Finite Group Databasebeta

← #6354846

Subgroups — #6354846

Order 512.

1777 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic, complemented
2#60803C211yesmaximal, characteristic
3C2×#3206C211yesmaximal, characteristic, complemented
4#60236C211yesmaximal, characteristic, complemented
5#19514C218yesmaximal, complemented
6#19428C214yesmaximal, complemented
7#19392C214yesmaximal, complemented
8#19198C214yesmaximal, complemented
9#19192C214yesmaximal, complemented
10#17803C211yesmaximal, characteristic, complemented
11#17742C211yesmaximal, characteristic, complemented
12#17717C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16#3226—22no—
17#3226—22no—
18#3226C2212yes—
19#3226C2212yes—
20#3226C2214yescomplemented
21#3214C2214yes—
22#3213—22no—
23#3207—22no—
24#3206—22no—
25#3206—22no—
26#3206C2214yes—
27#3206C2212yes—
28#3206C2214yescomplemented
29#3205C2212yes—
30C2×#500C2212yescomplemented
31#3151—21no—
32C2×#492—21no—
33#3141—21no—
34C2×#487—21no—
35C2×#486C2212yescomplemented
36C2×#480—22no—
37C2×#474—22no—
38#3036C2214yes—
39C2×#461C2214yescomplemented
40#3006—22no—
41C2×#448—22no—
42#2981C2214yescomplemented
43#2979—22no—
44#2975C2212yescomplemented
45#2974—22no—
46#2972C2212yescomplemented
47#2935—22no—
48#2931C2214yes—
49#2920C2211yescharacteristic
50#2920C2211yescharacteristic
51C2×#431C2211yescharacteristic
52C2×#431C2211yescharacteristic
53#2880—22no—
54#1701C22116yescomplemented
55#1693C22116yescomplemented
56#1681C22116yescomplemented
57#1673C22116yescomplemented
58#1572C2214yescomplemented
59#1571C2212yescomplemented
60#1570C2212yescomplemented
61#1564C2214yescomplemented
62#1556C2214yescomplemented
63#1553C2214yescomplemented
64#1549C2212yescomplemented
65#1549C2214yescomplemented
66#1546C2214yescomplemented
67#1545C2212yescomplemented
68#1540C2214yes—
69C22×#41—21no—
70C22×#41—21no—
71C23×Q8—21no—
72C2×AGL(1,ℤ/8)—24no—
73C22×SD16—24no—
74C2×#131—24no—
75#532—21no—
76#532—21no—
77#532—21no—
78#532—21no—
79#532—21no—
80#532—21no—
81#532—21no—
82#532—21no—
83#532—21no—
84#532—21no—
85#532D412yes—
86#532D412yes—
87#532C2312yes—
88#532C2312yes—
89#532C2312yes—
90#532C2312yes—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8D411yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127D411yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#124—21no—
108C2×#124—21no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121D412yes—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C42C2311yescharacteristic
120#500—24no—
121#500C2314yes—
122#500C2314yes—
123#496—24no—
124#493—22no—
125#492—24no—
126#492—24no—
127#492—22no—
128#489—24no—
129#487—24no—
130#487—24no—
131#487—22no—
132#486—22no—
133#486—24no—
134#486C2314yes—
135#486C2314yes—
136#480—24no—
137#480—24no—
138#480—28no—
139#474—24no—
140#474—24no—
141#474—28no—
142#463—28no—
143#461—28no—
144#461C23116yes—
145#461C2318yes—
146#459C2318yes—
147#449—28no—
148#448—28no—
149#448—28no—
150#448—24no—
151#447—24no—
152#436—24no—
153#436—24no—
154#436C2318yes—
155#435—28no—
156C8×D4—24no—
157C8×D4—24no—
158C8×D4C2318yes—
159#432—21no—
160#432—21no—
161#432C2312yes—
162#432C2312yes—
163#431—21no—
164#431—21no—
165#431—21no—
166#431—21no—
167#431D412yes—
168#431D412yes—
169#431C2312yes—
170#431C2312yes—
171#431C2312yes—
172#431C2312yes—
173#431C2311yescharacteristic
174#431C2311yescharacteristic
175#431C2312yes—
176C2×#107—24no—
177#423C2312yes—
178#423C2312yes—
179C2×C4⋊C8D412yes—
180C2×C4⋊C8C2312yes—
181#417—24no—
182C2×#103—24no—
