Finite Group Databasebeta

← #6506475

Subgroups — #6506475

Order 512.

2306 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37425C211yesmaximal, characteristic, complemented
2#35616C211yesmaximal, characteristic
3#29115C212yesmaximal, complemented
4#29108C212yesmaximal, complemented
5#27284C214yesmaximal, complemented
6#27271C214yesmaximal, complemented
7#27256C214yesmaximal, complemented
8#27255C214yesmaximal, complemented
9#25868C212yesmaximal, complemented
10#25859C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#20602C212yesmaximal
13#20530C211yesmaximal, characteristic, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C2×#491C2212yescomplemented
19C2×#488C2214yescomplemented
20C2×#488C2212yescomplemented
21C2×#417C2212yes—
22C2×#418C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#408C2212yes—
26#2851C2212yescomplemented
27#2809—22no—
28#2780—24no—
29#2748—22no—
30#2629C2212yescomplemented
31#2442C2214yescomplemented
32#2441C2214yescomplemented
33#2435—22no—
34#2435C2214yes—
35#2432—22no—
36#2432C2214yes—
37#2405—24no—
38#2311C2212yes—
39#2309C2211yescharacteristic
40#2297C2211yescharacteristic
41#2297C2211yescharacteristic
42#2074C2212yescomplemented
43#2073C2212yescomplemented
44#2071C2214yescomplemented
45#2043C2214yescomplemented
46#2042C2214yescomplemented
47#2042C2214yescomplemented
48#2042C2214yescomplemented
49#2026C2212yescomplemented
50#2025C2214yescomplemented
51#2025C2212yescomplemented
52#1959C2212yescomplemented
53#1958C2214yescomplemented
54#1956C2212yescomplemented
55#1936C2214yescomplemented
56#1934C2214yescomplemented
57#1934C2214yescomplemented
58#1933C2214yescomplemented
59#1894C2214yescomplemented
60#1892C2214yescomplemented
61#1873C2214yescomplemented
62#1872C2214yescomplemented
63#1825C2214yescomplemented
64#1788C2214yes—
65#1788C2214yes—
66#1787C2214yescomplemented
67#1786C2214yes—
68#1786C2214yes—
69#1753C2214yescomplemented
70C8×C22⋊C4C2214yescomplemented
71#1750C2212yescomplemented
72C4×C22⋊C8C2212yescomplemented
73C2×H(5,𝔽2)—28no—
74C2×H(5,𝔽2)—42no—
75C2×H(5,𝔽2)—44no—
76C2×H(5,𝔽2)—42no—
77C22×#41—21no—
78C22×#41C2312yes—
79C22×#41C2311yescharacteristic
80C22×#41C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4—22no—
83C23×D4—21no—
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C22×M16C2312yes—
87C23×C8C2312yes—
88#536—28no—
89#536—42no—
90#536—42no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8C2312yes—
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×#124—21no—
96C2×#124—22no—
97C2×#124—44no—
98C2×#124C2314yes—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×#123—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2314yes—
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—22no—
109C22×C42C2311yescharacteristic
110#491—44no—
111#488—44no—
112#488—48no—
113#418—44no—
114#417—44no—
115#416—44no—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102—44no—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#408—28no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2312yes—
133C2×C4×C8C2314yes—
134C2×C4×C8C2312yes—
135#400—24no—
136#399—24no—
137#398—24no—
138#398—24no—
139#396—24no—
140#396—24no—
141#394—24no—
142#394—24no—
143#394—24no—
144#394—24no—
145#391—22no—
146#391D412yes—
147#391C2312yes—
148#390—22no—
149#390—22no—
150#390D412yes—
151#390C2314yes—
152#390C2312yes—
153#389—24no—
154#389C2314yes—
155#389C2314yes—
156#387—24no—
157#387—24no—
158#386—22no—
159#386—22no—
160#386—22no—
161#386D414yes—
162#385—24no—
163#385—24no—
164#385—24no—
165#385—24no—
166#384C2312yes—
167#382C2312yes—
168#380—24no—
169#380C2314yes—
170#380C2314yes—
171#379—22no—
172#379—24no—
173#379—24no—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2312yes—
182#336C2312yes—
183C25—21no—
184C25—22no—
185H(5,𝔽2)—432no—
186H(5,𝔽2)—432no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—24no—
190C2×#41—24no—
191C2×#41—24no—
192C2×#41—24no—
193C2×#41—24no—
194C2×#41—42no—
195C2×#41—41no—
196C2×#41—41no—
197C2×#41—42no—
198C2×#41—42no—
199C2×#41—42no—
200C2×#41—44no—
201C2×#41—44no—
202C2×#41—44no—
203C2×#41#4118yes—
204C22×Q8—21no—
205C22×Q8—24no—
206C22×Q8—42no—
207C22×Q8—41no—
208C22×Q8—41no—
209C22×Q8C2×D411yescharacteristic
210C22×Q8C2412yes—
211C22×Q8C2411yescharacteristic
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—24no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—42no—
223C22×D4—42no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4—44no—
227C22×D4—48no—
228C22×D4—44no—
229C22×D4—48no—
230C22×D4—48no—
231C22×D4—44no—
232C22×D4C2×D412yes—
233C22×D4C2×D411yescharacteristic
234C22×D4C2411yescharacteristic
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C2×M16—24no—
