Finite Group Databasebeta

← #6357946

Subgroups — #6357946

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60969C212yesmaximal, complemented
2#60968C211yesmaximal, characteristic, complemented
3#60862C211yesmaximal, characteristic, complemented
4#60801C212yesmaximal, complemented
5#19726C214yesmaximal, complemented
6#19725C214yesmaximal, complemented
7#19723C214yesmaximal, complemented
8#19722C214yesmaximal, complemented
9#19719C214yesmaximal, complemented
10#19715C214yesmaximal, complemented
11#17338C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C2×#532C2212yes—
15#3254—24no—
16#3253—22no—
17#3240—24no—
18#3231C2214yescomplemented
19#3230C2214yescomplemented
20#3228C2214yescomplemented
21#3227C2214yes—
22#3226C2214yescomplemented
23#3219—24no—
24#3219—24no—
25#3210C2214yes—
26#3210C2214yescomplemented
27#3208C2214yescomplemented
28#3207C2214yes—
29#3207C2214yescomplemented
30#3206C2214yescomplemented
31#3205C2214yes—
32#3193—24no—
33#3192—22no—
34#3184C2214yes—
35#3182C2214yes—
36#3182C2214yes—
37#3165C2212yescomplemented
38#3152C2212yescomplemented
39#3151C2212yescomplemented
40#3141C2212yescomplemented
41#3139C2212yescomplemented
42#3138C2212yescomplemented
43#3112—24no—
44#3110—24no—
45#3109—24no—
46#3093—22no—
47#3080—22no—
48#3040—24no—
49#3005—24no—
50#2920C2211yescharacteristic
51#2920C2212yes—
52#1723C2214yescomplemented
53#1722C2214yescomplemented
54#1721C2214yescomplemented
55#1720C2214yescomplemented
56#1719C2214yescomplemented
57#1719C2214yescomplemented
58#1718C2214yescomplemented
59#1717C2214yescomplemented
60#1717C2214yescomplemented
61#1715C2214yescomplemented
62#1714C2214yescomplemented
63#1714C2214yescomplemented
64#1711C2214yescomplemented
65#1707C2214yescomplemented
66#1706C2214yescomplemented
67#1706C2214yescomplemented
68#1453C2214yescomplemented
69#1453C2214yescomplemented
70#1453C2214yescomplemented
71C22×#41—22no—
72C2×#137—24no—
73C2×AGL(1,ℤ/8)—24no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533C2312yes—
78#533C2311yescharacteristic
79#533C2314yes—
80#532—22no—
81#532—22no—
82#532—22no—
83#532—22no—
84#532—22no—
85#532—22no—
86#532C2311yescharacteristic
87#532C2312yes—
88#532C2314yes—
89#532C2314yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127C2312yes—
93#529—22no—
94#529—22no—
95#529—22no—
96C2×#125—21no—
97C2×#124—22no—
98#525—21no—
99#525—22no—
100C2×C22⋊Q8—22no—
101C2×#121—22no—
102C2×#121—22no—
103C4×#41—21no—
104C4×#41—22no—
105C2×C4×Q8—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×#117—21no—
109C22×C42C2311yescharacteristic
110#501—24no—
111#500C2314yes—
112#500C2314yes—
113#500C2314yes—
114#498C2314yes—
115#498C2314yes—
116#497—24no—
117#496—24no—
118#494C2314yes—
119#493C2314yes—
120#493C2314yes—
121#492C2314yes—
122#492C2314yes—
123#492C2314yes—
124#492C2314yes—
125#491—24no—
126#490—24no—
127#489—24no—
128#487C2314yes—
129#487C2314yes—
130#487C2314yes—
131#487C2314yes—
132#487C2314yes—
133#486C2314yes—
134#486C2314yes—
135#486C2314yes—
136#486C2314yes—
137#486C2314yes—
138#486C2314yes—
139#486C2314yes—
140#485—28no—
141#484—28no—
142#482—28no—
143#481—28no—
144#480—28no—
145#479—28no—
146#478—28no—
147#477—28no—
148#476—28no—
149#474—28no—
150#469—28no—
151#468—28no—
152#468—28no—
153#464—28no—
154#464—28no—
155#463—28no—
156#463—28no—
157#461—28no—
158#460—28no—
159#459—28no—
160#452—28no—
161#452—28no—
162#450—28no—
163#449—28no—
164#449—28no—
165#448—28no—
166#447—28no—
167#442—28no—
168#440—28no—
169#440—28no—
170#432D414yes—
171#432D414yes—
172#432D414yes—
173#431C2314yes—
174#431C2314yes—
175#431C2314yes—
176#428—24no—
177#423D412yes—
178#423D414yes—
179#419—24no—
180#418—24no—
181#417—24no—
182#407D412yes—
183#407D414yes—
184C4×M16C2312yes—
185C4×M16C2312yes—
186C4×M16C2312yes—
187#322C2314yes—
188#322C2314yes—
189#322C2314yes—
190C82C2314yes—
191C2×#41—22no—
192C2×#41—21no—
193C2×#41—24no—
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—88no—
208AGL(1,ℤ/8)—88no—
209C2×Q16—28no—
210C2×SD16—28no—
211C2×SD16—28no—
212C2×SD16—28no—
213C2×SD16—28no—
214C2×D8—28no—
215C2×D8—28no—
216C2×M16C2×D414yes—
217C2×M16C2×D414yes—
218C2×M16C2×D414yes—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8C2414yes—
226C4⋊Q8C2414yes—
227C4⋊Q8C2411yescharacteristic
228C4⋊Q8C2412yes—
229#127—22no—
230#127—22no—
231#127—22no—
232#127C2414yes—
233#127C2411yescharacteristic
234#126—42no—
235#125—24no—
236#125—22no—
237#125—21no—
238#125—41no—
239#125—41no—
240#124—24no—
241#124—24no—
242#124—21no—
243#124—22no—
244#124—42no—
245#124—42no—
246#123—41no—
