Finite Group Databasebeta

← #6480255

Subgroups — #6480255

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic
2#34221C211yesmaximal, characteristic, complemented
3#33039C214yesmaximal, complemented
4#30069C212yesmaximal, complemented
5#30051C212yesmaximal, complemented
6#29569C214yesmaximal, complemented
7#29358C212yesmaximal, complemented
8#28392C212yesmaximal, complemented
9#27588C212yesmaximal, complemented
10#24307C212yesmaximal, complemented
11#24211C212yesmaximal
12#23933C212yesmaximal, complemented
13#22049C212yesmaximal, complemented
14#22043C212yesmaximal, complemented
15#20779C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#460C2214yes—
21C2×#450C2212yes—
22C2×#450C2212yes—
23#2641—22no—
24#2641—22no—
25#2641C2212yes—
26#2641C2212yes—
27#2641C2212yes—
28#2641C2212yes—
29C2×#394—22no—
30C2×#394—22no—
31#2371—22no—
32#2368—22no—
33#2356—22no—
34#2343—22no—
35#2331C2211yescharacteristic, complemented
36#2331C2211yescharacteristic, complemented
37#2327C2211yescharacteristic, complemented
38#2327C2211yescharacteristic, complemented
39#2320C2212yescomplemented
40C4×C22⋊Q8C2212yes—
41C4×C22⋊Q8C2212yes—
42C2×#390C2212yes—
43C2×#390C2212yes—
44#2283C2211yescharacteristic
45#2276C2212yescomplemented
46C2×#376C2211yescharacteristic
47#2092C2214yescomplemented
48#2073C2214yescomplemented
49#2059C2214yescomplemented
50#2041C2214yes—
51#2026C2214yescomplemented
52#2023C2214yescomplemented
53#2014C2214yes—
54#2013C2214yescomplemented
55#1989C2214yescomplemented
56#1982C2214yes—
57#1973C2214yescomplemented
58#1968C2214yescomplemented
59#1936C2214yescomplemented
60#1934C2214yescomplemented
61#1926C2214yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1925C2214yescomplemented
65#1880C2214yescomplemented
66#1880C2214yescomplemented
67#1880C2214yescomplemented
68#1880C2214yescomplemented
69#1872C2212yes—
70#1872C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1840C2212yescomplemented
74#1832C2212yescomplemented
75#1817C2212yes—
76#1797C2212yes—
77#1786C2212yes—
78#1786C2212yes—
79#1768C2212yescomplemented
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yescomplemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87#533—28no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127—22no—
91C2×#127—44no—
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#125—21no—
95C2×#125—22no—
96C2×#125—44no—
97C2×#125C2312yes—
98C2×#125C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C2312yes—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×#117—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—22no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#479—48no—
116#460—48no—
117#450—28no—
118#450—28no—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×#97C2312yes—
133C2×C4×C8C2312yes—
134#394—24no—
135#394—24no—
136#394—24no—
137#394—24no—
138#394—24no—
139#394—24no—
140#394—24no—
141#394—24no—
142#391—22no—
143#391—22no—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2314yes—
152#390C2314yes—
153#390C2314yes—
154#389C2312yes—
155#389C2312yes—
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#387—22no—
165#387—22no—
166#385—22no—
167#385—22no—
168#383—21no—
169#383—22no—
170#382—22no—
171#382C2311yescharacteristic
172#382C2311yescharacteristic
173#381—21no—
174#380C2312yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—21no—
186#376—21no—
187#376—22no—
188#376D412yes—
189#376D412yes—
190#376C2311yescharacteristic
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193#376C2311yescharacteristic
194C2×#95D412yes—
195C2×#95D412yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203#336C2312yes—
204#336C2312yes—
205C25—22no—
206C22×Q8—21no—
207C22×Q8C2412yes—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—48no—
216C22×D4—84no—
217C22×D4—84no—
218C22×D4C2414yes—
219C23×C4—22no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C2×SD16—28no—
226C2×SD16—28no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—44no—
230C2×SD16—44no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235#127—28no—
236#127—42no—