183C2×#102—24no—
184#408—24no—
185#408—28no—
186#407—21no—
187#407—21no—
188#407—24no—
189C2×C22⋊C8—21no—
190C2×C22⋊C8—24no—
191C2×C22⋊C8D412yes—
192C4×M16—21no—
193C4×M16—21no—
194C2×C4×C8—21no—
195C2×C4×C8C2312yes—
196#335C2318yes—
197#334C2318yes—
198#332C2318yes—
199#332C23116yes—
200#331C2318yes—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—22no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—41no—
208C2×#41—41no—
209C2×#41—84no—
210C2×#41—84no—
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8—41no—
219C22×Q8—84no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—41no—
227C22×D4C2×D412yes—
228C23×C4—22no—
229C23×C4—41no—
230C23×C4—41no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4#4111yescharacteristic
234AGL(1,ℤ/8)—88no—
235C2×SD16—216no—
236C2×SD16—28no—
237C2×SD16—88no—
238C2×D8—28no—
239#131—88no—
240C2×M16—22no—
241C2×M16—22no—
242C2×M16—28no—
243C22×C8—22no—
244C22×C8—28no—
245C22×C8C2×D414yes—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—21no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—21no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—21no—
260C4⋊Q8—41no—
261C4⋊Q8—41no—
262C4⋊Q8—41no—
263C4⋊Q8—41no—
264C4⋊Q8—41no—
265C4⋊Q8—41no—
266C4⋊Q8C2×D412yes—
267C4⋊Q8C2×D412yes—
268C4⋊Q8C2414yes—
269C4⋊Q8C2412yes—
270#127—21no—
271#127—21no—
272#127—21no—
273#127—21no—
274#127—21no—
275#127C2×D412yes—
276#127C2×D412yes—
277#127C2412yes—
278#124—22no—
279#124—22no—
280#124—22no—
281#124—22no—
282#124—22no—
283#124—22no—
284#124—41no—
285#124—41no—
286#121—22no—
287#121—22no—
288#121—22no—
289#121—22no—
290#121—22no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—22no—
295#121—22no—
296#121—41no—
297#121—41no—
298#121—41no—
299#121—41no—
300#121C2×D414yes—
301#121C2×D414yes—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—41no—
313C4×D4—41no—
314C4×D4—41no—
315C4×D4—41no—
316C4×D4C2×D414yes—
317C4×D4C2414yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—41no—
329C2×C4⋊C4—41no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4C2×D412yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—41no—
337C2×C22⋊C4—41no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4111yescharacteristic
345C2×C42#4111yescharacteristic
346C2×C42#4111yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—28no—
351#107—28no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355C4⋊C8C2×D412yes—
356C4⋊C8C2×D414yes—
357C4⋊C8C2×D412yes—
358C4⋊C8C2414yes—
359C4⋊C8C2414yes—
360C4⋊C8C2414yes—
361#103—216no—
362#103—28no—
363#102—28no—
364#102—28no—
365#102—28no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—28no—
373C22⋊C8—28no—
374C22⋊C8C2×D412yes—
375C22⋊C8C2×D412yes—
376C22⋊C8C2×D414yes—
377#97—22no—
378#97—22no—
379C4×C8—22no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8C2×D414yes—
384C4×C8C2412yes—
385C4×C8C2412yes—
386C24—22no—
387C24—41no—
388C24#12111yescharacteristic
389#41—48no—
390#41—48no—
391#41—84no—
392#41—84no—
393#41—84no—
394#41—84no—
395#41—84no—
396#41—84no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—41no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—48no—
406C2×Q8—84no—
407C2×Q8—84no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—84no—
411C2×Q8—84no—
412C2×Q8—84no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—28no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—44no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4C22×D414yes—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4#12111yescharacteristic
