Finite Group Databasebeta

← #6479629

Subgroups — #6479629

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35315C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#32959C212yesmaximal, complemented
4#32905C212yesmaximal, complemented
5#29913C212yesmaximal, complemented
6#29625C212yesmaximal
7#29504C214yesmaximal, complemented
8#29377C214yesmaximal, complemented
9#29133C212yesmaximal, complemented
10#28388C212yesmaximal, complemented
11#28214C212yesmaximal, complemented
12#28166C212yesmaximal, complemented
13#22049C212yesmaximal, complemented
14#22043C212yesmaximal, complemented
15#20779C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2212yes—
18C2×#478C2214yescomplemented
19C2×#461C2214yescomplemented
20C2×#450C2212yes—
21C2×#450C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2851C2211yescharacteristic
25#2647C2211yescharacteristic, complemented
26#2645C2212yescomplemented
27#2642C2211yescharacteristic, complemented
28#2533—22no—
29#2528—22no—
30#2505—22no—
31#2493—22no—
32#2330C2211yescharacteristic, complemented
33#2328—22no—
34#2328—22no—
35#2328C2212yes—
36#2328C2212yes—
37#2328C2212yes—
38#2328C2212yes—
39#2327C2211yescharacteristic, complemented
40#2320C2212yescomplemented
41C4×C22⋊Q8C2212yes—
42C2×#390C2212yes—
43C2×#386—22no—
44C2×#386—22no—
45C2×#382C2211yescharacteristic
46C2×#376C2211yescharacteristic
47#2101C2214yescomplemented
48#2097C2214yescomplemented
49#2085C2214yescomplemented
50#2074C2214yescomplemented
51#2041C2214yes—
52#2041C2214yes—
53#2014C2214yes—
54#2014C2214yes—
55#1973C2214yescomplemented
56#1968C2214yescomplemented
57#1936C2214yescomplemented
58#1934C2214yescomplemented
59#1926C2214yescomplemented
60#1925C2214yescomplemented
61#1880C2214yescomplemented
62#1880C2214yescomplemented
63#1877C2214yescomplemented
64#1877C2214yescomplemented
65#1872C2212yes—
66#1872C2212yes—
67#1864C2214yescomplemented
68#1862C2214yescomplemented
69#1853C2212yes—
70#1852C2212yes—
71#1840C2212yescomplemented
72#1832C2212yescomplemented
73#1817C2212yes—
74#1804C2214yescomplemented
75#1802C2214yescomplemented
76#1797C2212yes—
77#1786C2212yes—
78#1786C2212yes—
79#1768C2212yescomplemented
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yescomplemented
83C23×D4—21no—
84C23×D4—21no—
85C23×D4C2312yes—
86C24×C4—22no—
87C22×SD16C2314yes—
88C22×SD16C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127—28no—
91C2×#127—21no—
92C2×#127—21no—
93C2×#127C2312yes—
94C2×#127C2312yes—
95C2×#125C2311yescharacteristic
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—48no—
101C2×C4×D4C2314yes—
102C2×C4×D4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#478—48no—
111#461—48no—
112#450—28no—
113#450—28no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129#401—22no—
130#400—22no—
131#400—22no—
132#400—22no—
133#400—22no—
134#400—22no—
135#400—22no—
136#398—22no—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#391C2312yes—
141#390—22no—
142#390—22no—
143#390C2314yes—
144#390C2314yes—
145#389—22no—
146#389—22no—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#386—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#386—24no—
165#386—24no—
166#386—24no—
167#386—24no—
168#385—22no—
169#385—22no—
170#385—22no—
171#384C2311yescharacteristic
172#383—22no—
173#382—21no—
174#382—21no—
175#382—22no—
176#382C2311yescharacteristic
177#382C2312yes—
178#382C2312yes—
179#381—22no—
180#380—22no—
181#380C2312yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C22⋊C4—22no—
185#376D412yes—
186#376D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189#376C2311yescharacteristic
190#376C2311yescharacteristic
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D412yes—
194C2×#95D412yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C25—22no—
205C22×Q8C2412yes—
206C22×D4—28no—
207C22×D4—28no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—84no—
217C22×D4—84no—
218C22×D4#4118yes—
219C22×D4C2414yes—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—48no—
224C23×C4—84no—
225C23×C4—84no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C2×SD16—28no—
232C2×SD16—28no—
233C2×SD16—44no—
234C2×SD16—44no—
235C2×SD16—44no—
