Finite Group Databasebeta

← #6506504

Subgroups — #6506504

Order 512.

2338 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic, complemented
2#35616C211yesmaximal, characteristic
3#29115C212yesmaximal, complemented
4#29113C212yesmaximal, complemented
5#27278C214yesmaximal, complemented
6#27271C214yesmaximal, complemented
7#27267C214yesmaximal, complemented
8#27262C214yesmaximal, complemented
9#25868C212yesmaximal, complemented
10#25864C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#20602C212yesmaximal
13#20532C211yesmaximal, characteristic, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C2×#491C2212yescomplemented
19C2×#489C2214yescomplemented
20C2×#488C2212yescomplemented
21C2×#417C2212yes—
22C2×#418C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#408C2212yes—
26#2857C2212yescomplemented
27#2806—22no—
28#2748—22no—
29#2732—22no—
30#2683—22no—
31#2629C2212yescomplemented
32#2441C2214yescomplemented
33#2440C2214yescomplemented
34#2435—22no—
35#2435C2214yes—
36#2432—22no—
37#2432C2214yes—
38#2405—22no—
39#2405—22no—
40#2311C2212yes—
41#2309C2211yescharacteristic
42#2297C2211yescharacteristic
43#2297C2211yescharacteristic
44#2072C2212yescomplemented
45#2071C2214yescomplemented
46#2071C2212yescomplemented
47#2043C2214yescomplemented
48#2042C2214yescomplemented
49#2042C2214yescomplemented
50#2042C2214yescomplemented
51#2025C2214yescomplemented
52#2024C2212yescomplemented
53#2023C2212yescomplemented
54#1959C2214yescomplemented
55#1959C2212yescomplemented
56#1956C2212yescomplemented
57#1936C2214yescomplemented
58#1936C2214yescomplemented
59#1935C2214yescomplemented
60#1934C2214yescomplemented
61#1894C2214yescomplemented
62#1892C2214yescomplemented
63#1873C2214yescomplemented
64#1872C2214yescomplemented
65#1825C2214yescomplemented
66#1788C2214yes—
67#1788C2214yes—
68#1787C2214yescomplemented
69#1786C2214yes—
70#1786C2214yes—
71#1753C2214yescomplemented
72#1752C2214yescomplemented
73#1750C2212yescomplemented
74#1749C2212yescomplemented
75C2×H(5,𝔽2)—28no—
76C2×H(5,𝔽2)—44no—
77C2×H(5,𝔽2)—42no—
78C2×H(5,𝔽2)—42no—
79C22×#41—21no—
80C22×#41C2312yes—
81C22×#41C2311yescharacteristic
82C22×#41C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4—22no—
85C23×D4—21no—
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C22×M16C2312yes—
89C23×C8C2312yes—
90#536—28no—
91#536—42no—
92#536—42no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×#125C2312yes—
97C2×#124—21no—
98C2×#124—22no—
99C2×#124—44no—
100C2×#124C2312yes—
101C2×#124C2314yes—
102C2×#123—22no—
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8C2314yes—
108C2×#117C2311yescharacteristic
109C2×#117C2312yes—
110C2×#117C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C42C2311yescharacteristic
115#491—44no—
116#489—48no—
117#488—44no—
118#418—44no—
119#417—44no—
120#416—44no—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102—44no—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134#408—28no—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2314yes—
138C2×#97C2312yes—
139C2×C4×C8C2312yes—
140#399—24no—
141#398—24no—
142#398—24no—
143#396—24no—
144#396—24no—
145#395—24no—
146#394—24no—
147#394—24no—
148#394—24no—
149#394—24no—
150#391—22no—
151#391—22no—
152#391D412yes—
153#391C2312yes—
154#390—22no—
155#390—22no—
156#390D412yes—
157#390C2314yes—
158#390C2312yes—
159#389—22no—
160#389—24no—
161#389C2314yes—
162#389C2314yes—
163#387—24no—
164#386—22no—
165#386—22no—
166#386—21no—
167#386—21no—
168#386D412yes—
169#386D412yes—
170#385—24no—
171#385—24no—
172#385—24no—
173#384C2312yes—
174#384C2312yes—
175#380—22no—
176#380C2314yes—
177#380C2314yes—
178#379—22no—
179#379—24no—
180#379—24no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2314yes—
189#336C2312yes—
190C25—21no—
191C25—22no—
192H(5,𝔽2)—432no—
193H(5,𝔽2)—432no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—42no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—42no—
207C2×#41—44no—
208C2×#41—44no—
209C2×#41—44no—
210C2×#41#4118yes—
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—24no—
214C22×Q8—42no—
215C22×Q8—41no—
216C22×Q8—41no—
217C22×Q8C2×D411yescharacteristic
218C22×Q8C2412yes—
219C22×Q8C2411yescharacteristic
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—21no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—41no—
233C22×D4—41no—
234C22×D4—44no—
235C22×D4—48no—
236C22×D4—44no—
237C22×D4—48no—
238C22×D4—48no—
239C22×D4—44no—
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—42no—
254C2×M16—42no—
255C22×C8—24no—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264#125—42no—
265#125—42no—
