—28no—
144#478—28no—
145#478—28no—
146#476—28no—
147#475—24no—
148#474—24no—
149#469—28no—
150#469—28no—
151#468—28no—
152#464—28no—
153#464—28no—
154#463—28no—
155#462—28no—
156#460—24no—
157#460—28no—
158#459—28no—
159#459—24no—
160#452—28no—
161#452—28no—
162#451—28no—
163#450—24no—
164#450—28no—
165#449—28no—
166#447—28no—
167#447—24no—
168#442—28no—
169#442—28no—
170#440—28no—
171#432D414yes—
172#432D414yes—
173#432D414yes—
174#431C2314yes—
175#431C2312yes—
176#431C2314yes—
177#431C2312yes—
178#428—24no—
179#423D412yes—
180#423D414yes—
181#417—24no—
182#417—28no—
183#407D412yes—
184#407D414yes—
185C4×M16C2314yes—
186C4×M16C2312yes—
187#322C2314yes—
188#322C2314yes—
189#322C2314yes—
190C82C2314yes—
191C2×#41—21no—
192C2×#41—24no—
193C2×#41—22no—
194C2×#41—24no—
195C2×#41—42no—
196C2×#41—42no—
197C2×#41—88no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—42no—
204C23×C4—42no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D412yes—
207#137—816no—
208C2×Q16—28no—
209C2×Q16—28no—
210C2×SD16—28no—
211C2×SD16—28no—
212C2×SD16—28no—
213C2×D8—28no—
214C2×D8—28no—
215C2×M16C2×D418yes—
216C2×M16C2×D414yes—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8C2414yes—
224C4⋊Q8C2414yes—
225C4⋊Q8C2411yescharacteristic
226C4⋊Q8C2412yes—
227#127—22no—
228#127—22no—
229#127—22no—
230#127C2414yes—
231#127C2411yescharacteristic
232#126—42no—
233#125—24no—
234#125—22no—
235#125—21no—
236#125—41no—
237#125—41no—
238#124—24no—
239#124—24no—
240#124—22no—
241#124—21no—
242#124—42no—
243#124—42no—
244#123—41no—
245#123—42no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—42no—
252C22⋊Q8—42no—
253#121—24no—
254#121—24no—
255#121—24no—
256#121—24no—
257#121—22no—
258#121—22no—
259#121—22no—
260#121—42no—
261#121—42no—
262#121—42no—
263#121—42no—
264#120—41no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—42no—
271C4×Q8—42no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—42no—
280C4×D4—42no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—42no—
286#117—24no—
287#117—21no—
288#117—22no—
289#117—42no—
290#117—41no—
291#117—42no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C42—41no—
301C2×C42—41no—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42C2412yes—
308C2×C42C2411yescharacteristic
309#107—28no—
310#107—28no—
311#106—28no—
312#106—28no—
313#106—28no—
314C4⋊C8C2×D414yes—
315C4⋊C8C2×D418yes—
316C4⋊C8C2×D418yes—
317C4⋊C8C2×D414yes—
318#103—28no—
319#103—28no—
320#103—28no—
321#103—28no—
322#102—28no—
323#102—28no—
324#102—28no—
325#102—28no—
326#102—28no—
327#102—28no—
328#102—28no—
329#102—28no—
330C22⋊C8C2×D414yes—
331C22⋊C8C2×D418yes—
332C22⋊C8C2×D418yes—
333C22⋊C8C2×D414yes—
334#97C2×D418yes—
335#97C2×D414yes—
336C4×C8C2414yes—
337C4×C8C2414yes—
338C4×C8C2414yes—
339C24—42no—
340C24#12711yescharacteristic
341#41—416no—
342#41—44no—
343#41—84no—
344#41—88no—
345#41—88no—
346#41—88no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—48no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—84no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—88no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—88no—
380C22×C4#12711yescharacteristic
381C22×C4#12712yes—
382C22×C4#12712yes—
383C22×C4#12712yes—
384C22×C4#12712yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390Q16—88no—
391Q16—88no—
392SD16—88no—
393SD16—88no—
394SD16—88no—
395D8—88no—
396D8—88no—
397M16—44no—
398M16—48no—
399C2×C8C22×D418yes—
400C2×C8C22×D418yes—
401C2×C8C22×D418yes—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—42no—
420C22⋊C4—41no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C42—41no—
429C42—42no—
430C42—44no—
431C42#12711yescharacteristic
432C42#12711yescharacteristic
433C42#12712yes—
434C42#12712yes—
435C42C22×D411yescharacteristic
436C42C22×D412yes—
437C42C22×D412yes—
438C42C22×D412yes—
439C42C22×D412yes—
440C42C2511yescharacteristic
441C23—42no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—84no—
446C23—84no—
447C23—84no—
448C23C2×#12712yes—
449C23C2×#12711yescharacteristic
450Q8—48no—
451Q8—44no—
452Q8—48no—
453Q8—42no—
454Q8—84no—
455Q8—84no—
456D4—44no—
457D4—48no—
458D4—48no—
459D4—48no—
460D4—42no—
461D4—84no—
462D4—84no—
463D4—84no—
464D4—88no—
465D4—88no—
466D4—82no—
467D4—84no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—88no—
483C2×C4—88no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4C2×#12712yes—
489C2×C4C2×#12711yescharacteristic
490C2×C4C2×#12712yes—
491C2×C4C2×#12712yes—
492C2×C4C2×#12712yes—
493C2×C4C23×D412yes—
494C2×C4C23×D411yescharacteristic
495C8—28no—
496C8—28no—
497C8—28no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—21no—
502C22—22no—
503C22—82no—
504C22—84no—
505C22—82no—
506C22—84no—
507C22—81no—
508C22—84no—
509C22—168no—
510C22C22×#12711yescharacteristic
511C4—22no—
512C4—22no—
513C4—22no—
514C4—21no—
515C4—22no—
516C4—84no—
517C4—82no—
518C4—84no—
519C4—81no—
520C4#358514yes—
521C4#358612yes—
522C2—41no—
523C2—42no—
524C2—161no—
525C2—164no—
526C2#6064412yes—
527C2#6064511yescharacteristic
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. #60965C2semidirect
3. #60862C2semidirect
4. #60804C2semidirect
5. #19750C2semidirect
6. #19748C2semidirect
7. #19741C2semidirect
8. #19739C2semidirect
9. #19737C2semidirect
10. #19735C2semidirect
11. #17339C2semidirect
18. #3231C22semidirect
19. #3229C22semidirect
21. #3228C22semidirect
23. #3226C22semidirect
28. #3209C22semidirect
29. #3208C22semidirect
30. #3208C22semidirect
33. #3205C22semidirect
37. #3184C22semidirect
38. #3182C22semidirect
39. #3165C22semidirect
40. #3152C22semidirect
41. #3141C22semidirect
42. #3141C22semidirect
52. #1723C22semidirect
53. #1722C22semidirect
54. #1721C22semidirect
55. #1720C22semidirect
56. #1720C22semidirect
57. #1720C22semidirect
58. #1719C22semidirect
59. #1717C22semidirect
60. #1717C22semidirect
61. #1716C22semidirect
62. #1715C22semidirect
63. #1714C22semidirect
64. #1709C22semidirect
65. #1708C22semidirect
66. #1707C22semidirect
67. #1706C22semidirect
68. #1455C22semidirect
69. #1454C22semidirect
70. #1454C22semidirect

Maximal subgroups

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

subgroupindex
#609682
#609652
#608622
#608042
#197502
#197482
#197412
#197392
#197372
#197352
#173392