244C2×M16—42no—
245C2×M16—42no—
246C22×C8—24no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—42no—
253C4⋊Q8—44no—
254#125—44no—
255#125—42no—
256#125—42no—
257#124—28no—
258#124—24no—
259#124—44no—
260#124—42no—
261#124—48no—
262#124—44no—
263#123—48no—
264#123—48no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—48no—
268#117—24no—
269#117—41no—
270#117—41no—
271#117—42no—
272#117—42no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42C2×D412yes—
306C2×C42C2412yes—
307C2×C42C2412yes—
308#103—28no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314#102—28no—
315#102—28no—
316#102—28no—
317#102—44no—
318#102—44no—
319#102—44no—
320C22⋊C8—28no—
321C22⋊C8—28no—
322#97—42no—
323C4×C8—44no—
324C4×C8—42no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24—24no—
334C24—21no—
335C24—22no—
336C24—44no—
337C24—44no—
338C24—42no—
339C24—44no—
340C24—42no—
341C24—84no—
342C24—84no—
343C24—84no—
344C24C22×D412yes—
345C24C22×D411yescharacteristic
346#41—216no—
347#41—48no—
348#41—48no—
349#41—48no—
350#41—48no—
351#41—48no—
352#41—48no—
353#41—48no—
354#41—432no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8#12414yes—
370C2×Q8H(5,𝔽2)12yes—
371C2×D4—216no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—41no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—42no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—432no—
398C2×D4—464no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4#12418yes—
404C2×D4#12414yes—
405C2×D4H(5,𝔽2)12yes—
406C2×D4H(5,𝔽2)14yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4—84no—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2511yescharacteristic
440M16—84no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—84no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—42no—
469C22⋊C4—48no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—82no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—42no—
480C42—42no—
481C42—82no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—21no—
486C23—22no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—416no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—88no—
501C23—164no—
502C23—168no—
503C23—168no—
504C23—164no—
505C23—164no—
506C23—164no—
507C23C2×#12414yes—
508C23C2×#12412yes—
509C23C23×D411yescharacteristic
510Q8—24no—
511Q8—44no—
512Q8—42no—
513Q8—44no—
514Q8—44no—
515Q8—84no—
516D4—24no—
517D4—24no—
518D4—24no—
519D4—48no—
520D4—44no—
521D4—416no—
522D4—48no—
523D4—48no—
524D4—416no—
525D4—42no—
526D4—42no—
527D4—42no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—416no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—164no—
563C2×C4C2×#12412yes—
564C2×C4C2×#12414yes—
565C2×C4C22×#4112yes—
566C2×C4C2×H(5,𝔽2)11yescharacteristic
567C2×C4C2×H(5,𝔽2)11yescharacteristic
568C2×C4C2×H(5,𝔽2)12yes—
569C8—44no—
570C8—44no—
571C22—22no—
572C22—24no—
573C22—28no—
574C22—24no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—48no—
583C22—44no—
584C22—82no—
585C22—84no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—168no—
590C22—1616no—
591C22C22×#12411yescharacteristic
592C22#348411yescharacteristic
593C22#348411yescharacteristic
594C22#358414yes—
595C4—24no—
596C4—22no—
597C4—42no—
598C4—44no—
599C4—44no—
600C4—82no—
601C4—84no—
602C4—82no—
603C4—84no—
604C4#349012yes—
605C4#349012yes—
606C2—42no—
607C2—44no—
608C2—84no—
609C2—82no—
610C2#3536711yescharacteristic
611C2#6062412yes—
612C2#6163512yes—
613C2#6166312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37425C2semidirect
3. #29115C2semidirect
4. #29108C2semidirect
5. #27284C2semidirect
6. #27271C2semidirect
7. #27256C2semidirect
8. #27255C2semidirect
9. #25868C2semidirect
10. #25859C2semidirect
11. #21771C2semidirect
13. #20530C2semidirect
18. C2×#491C22semidirect
19. C2×#488C22semidirect
20. C2×#488C22semidirect
26. #2851C22semidirect
30. #2629C22semidirect
31. #2442C22semidirect
32. #2441C22semidirect
42. #2074C22semidirect
43. #2073C22semidirect
44. #2071C22semidirect
45. #2043C22semidirect
46. #2042C22semidirect
47. #2042C22semidirect
48. #2042C22semidirect
49. #2026C22semidirect
50. #2025C22semidirect
51. #2025C22semidirect
52. #1959C22semidirect
53. #1958C22semidirect
54. #1956C22semidirect
55. #1936C22semidirect
56. #1934C22semidirect
57. #1934C22semidirect
58. #1933C22semidirect
59. #1894C22semidirect
60. #1892C22semidirect
61. #1873C22semidirect
62. #1872C22semidirect
63. #1825C22semidirect
66. #1787C22semidirect
69. #1753C22semidirect
70. C8×C22⋊C4C22semidirect
71. #1750C22semidirect
72. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#374252
#356162
#291152
#291082
#272842
#272712
#272562
#272552
#258682
#258592
#217712
#206022
#205302