266#124—28no—
267#124—24no—
268#124—44no—
269#124—42no—
270#124—48no—
271#124—44no—
272#123—48no—
273#123—48no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—48no—
277C22⋊Q8—88no—
278C22⋊Q8—84no—
279#117—24no—
280#117—41no—
281#117—41no—
282#117—42no—
283#117—42no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2414yes—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D414yes—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—42no—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#103—28no—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#102—28no—
330#102—28no—
331#102—28no—
332#102—44no—
333#102—44no—
334#102—44no—
335C22⋊C8—28no—
336C22⋊C8—28no—
337#97—44no—
338#97—42no—
339C4×C8—42no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95C2×D414yes—
347#95C2×D414yes—
348C24—24no—
349C24—21no—
350C24—22no—
351C24—44no—
352C24—44no—
353C24—42no—
354C24—44no—
355C24—42no—
356C24—82no—
357C24—82no—
358C24—84no—
359C24—84no—
360C24C22×D412yes—
361C24C22×D411yescharacteristic
362#41—216no—
363#41—48no—
364#41—48no—
365#41—48no—
366#41—48no—
367#41—48no—
368#41—48no—
369#41—48no—
370#41—432no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8#12414yes—
389C2×Q8H(5,𝔽2)12yes—
390C2×D4—216no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—432no—
417C2×D4—464no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4#12418yes—
423C2×D4#12414yes—
424C2×D4H(5,𝔽2)12yes—
425C2×D4H(5,𝔽2)14yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—82no—
452C22×C4—84no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4C2511yescharacteristic
462M16—84no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—84no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—42no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—82no—
496C22⋊C4—84no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—42no—
505C42—42no—
506C42—82no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—21no—
512C23—22no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—416no—
518C23—48no—
519C23—44no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—164no—
530C23—168no—
531C23—164no—
532C23—164no—
533C23—168no—
534C23—164no—
535C23C2×#12414yes—
536C23C2×#12412yes—
537C23C23×D411yescharacteristic
538Q8—24no—
539Q8—44no—
540Q8—42no—
541Q8—44no—
542Q8—44no—
543Q8—84no—
544Q8—164no—
545D4—24no—
546D4—24no—
547D4—24no—
548D4—48no—
549D4—44no—
550D4—416no—
551D4—48no—
552D4—48no—
553D4—416no—
554D4—42no—
555D4—42no—
556D4—42no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—416no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—164no—
597C2×C4C2×#12412yes—
598C2×C4C2×#12414yes—
599C2×C4C22×#4112yes—
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×H(5,𝔽2)12yes—
603C8—44no—
604C8—44no—
605C22—22no—
606C22—24no—
607C22—28no—
608C22—24no—
609C22—41no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—81no—
621C22—84no—
622C22—82no—
623C22—81no—
624C22—84no—
625C22—168no—
626C22—1616no—
627C22C22×#12411yescharacteristic
628C22#348411yescharacteristic
629C22#348411yescharacteristic
630C22#358512yes—
631C22#358612yes—
632C4—24no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—84no—
642C4#349012yes—
643C4#349012yes—
644C2—42no—
645C2—44no—
646C2—84no—
647C2—82no—
648C2#3536711yescharacteristic
649C2#6062512yes—
650C2#6164811yescharacteristic
651C2#6164911yescharacteristic
652C2#6167712yes—
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. #37307C2semidirect
3. #29115C2semidirect
4. #29113C2semidirect
5. #27278C2semidirect
6. #27271C2semidirect
7. #27267C2semidirect
8. #27262C2semidirect
9. #25868C2semidirect
10. #25864C2semidirect
11. #21771C2semidirect
13. #20532C2semidirect
18. C2×#491C22semidirect
19. C2×#489C22semidirect
20. C2×#488C22semidirect
26. #2857C22semidirect
31. #2629C22semidirect
32. #2441C22semidirect
33. #2440C22semidirect
44. #2072C22semidirect
45. #2071C22semidirect
46. #2071C22semidirect
47. #2043C22semidirect
48. #2042C22semidirect
49. #2042C22semidirect
50. #2042C22semidirect
51. #2025C22semidirect
52. #2024C22semidirect
53. #2023C22semidirect
54. #1959C22semidirect
55. #1959C22semidirect
56. #1956C22semidirect
57. #1936C22semidirect
58. #1936C22semidirect
59. #1935C22semidirect
60. #1934C22semidirect
61. #1894C22semidirect
62. #1892C22semidirect
63. #1873C22semidirect
64. #1872C22semidirect
65. #1825C22semidirect
68. #1787C22semidirect
71. #1753C22semidirect
72. #1752C22semidirect
73. #1750C22semidirect
74. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#373072
#356162
#291152
#291132
#272782
#272712
#272672
#272622
#258682
#258642
#217712
#206022
#205322