453C22×C4#12111yescharacteristic
454C22×C4#12111yescharacteristic
455C22×C4#12111yescharacteristic
456C22×C4#12111yescharacteristic
457C22×C4#12111yescharacteristic
458C22×C4#12111yescharacteristic
459C22×C4#12111yescharacteristic
460C22×C4#12111yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470SD16—816no—
471SD16—88no—
472D8—88no—
473M16—44no—
474M16—44no—
475M16—88no—
476C2×C8—28no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—44no—
481C2×C8—88no—
482C2×C8C22×D414yes—
483C2×C8C22×D414yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C22×D414yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C42#12111yescharacteristic
509C42#12111yescharacteristic
510C42#12111yescharacteristic
511C42#12111yescharacteristic
512C42#12111yescharacteristic
513C42#12111yescharacteristic
514C42C22×D411yescharacteristic
515C42C22×D411yescharacteristic
516C42C22×D411yescharacteristic
517C42C22×D411yescharacteristic
518C42C22×D411yescharacteristic
519C42C22×D411yescharacteristic
520C42C2×#4111yescharacteristic
521C42C2×#4111yescharacteristic
522C42C2×#4111yescharacteristic
523C42C2511yescharacteristic
524C23—22no—
525C23—22no—
526C23—24no—
527C23—41no—
528C23—44no—
529C23—42no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23C2×#12111yescharacteristic
536C23C2×#12111yescharacteristic
537C23C2×#12111yescharacteristic
538Q8—44no—
539Q8—48no—
540Q8—44no—
541Q8—82no—
542Q8—84no—
543Q8—84no—
544Q8—82no—
545Q8—84no—
546D4—28no—
547D4—48no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—82no—
554D4—82no—
555D4—82no—
556D4—82no—
557D4—84no—
558D4—84no—
559D4—84no—
560D4—82no—
561D4—82no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4C2×#12111yescharacteristic
591C2×C4C2×#12111yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4C2×#12111yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4#57614yes—
600C2×C4#57714yes—
601C2×C4C23×D411yescharacteristic
602C2×C4C23×D411yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C8—24no—
605C8—24no—
606C8—44no—
607C8—44no—
608C8—44no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—164no—
624C22—164no—
625C22#304212yes—
626C22#305812yes—
627C22C22×#12111yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—44no—
636C4—82no—
637C4—84no—
638C4#304212yes—
639C4#305812yes—
640C4#306912yes—
641C4C2×#57612yes—
642C4C2×#57712yes—
643C2—22no—
644C2—22no—
645C2—41no—
646C2—84no—
647C2—162no—
648C2C2×#304211yescharacteristic
649C2C2×#305811yescharacteristic
650C2C2×#306911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60803C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3226C2semidirect
3. C2×#3206C2semidirect
4. #60236C2semidirect
5. #19514C2semidirect
6. #19428C2semidirect
7. #19392C2semidirect
8. #19198C2semidirect
9. #19192C2semidirect
10. #17803C2semidirect
11. #17742C2semidirect
12. #17717C2semidirect
20. #3226C22semidirect
28. #3206C22semidirect
30. C2×#500C22semidirect
35. C2×#486C22semidirect
39. C2×#461C22semidirect
42. #2981C22semidirect
44. #2975C22semidirect
46. #2972C22semidirect
54. #1701C22semidirect
55. #1693C22semidirect
56. #1681C22semidirect
57. #1673C22semidirect
58. #1572C22semidirect
59. #1571C22semidirect
60. #1570C22semidirect
61. #1564C22semidirect
62. #1556C22semidirect
63. #1553C22semidirect
64. #1549C22semidirect
65. #1549C22semidirect
66. #1546C22semidirect
67. #1545C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#608032
C2×#32062
#602362
#195142
#194282
#193922
#191982
#191922
#178032
#177422
#177172