247#123—42no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255#121—24no—
256#121—24no—
257#121—24no—
258#121—24no—
259#121—22no—
260#121—22no—
261#121—22no—
262#121—42no—
263#121—42no—
264#121—42no—
265#121—42no—
266#120—41no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—42no—
273C4×Q8—42no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—42no—
288#117—24no—
289#117—22no—
290#117—21no—
291#117—42no—
292#117—41no—
293#117—42no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—42no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D411yescharacteristic
309C2×C42C2412yes—
310C2×C42C2411yescharacteristic
311#107—28no—
312#107—28no—
313#107—28no—
314#106—28no—
315#106—28no—
316C4⋊C8C2×D418yes—
317C4⋊C8C2×D418yes—
318C4⋊C8C2×D418yes—
319#103—28no—
320#103—28no—
321#103—28no—
322#103—28no—
323#103—28no—
324#102—28no—
325#102—28no—
326#102—28no—
327#102—28no—
328#102—28no—
329#102—28no—
330#102—28no—
331C22⋊C8C2×D418yes—
332C22⋊C8C2×D418yes—
333C22⋊C8C2×D418yes—
334#97C2×D414yes—
335#97C2×D414yes—
336#97C2×D414yes—
337C4×C8C2414yes—
338C4×C8C2414yes—
339C4×C8C2414yes—
340C24—42no—
341C24#12711yescharacteristic
342#41—48no—
343#41—44no—
344#41—88no—
345#41—88no—
346#41—88no—
347#41—88no—
348C2×Q8—21no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—48no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—84no—
356C2×D4—21no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—48no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4—88no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—88no—
381C22×C4#12711yescharacteristic
382C22×C4#12712yes—
383C22×C4#12712yes—
384C22×C4#12712yes—
385C22×C4#12712yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391Q16—88no—
392SD16—88no—
393SD16—88no—
394SD16—88no—
395SD16—88no—
396D8—88no—
397D8—88no—
398M16—44no—
399M16—44no—
400M16—44no—
401C2×C8C22×D418yes—
402C2×C8C22×D418yes—
403C2×C8C22×D418yes—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C42—41no—
431C42—42no—
432C42—44no—
433C42#12711yescharacteristic
434C42#12711yescharacteristic
435C42#12712yes—
436C42#12712yes—
437C42C22×D411yescharacteristic
438C42C22×D412yes—
439C42C22×D412yes—
440C42C22×D412yes—
441C42C22×D412yes—
442C42C2511yescharacteristic
443C23—42no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—84no—
448C23—84no—
449C23—84no—
450C23C2×#12712yes—
451C23C2×#12711yescharacteristic
452Q8—44no—
453Q8—48no—
454Q8—48no—
455Q8—42no—
456Q8—84no—
457Q8—84no—
458D4—44no—
459D4—48no—
460D4—48no—
461D4—48no—
462D4—42no—
463D4—84no—
464D4—84no—
465D4—84no—
466D4—88no—
467D4—88no—
468D4—84no—
469D4—82no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—88no—
485C2×C4—88no—
486C2×C4—84no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4C2×#12712yes—
491C2×C4C2×#12711yescharacteristic
492C2×C4C2×#12712yes—
493C2×C4C2×#12712yes—
494C2×C4C2×#12712yes—
495C2×C4C23×D412yes—
496C2×C4C23×D411yescharacteristic
497C8—28no—
498C8—28no—
499C8—28no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—21no—
504C22—22no—
505C22—82no—
506C22—84no—
507C22—82no—
508C22—84no—
509C22—81no—
510C22—84no—
511C22—168no—
512C22C22×#12711yescharacteristic
513C4—22no—
514C4—22no—
515C4—22no—
516C4—21no—
517C4—22no—
518C4—84no—
519C4—82no—
520C4—84no—
521C4—81no—
522C4#358514yes—
523C4#358612yes—
524C2—41no—
525C2—42no—
526C2—161no—
527C2—164no—
528C2#6064412yes—
529C2#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. #60969C2semidirect
2. #60968C2semidirect
3. #60862C2semidirect
4. #60801C2semidirect
5. #19726C2semidirect
6. #19725C2semidirect
7. #19723C2semidirect
8. #19722C2semidirect
9. #19719C2semidirect
10. #19715C2semidirect
11. #17338C2semidirect
18. #3231C22semidirect
19. #3230C22semidirect
20. #3228C22semidirect
22. #3226C22semidirect
26. #3210C22semidirect
27. #3208C22semidirect
29. #3207C22semidirect
30. #3206C22semidirect
37. #3165C22semidirect
38. #3152C22semidirect
39. #3151C22semidirect
40. #3141C22semidirect
41. #3139C22semidirect
42. #3138C22semidirect
52. #1723C22semidirect
53. #1722C22semidirect
54. #1721C22semidirect
55. #1720C22semidirect
56. #1719C22semidirect
57. #1719C22semidirect
58. #1718C22semidirect
59. #1717C22semidirect
60. #1717C22semidirect
61. #1715C22semidirect
62. #1714C22semidirect
63. #1714C22semidirect
64. #1711C22semidirect
65. #1707C22semidirect
66. #1706C22semidirect
67. #1706C22semidirect
68. #1453C22semidirect
69. #1453C22semidirect
70. #1453C22semidirect

Maximal subgroups

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

subgroupindex
#609692
#609682
#608622
#608012
#197262
#197252
#197232
#197222
#197192
#197152
#173382