Finite Group Databasebeta

← #6489093

Subgroups — #6489093

Order 512.

2038 conjugacy classes of subgroups.

The subgroup lattice has 575 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#30138C214yesmaximal, complemented
6#29136C212yesmaximal, complemented
7#28388C212yesmaximal, complemented
8#28292C212yesmaximal, complemented
9#28260C212yesmaximal, complemented
10#27686C212yesmaximal, complemented
11#24507C212yesmaximal, complemented
12#24501C212yesmaximal, complemented
13#24211C212yesmaximal
14#22160C214yesmaximal, complemented
15#20814C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2212yes—
18C2×#478C2214yescomplemented
19C2×#460C2214yes—
20C2×#461C2214yescomplemented
21C2×#450C2212yes—
22C2×#450C2212yes—
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2851C2211yescharacteristic
28#2647C2211yescharacteristic, complemented
29#2645C2212yescomplemented
30#2642C2211yescharacteristic, complemented
31#2533—22no—
32#2528—22no—
33#2505—22no—
34#2493—22no—
35#2330C2211yescharacteristic, complemented
36#2328—22no—
37#2328—22no—
38#2328C2212yes—
39#2328C2212yes—
40#2328C2212yes—
41#2328C2212yes—
42#2327C2211yescharacteristic, complemented
43#2320C2212yescomplemented
44C4×C22⋊Q8C2212yes—
45C2×#390C2212yes—
46C2×#386—22no—
47C2×#386—22no—
48C2×#382C2211yescharacteristic
49C2×#376C2211yescharacteristic
50#2101C2214yescomplemented
51#2097C2214yescomplemented
52#2067C2214yescomplemented
53#2065C2214yescomplemented
54#2055C2214yescomplemented
55#2055C2214yescomplemented
56#2041C2214yes—
57#2014C2214yes—
58#1982C2214yes—
59#1973C2214yescomplemented
60#1968C2214yescomplemented
61#1949C2214yescomplemented
62#1936C2214yescomplemented
63#1934C2214yescomplemented
64#1877C2214yescomplemented
65#1877C2214yescomplemented
66#1872C2212yes—
67#1872C2212yes—
68#1864C2214yescomplemented
69#1862C2214yescomplemented
70#1853C2212yes—
71#1852C2212yes—
72#1843C2212yescomplemented
73#1832C2212yescomplemented
74#1805C2214yescomplemented
75#1804C2214yescomplemented
76#1802C2214yescomplemented
77#1797C2212yes—
78#1797C2212yes—
79#1786C2212yes—
80#1786C2212yes—
81#1768C2212yescomplemented
82#1768C2212yescomplemented
83C23×D4—22no—
84C23×D4C2312yes—
85C24×C4—22no—
86C22×SD16C2314yes—
87C22×SD16C2314yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127—28no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×#125C2311yescharacteristic
94C2×C22⋊Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—48no—
99C2×C4×D4C2314yes—
100C2×C4×D4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#478—48no—
109#461—48no—
110#460—216no—
111#450—44no—
112#450—44no—
113#432—42no—
114#431—42no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2312yes—
132C2×C4×C8C2312yes—
133#401—22no—
134#400—22no—
135#400—22no—
136#400—22no—
137#400—22no—
138#400—22no—
139#400—22no—
140#398—22no—
141#391C2312yes—
142#391C2312yes—
143#391C2312yes—
144#391C2312yes—
145#390—22no—
146#390—22no—
147#390C2314yes—
148#390C2314yes—
149#389—22no—
150#389—22no—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#386—24no—
165#386—24no—
166#386—24no—
167#386—24no—
168#386—24no—
169#386—24no—
170#386—24no—
171#386—24no—
172#385—22no—
173#385—22no—
174#385—22no—
175#384C2311yescharacteristic
176#383—22no—
177#382—22no—
178#382—22no—
179#382C2311yescharacteristic
180#382C2312yes—
181#382C2312yes—
182#381—22no—
183#380—22no—
184#380C2312yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—22no—
188#376D412yes—
189#376D412yes—
190#376C2311yescharacteristic
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193#376C2311yescharacteristic
194C2×#95—22no—
195C2×#95D412yes—
196C2×#95D412yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—22no—
204C22×Q8C2412yes—
205C22×D4—216no—
206C22×D4—28no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—44no—
213C22×D4—84no—
214C22×D4—84no—
215C22×D4C2414yes—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—48no—
220C23×C4—84no—
221C23×C4—84no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C2×SD16—28no—
228C2×SD16—28no—
229C2×SD16—44no—
230C2×SD16—44no—
231C2×SD16—44no—
232C2×SD16—44no—
233C22×C8C2414yes—
234C22×C8C2414yes—
235C4⋊Q8—24no—
236#127—216no—
