Finite Group Databasebeta

← #6503199

Subgroups — #6503199

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35076C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#27862C214yesmaximal, complemented
4#26209C214yesmaximal, complemented
5#26137C214yesmaximal, complemented
6#24964C214yesmaximal, complemented
7#24890C214yesmaximal, complemented
8#24560C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic, complemented
10#21767C211yesmaximal, characteristic, complemented
11#20973C211yesmaximal, characteristic, complemented
12#20898C211yesmaximal, characteristic, complemented
13#20660C211yesmaximal, characteristic, complemented
14C23×M16C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#482C2214yescomplemented
18C2×#427C2212yescomplemented
19C2×#418C2212yescomplemented
20C2×#407C2211yescharacteristic
21C2×#407C2212yescomplemented
22#2854C2212yes—
23#2847C2212yes—
24#2635—22no—
25#2591—21no—
26#2539—21no—
27#2487C2212yes—
28#2414C2214yes—
29#2408C2212yes—
30#2405—21no—
31#2403—21no—
32#2357—22no—
33#2357—22no—
34#2354C2214yes—
35#2354C2214yescomplemented
36#2321—22no—
37#2312—21no—
38#2312—21no—
39#2309—21no—
40#2285C2212yescomplemented
41#2280C2211yescharacteristic
42#2280C2211yescharacteristic
43#2280C2211yescharacteristic
44#2280C2212yescomplemented
45#2274—21no—
46#2099C2214yescomplemented
47#2093C2214yescomplemented
48#2084C2214yescomplemented
49#2077C2214yescomplemented
50#2075C2214yescomplemented
51#2071C2214yescomplemented
52#2034C2214yescomplemented
53#2033C2214yescomplemented
54#1926C2214yescomplemented
55#1926C2214yescomplemented
56#1921C2214yescomplemented
57#1921C2214yescomplemented
58#1895C2214yes—
59#1895C2214yes—
60#1893C2212yescomplemented
61#1893C2212yescomplemented
62#1875C2214yescomplemented
63#1875C2214yescomplemented
64#1870C2214yes—
65#1869C2212yescomplemented
66#1869C2212yescomplemented
67#1854C2214yescomplemented
68#1854C2214yescomplemented
69#1854C2214yescomplemented
70#1825C2214yescomplemented
71#1823C2212yescomplemented
72#1809C2214yescomplemented
73#1809C2214yescomplemented
74#1809C2214yescomplemented
75#1786C2212yescomplemented
76#1736C2212yescomplemented
77#1735C2212yescomplemented
78#1728C2212yescomplemented
79C23×D4—21no—
80C23×D4D411yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16—24no—
85C22×M16—44no—
86C22×M16C2312yes—
87C22×M16C2312yes—
88C22×M16C2312yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×#123—22no—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C2×#121C2314yes—
99C2×#120—22no—
100C2×#120—22no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—22no—
108C2×#117C2314yes—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42C2311yescharacteristic
118#482—48no—
119#427—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124#418—44no—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#407—24no—
130#407—44no—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2314yes—
135#400—22no—
136#399—22no—
137#394—22no—
138#394—24no—
139#392—24no—
140#392—22no—
141#390—24no—
142#390—24no—
143#390—24no—
144#390C2314yes—
145#389C2314yes—
146#388—24no—
147#388—24no—
148#388—24no—
149#388—24no—
150#388—24no—
151#388—24no—
152#386—22no—
153#386—24no—
154#386—22no—
155#386—22no—
156#385—22no—
157#384C2314yes—
158#382—22no—
159#381—21no—
160#381—21no—
161#381—22no—
162#381—21no—
163#380—22no—
164#380C2312yes—
165#380C2312yes—
166#380C2312yes—
167#380C2312yes—
168#380C2312yes—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172#379—21no—
173#379—21no—
174#379—22no—
175#379—21no—
176#379—22no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—22no—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C25C2×D411yescharacteristic
201C25C2×D411yescharacteristic
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—44no—
213C22×D4—84no—
214C22×D4C2×D412yes—
215C22×D4C2×D412yes—
216C22×D4C2412yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4#4114yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—42no—
243C2×M16—42no—
244C2×M16—42no—
245C2×M16—42no—
246C2×M16—416no—
247C22×C8—22no—
248C22×C8—24no—
249C22×C8—22no—
250C22×C8—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254#123—84no—
255#121—24no—
256#121—24no—
257#121—24no—
258#121—24no—
259#121—44no—
260#121—44no—
261#121—44no—
262#121—44no—
263#120—48no—
264#120—84no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271#117—48no—