236C2×SD16—44no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—24no—
240#127—28no—
241#127—28no—
242#127—28no—
243#127—28no—
244#125—24no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C4×Q8—44no—
248C4×D4—28no—
249C4×D4—28no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—48no—
253C4×D4—48no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—84no—
279C2×C22⋊C4—84no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—42no—
285C2×C42C2×D412yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295#107—44no—
296#106—44no—
297C4⋊C8—28no—
298#103—44no—
299#103—44no—
300#102—28no—
301#102—28no—
302#102—44no—
303#102—44no—
304#102—44no—
305#102—44no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308#97—42no—
309C4×C8—42no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95C2×D412yes—
330#95C2×D412yes—
331#95C2×D412yes—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337C24—28no—
338C24—24no—
339C24—44no—
340C24—44no—
341C24—44no—
342C24—44no—
343C24—44no—
344C24—44no—
345C24C22×D411yescharacteristic
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—416no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—416no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—816no—
369C2×D4—816no—
370C2×D4C2×#4118yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—832no—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C2×#4114yes—
408C22×C4C2511yescharacteristic
409SD16—88no—
410SD16—88no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—22no—
442C42—22no—
443C42—24no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—44no—
451C42—82no—
452C42H(5,𝔽2)14yes—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—28no—
457C23—416no—
458C23—416no—
459C23—44no—
460C23—44no—
461C23—416no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—88no—
467C23—88no—
468C23—88no—
469C23—88no—
470C23#52512yes—
471C23#57714yes—
472C23C23×D411yescharacteristic
473Q8—44no—
474Q8—44no—
475D4—28no—
476D4—28no—
477D4—416no—
478D4—44no—
479D4—44no—
480D4—88no—
481D4—88no—
482D4—88no—
483D4—88no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—816no—
525C2×C4—84no—
526C2×C4—816no—
527C2×C4—88no—
528C2×C4#52512yes—
529C2×C4#53514yes—
530C2×C4C22×#4112yes—
531C2×C4C2×H(5,𝔽2)12yes—
532C2×C4#58512yes—
533C8—44no—
534C8—44no—
535C22—22no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—48no—
540C22—48no—
541C22—48no—
542C22—832no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22#312314yes—
548C22C2×#52511yescharacteristic
549C22C2×#53511yescharacteristic
550C22#344911yescharacteristic
551C22C2×#57712yes—
552C22#358512yes—
553C4—22no—
554C4—24no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—41no—
559C4—42no—
560C4—41no—
561C4—41no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—84no—
566C4—82no—
567C4#350514yes—
568C2—24no—
569C2—88no—
570C2#3510211yescharacteristic
571C2C2×#312311yescharacteristic
572C2#6056411yescharacteristic
573C2#6081212yes—
574C2#6097912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35315C2semidirect
3. #32959C2semidirect
4. #32905C2semidirect
5. #29913C2semidirect
7. #29504C2semidirect
8. #29377C2semidirect
9. #29133C2semidirect
10. #28388C2semidirect
11. #28214C2semidirect
12. #28166C2semidirect
13. #22049C2semidirect
14. #22043C2semidirect
15. #20779C2semidirect
18. C2×#478C22semidirect
19. C2×#461C22semidirect
25. #2647C22semidirect
26. #2645C22semidirect
27. #2642C22semidirect
32. #2330C22semidirect
39. #2327C22semidirect
40. #2320C22semidirect
47. #2101C22semidirect
48. #2097C22semidirect
49. #2085C22semidirect
50. #2074C22semidirect
55. #1973C22semidirect
56. #1968C22semidirect
57. #1936C22semidirect
58. #1934C22semidirect
59. #1926C22semidirect
60. #1925C22semidirect
61. #1880C22semidirect
62. #1880C22semidirect
63. #1877C22semidirect
64. #1877C22semidirect
67. #1864C22semidirect
68. #1862C22semidirect
71. #1840C22semidirect
72. #1832C22semidirect
74. #1804C22semidirect
75. #1802C22semidirect
79. #1768C22semidirect
82. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#353152
C2×#23282
#329592
#329052
#299132
#296252
#295042
#293772
#291332
#283882
#282142
#281662
#220492
#220432
#207792