237#127—28no—
238#127—24no—
239#127#4118yes—
240#125—24no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C4×Q8—28no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—48no—
249C4×D4—48no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
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—44no—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—84no—
275C2×C22⋊C4—84no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292#107—44no—
293#106—44no—
294C4⋊C8—28no—
295C4⋊C8—28no—
296C4⋊C8—42no—
297C4⋊C8—42no—
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—24no—
311#95—24no—
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#95C2×D412yes—
328#95C2×D412yes—
329#95C2×D412yes—
330#95C2×D412yes—
331#95C2×D412yes—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335C24—28no—
336C24—24no—
337C24—44no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—44no—
342C24—44no—
343C24C22×D411yescharacteristic
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×D4—28no—
349C2×D4—216no—
350C2×D4—216no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—84no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—832no—
365C2×D4C2×#4118yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—832no—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C22×C4C2511yescharacteristic
405SD16—88no—
406SD16—88no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—24no—
438C42—24no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—21no—
443C42—21no—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—44no—
448C42—82no—
449C42C2×#4114yes—
450C42H(5,𝔽2)14yes—
451C23—22no—
452C23—21no—
453C23—21no—
454C23—28no—
455C23—416no—
456C23—416no—
457C23—416no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—88no—
465C23—88no—
466C23—88no—
467C23—88no—
468C23#53712yes—
469C23#57714yes—
470C23C23×D411yescharacteristic
471Q8—44no—
472Q8—44no—
473D4—28no—
474D4—28no—
475D4—416no—
476D4—44no—
477D4—44no—
478D4—816no—
479D4—816no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
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—48no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—816no—
521C2×C4—82no—
522C2×C4—816no—
523C2×C4—88no—
524C2×C4#52514yes—
525C2×C4#53514yes—
526C2×C4#53712yes—
527C2×C4C22×#4112yes—
528C2×C4C2×H(5,𝔽2)11yescharacteristic
529C2×C4C2×H(5,𝔽2)12yes—
530C2×C4C2×2−1+411yescharacteristic
531C8—44no—
532C8—44no—
533C22—24no—
534C22—24no—
535C22—22no—
536C22—22no—
537C22—48no—
538C22—48no—
539C22—48no—
540C22—832no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22#326014yes—
546C22C2×#52511yescharacteristic
547C22C2×#53511yescharacteristic
548C22C2×#53711yescharacteristic
549C22C2×#57712yes—
550C22#358512yes—
551C4—24no—
552C4—24no—
553C4—22no—
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—82no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4#351912yes—
566C4#352012yes—
567C2—24no—
568C2—88no—
569C2#3523611yescharacteristic
570C2#6057912yes—
571C2#6081212yes—
572C2C2×#326011yescharacteristic
573C2#6092011yescharacteristic
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. #30138C2semidirect
6. #29136C2semidirect
7. #28388C2semidirect
8. #28292C2semidirect
9. #28260C2semidirect
10. #27686C2semidirect
11. #24507C2semidirect
12. #24501C2semidirect
14. #22160C2semidirect
15. #20814C2semidirect
18. C2×#478C22semidirect
20. C2×#461C22semidirect
28. #2647C22semidirect
29. #2645C22semidirect
30. #2642C22semidirect
35. #2330C22semidirect
42. #2327C22semidirect
43. #2320C22semidirect
50. #2101C22semidirect
51. #2097C22semidirect
52. #2067C22semidirect
53. #2065C22semidirect
54. #2055C22semidirect
55. #2055C22semidirect
59. #1973C22semidirect
60. #1968C22semidirect
61. #1949C22semidirect
62. #1936C22semidirect
63. #1934C22semidirect
64. #1877C22semidirect
65. #1877C22semidirect
68. #1864C22semidirect
69. #1862C22semidirect
72. #1843C22semidirect
73. #1832C22semidirect
74. #1805C22semidirect
75. #1804C22semidirect
76. #1802C22semidirect
81. #1768C22semidirect
82. #1768C22semidirect

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
#301382
#291362
#283882
#282922
#282602
#276862
#245072
#245012
#242112
#221602
#208142