237#127—42no—
238#127—48no—
239#125—28no—
240#125—28no—
241#125—42no—
242#125—42no—
243#125—48no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8#4118yes—
247C4×Q8—28no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—28no—
251C4×Q8—44no—
252#117—28no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—84no—
282C2×C22⋊C4—84no—
283C2×C22⋊C4—84no—
284C2×C22⋊C4—84no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—42no—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D412yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—44no—
305#106—44no—
306C4⋊C8—28no—
307#103—44no—
308#103—44no—
309#102—28no—
310#102—28no—
311#102—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317#97—42no—
318C4×C8—42no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—21no—
324#95—21no—
325#95—22no—
326#95—22no—
327#95—21no—
328#95—21no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341C24—24no—
342C24—44no—
343C24—44no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—44no—
348C24—44no—
349C24C22×D411yescharacteristic
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—82no—
355C2×Q8C2×#4114yes—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—832no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C2×#4114yes—
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—24no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—28no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4C2×#4114yes—
431C22⋊C4—28no—
432C22⋊C4—44no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—28no—
442C42—22no—
443C42—22no—
444C42—24no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—42no—
451C42—44no—
452C42—44no—
453C42—82no—
454C42—82no—
455C42—82no—
456C42H(5,𝔽2)14yes—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—88no—
468C23—88no—
469C23—88no—
470C23—88no—
471C23—88no—
472C23—88no—
473C23—88no—
474C23#52512yes—
475C23#57714yes—
476C23C23×D411yescharacteristic
477Q8—28no—
478Q8—44no—
479Q8—84no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—88no—
485D4—816no—
486D4—816no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—84no—
527C2×C4—82no—
528C2×C4—88no—
529C2×C4#52512yes—
530C2×C4#53514yes—
531C2×C4C22×#4112yes—
532C2×C4C2×H(5,𝔽2)12yes—
533C2×C4#58512yes—
534C8—44no—
535C8—44no—
536C22—24no—
537C22—22no—
538C22—24no—
539C22—22no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22—1616no—
548C22#312614yes—
549C22C2×#52511yescharacteristic
550C22C2×#53511yescharacteristic
551C22#344911yescharacteristic
552C22C2×#57712yes—
553C22#358512yes—
554C4—22no—
555C4—24no—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—41no—
560C4—41no—
561C4—42no—
562C4—41no—
563C4—41no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4#351414yes—
569C2—24no—
570C2—164no—
571C2#3512911yescharacteristic
572C2C2×#312611yescharacteristic
573C2#6056811yescharacteristic
574C2#6081212yes—
575C2#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
2. #34221C2semidirect
3. #33039C2semidirect
4. #30069C2semidirect
5. #30051C2semidirect
6. #29569C2semidirect
7. #29358C2semidirect
8. #28392C2semidirect
9. #27588C2semidirect
10. #24307C2semidirect
12. #23933C2semidirect
13. #22049C2semidirect
14. #22043C2semidirect
15. #20779C2semidirect
19. C2×#479C22semidirect
35. #2331C22semidirect
36. #2331C22semidirect
37. #2327C22semidirect
38. #2327C22semidirect
39. #2320C22semidirect
45. #2276C22semidirect
47. #2092C22semidirect
48. #2073C22semidirect
49. #2059C22semidirect
51. #2026C22semidirect
52. #2023C22semidirect
54. #2013C22semidirect
55. #1989C22semidirect
57. #1973C22semidirect
58. #1968C22semidirect
59. #1936C22semidirect
60. #1934C22semidirect
61. #1926C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
64. #1925C22semidirect
65. #1880C22semidirect
66. #1880C22semidirect
67. #1880C22semidirect
68. #1880C22semidirect
73. #1840C22semidirect
74. #1832C22semidirect
79. #1768C22semidirect
82. #1746C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
#342212
#330392
#300692
#300512
#295692
#293582
#283922
#275882
#243072
#242112
#239332
#220492
#220432
#207792