Finite Group Databasebeta

← #6344736

Subgroups — #6344736

Order 512.

1687 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#60561C211yesmaximal, characteristic
2#60277C211yesmaximal, characteristic, complemented
3C2×#2981C211yesmaximal, characteristic, complemented
4#17951C214yesmaximal, complemented
5#17950C214yesmaximal, complemented
6#17948C218yesmaximal, complemented
7#17891C214yesmaximal, complemented
8#17869C214yesmaximal, complemented
9#17810C211yesmaximal, characteristic, complemented
10#17463C211yesmaximal, characteristic, complemented
11#17452C211yesmaximal, characteristic, complemented
12#17324C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#3115C414yes—
16#3112C412yes—
17#3109C412yes—
18#3106C2214yes—
19#3103C2212yes—
20#3100C2212yes—
21#3094C412yes—
22#3079C412yes—
23#3038C412yes—
24#3006C412yes—
25#2989—22no—
26#2989C2214yescomplemented
27#2983C2212yescomplemented
28#2981—22no—
29#2981—22no—
30#2981C2214yes—
31#2981C2214yescomplemented
32#2975C2212yescomplemented
33#2973—22no—
34#2965—22no—
35#2965C2212yes—
36#2965C2212yes—
37C4×AGL(1,ℤ/8)C2214yes—
38C2×#436C2212yescomplemented
39C2×#435—22no—
40C2×C8×D4C2212yescomplemented
41#2920—22no—
42#2920C2211yescharacteristic
43#2920C2211yescharacteristic
44C2×#432C2211yescharacteristic
45C2×#431C2211yescharacteristic
46C2×#431C2211yescharacteristic
47C2×#408—22no—
48#1594C2218yescomplemented
49#1592C2218yescomplemented
50#1590C2218yescomplemented
51#1589C2218yescomplemented
52#1586C2218yescomplemented
53#1585C2218yescomplemented
54#1582C2218yescomplemented
55#1581C2218yescomplemented
56#1573C2214yescomplemented
57#1572C2214yescomplemented
58#1565C2214yescomplemented
59#1491C2212yescomplemented
60#1490C2212yescomplemented
61#1488C2214yescomplemented
62#1484C2214yes—
63#1480C2214yescomplemented
64#1478C2214yescomplemented
65#1477C2214yes—
66#1454C2214yescomplemented
67#1453C2214yescomplemented
68#1451C2212yescomplemented
69C8×M16C2212yescomplemented
70C22×#41—21no—
71C22×#41C2×C411yescharacteristic
72C2×AGL(1,ℤ/8)C2×C414yes—
73C22×M16—22no—
74C23×C8—22no—
75#532—21no—
76#532—21no—
77#532—21no—
78#532—21no—
79#532C2×C412yes—
80#532C2×C412yes—
81#532C2312yes—
82#532C2312yes—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×#127C2×C411yescharacteristic
85C2×#124—21no—
86C2×#121—21no—
87C2×#121C2×C412yes—
88C4×#41—21no—
89C4×#41—21no—
90C4×#41—21no—
91C4×#41—21no—
92C4×#41C2×C412yes—
93C4×#41C2×C412yes—
94C4×#41C2312yes—
95C4×#41C2312yes—
96C2×C4×Q8C2×C411yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×C4×D4C2312yes—
104C2×#117—21no—
105C2×#117C2×C411yescharacteristic
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#482C2×C418yes—
109#481C2×C414yes—
110#480C2×C414yes—
111#477C2×C414yes—
112#476C2×C414yes—
113#475C2×C414yes—
114#474C2×C414yes—
115#460C2×C414yes—
116#460C2×C414yes—
117#459C2×C414yes—
118#459C2×C414yes—
119#449C2×C418yes—
120#448C2×C414yes—
121#447C2×C414yes—
122#446—24no—
123#446C2314yes—
124#446C2314yes—
125#442C2×C418yes—
126#440C2×C418yes—
127C4×SD16C2318yes—
128C4×D8C2318yes—
129#436—24no—
130#436—24no—
131#436C2314yes—
132#436C2314yes—
133#436C2318yes—
134#435—24no—
135#435—24no—
136#435—28no—
137#435—28no—
138#435—24no—
139#435—24no—
140C8×D4—24no—
141C8×D4C2318yes—
142#432—21no—
143#432—21no—
144#432—21no—
145#432—21no—
146#432—24no—
147#432—24no—
148#432C2×C412yes—
149#432C2×C412yes—
150#432C2312yes—
151#432C2312yes—
152#431—21no—
153#431—21no—
154#431—22no—
155#431—22no—
156#431D412yes—
157#431D412yes—
158#431C2312yes—
159#431C2312yes—
160#431C2312yes—
161#431C2312yes—
162#431C2312yes—
163#431C2312yes—
164#431C2312yes—
165#431C2312yes—
166#428C2×C414yes—
167#423—21no—
168#423—24no—
169#423C2×C412yes—
170C2×C4⋊C8D412yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173#418C2×C414yes—
174#417C2×C414yes—
175#408—28no—
176#408—28no—
177#407—21no—
178#407—24no—
179#407C2×C412yes—
180C2×C22⋊C8—21no—
181C2×C22⋊C8—21no—
182C2×C22⋊C8—22no—
183C2×C22⋊C8—22no—
184C2×C22⋊C8D412yes—
185C4×M16—22no—
186C4×M16—22no—
187C4×M16C2312yes—
188C4×M16C2312yes—
189C2×#97—21no—
190C2×C4×C8—21no—
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193#335C2318yes—
194#334C2318yes—
195Q8⋊C8C2318yes—
196D4⋊C8C2318yes—
197D4⋊C8C2318yes—
198D4⋊C8C2318yes—
199#322C2314yes—
200#322C2314yes—
201#322C2314yes—
202C82C2314yes—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—44no—
210C2×#41—84no—
211C2×#41C22×C412yes—
212C2×#41C22×C412yes—
213C22×Q8—21no—
214C22×Q8C22×C411yescharacteristic
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4C22×C412yes—
220C22×D4C22×C411yescharacteristic
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4C22×C411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4#4111yescharacteristic
228AGL(1,ℤ/8)—48no—
229C2×SD16C22×C418yes—