272#117—42no—
273#117—44no—
274#117—42no—
275#117—42no—
276#117—42no—
277#117—42no—
278#117—42no—
279#117—44no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2414yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
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⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326#107—28no—
327#107—44no—
328#106—44no—
329#106—44no—
330#102—28no—
331#102—44no—
332#102—44no—
333#102—44no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—28no—
337C22⋊C8—44no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348C24—22no—
349C24—22no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—44no—
357C24—42no—
358C24—82no—
359C24—82no—
360C24#12314yes—
361C24C22×D412yes—
362C24C22×D411yescharacteristic
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—416no—
376C2×D4—48no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—42no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—816no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—84no—
423C22×C4—82no—
424C22×C4#12314yes—
425C22×C4#12314yes—
426C22×C4#12314yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2511yescharacteristic
440M16—48no—
441M16—48no—
442M16—84no—
443C2×C8—28no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—44no—
453C2×C8H(5,𝔽2)14yes—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—42no—
465C4⋊C4—48no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—84no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—84no—
487C22⋊C4—82no—
488C22⋊C4—84no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—48no—
493C42—82no—
494C42—82no—
495C42—82no—
496C42—84no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—82no—
521C23—82no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—88no—
526C23—88no—
527C23—168no—
528C23C2×#12312yes—
529C23C2×#12312yes—
530C23C2×#12312yes—
531C23C23×D411yescharacteristic
532D4—48no—
533D4—48no—
534D4—42no—
535D4—42no—
536D4—44no—
537D4—84no—
538D4—88no—
539D4—88no—
540D4—168no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—84no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—84no—
582C2×C4C2×#12312yes—
583C2×C4C2×#12312yes—
584C2×C4C2×#12312yes—
585C2×C4#57514yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C2×H(5,𝔽2)12yes—
589C2×C4#58512yes—
590C8—24no—
591C8—42no—
592C8—44no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—88no—
611C22—88no—
612C22—82no—
613C22—82no—
614C22—84no—
615C22—84no—
616C22—82no—
617C22—168no—
618C22C22×#12311yescharacteristic
619C22#349612yes—
620C22C2×#57512yes—
621C22#358512yes—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—84no—
629C4—84no—
630C4—84no—
631C4—82no—
632C4#349011yescharacteristic
633C4#349111yescharacteristic
634C4#350512yes—
635C2—42no—
636C2—42no—
637C2—42no—
638C2—82no—
639C2—162no—
640C2#3536011yescharacteristic
641C2#6051111yescharacteristic
642C2#6051811yescharacteristic
643C2#6230514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35076C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #27862C2semidirect
4. #26209C2semidirect
5. #26137C2semidirect
6. #24964C2semidirect
7. #24890C2semidirect
8. #24560C2semidirect
9. #21943C2semidirect
10. #21767C2semidirect
11. #20973C2semidirect
12. #20898C2semidirect
13. #20660C2semidirect
17. C2×#482C22semidirect
18. C2×#427C22semidirect
19. C2×#418C22semidirect
21. C2×#407C22semidirect
35. #2354C22semidirect
40. #2285C22semidirect
44. #2280C22semidirect
46. #2099C22semidirect
47. #2093C22semidirect
48. #2084C22semidirect
49. #2077C22semidirect
50. #2075C22semidirect
51. #2071C22semidirect
52. #2034C22semidirect
53. #2033C22semidirect
54. #1926C22semidirect
55. #1926C22semidirect
56. #1921C22semidirect
57. #1921C22semidirect
60. #1893C22semidirect
61. #1893C22semidirect
62. #1875C22semidirect
63. #1875C22semidirect
65. #1869C22semidirect
66. #1869C22semidirect
67. #1854C22semidirect
68. #1854C22semidirect
69. #1854C22semidirect
70. #1825C22semidirect
71. #1823C22semidirect
72. #1809C22semidirect
73. #1809C22semidirect
74. #1809C22semidirect
75. #1786C22semidirect
76. #1736C22semidirect
77. #1735C22semidirect
78. #1728C22semidirect

Maximal subgroups

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

subgroupindex
#350762
#341862
#278622
#262092
#261372
#249642
#248902
#245602
#219432
#217672
#209732
#208982
#206602