230C2×D8C22×C418yes—
231C2×M16—22no—
232C2×M16—28no—
233C2×M16—24no—
234C2×M16—24no—
235C2×M16—84no—
236C2×M16C22×C414yes—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—28no—
240C22×C8—84no—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8C22×C412yes—
246C4⋊Q8C22×C412yes—
247#127—21no—
248#127—21no—
249#127C22×C412yes—
250#127C22×C412yes—
251#124—22no—
252#124—22no—
253#124—22no—
254#124—22no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—22no—
260#121—22no—
261#121C22×C414yes—
262#121C22×C414yes—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8C22×C412yes—
268C4×Q8C22×C412yes—
269C4×Q8C22×C412yes—
270C4×Q8C2412yes—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—21no—
286C4×D4—22no—
287C4×D4—21no—
288C4×D4—22no—
289C4×D4C22×C412yes—
290C4×D4C22×C412yes—
291C4×D4C22×C412yes—
292C4×D4C22×C414yes—
293C4×D4C2412yes—
294C4×D4C2414yes—
295#117—21no—
296#117—21no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117C22×C412yes—
302#117C22×C412yes—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C412yes—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42#4111yescharacteristic
328C2×C42#4111yescharacteristic
329C2×C42#4111yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107C22×C418yes—
334#106C22×C418yes—
335C4⋊C8—24no—
336C4⋊C8—24no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8C22×C414yes—
342C4⋊C8C22×C414yes—
343C4⋊C8C2×D414yes—
344C4⋊C8C2×D414yes—
345C4⋊C8C2414yes—
346C4⋊C8C2414yes—
347#103C22×C418yes—
348#102C22×C418yes—
349#102C22×C418yes—
350#102C22×C418yes—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—22no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C22⋊C8—22no—
357C22⋊C8—28no—
358C22⋊C8—28no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8C22×C414yes—
362C22⋊C8C22×C414yes—
363C22⋊C8C2×D414yes—
364C22⋊C8C2×D414yes—
365#97—24no—
366#97—24no—
367#97—22no—
368#97—22no—
369#97—22no—
370#97C22×C414yes—
371C4×C8—22no—
372C4×C8—22no—
373C4×C8C2×D412yes—
374C4×C8C2×D412yes—
375C4×C8C2414yes—
376C4×C8C2414yes—
377C4×C8C2412yes—
378C4×C8C2412yes—
379C24—22no—
380C24—21no—
381C24C4×D411yescharacteristic
382#41—24no—
383#41—24no—
384#41—42no—
385#41—42no—
386#41—44no—
387#41—44no—
388#41—84no—
389#41—84no—
390#41—84no—
391#41—84no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—82no—
397C2×Q8C23×C412yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—28no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—84no—
414C2×D4C23×C412yes—
415C2×D4C23×C414yes—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—82no—
434C22×C4—81no—
435C22×C4—82no—
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C23×C411yescharacteristic
446C22×C4C23×C411yescharacteristic
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454SD16—48no—
455D8—48no—
456M16—28no—
457M16—84no—
458M16—84no—
459M16—88no—
460M16—82no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—44no—
467C2×C8—88no—
468C2×C8—82no—
469C2×C8C23×C418yes—
470C2×C8C22×D414yes—
471C2×C8C22×D414yes—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4C23×C412yes—
482C4⋊C4C23×C412yes—
483C4⋊C4C23×C414yes—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42C4×D411yescharacteristic
501C42C4×D411yescharacteristic
502C42C4×D411yescharacteristic
503C42C4×D411yescharacteristic
504C42C4×D411yescharacteristic
505C42C4×D411yescharacteristic
506C42C23×C411yescharacteristic
507C42C23×C411yescharacteristic
508C42C23×C411yescharacteristic
509C42C22×D411yescharacteristic
510C42C22×D411yescharacteristic
511C42C22×D411yescharacteristic
512C42C2×#4111yescharacteristic
513C42C2×#4111yescharacteristic
514C42C2×#4111yescharacteristic
515C42C2511yescharacteristic
516C23—22no—
517C23—22no—
518C23—24no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—81no—
526C23—81no—
527C23C2×C4×D411yescharacteristic
528C23C2×C4×D411yescharacteristic
529C23C2×C4×D411yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—42no—
533Q8—82no—
534Q8—82no—
535D4—22no—
536D4—28no—
537D4—22no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—82no—
543D4—82no—
544D4—82no—
545D4—82no—
546D4—84no—
547D4—84no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×C4×D411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×C4×D411yescharacteristic
590C2×C4C2×C4×D411yescharacteristic
591C2×C4#56814yes—
592C2×C4C24×C411yescharacteristic
593C2×C4C23×D411yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C8—24no—
596C8—24no—
597C8—28no—
598C8—44no—
599C8—44no—
600C8—44no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—44no—
615C22—82no—
616C22—88no—
617C22—84no—
618C22#293012yes—
619C22C22×C4×D411yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4#293012yes—
634C4C2×#56812yes—
635C4#358512yes—
636C2—22no—
637C2—41no—
638C2—41no—
639C2—84no—
640C2—82no—
641C2C2×#293011yescharacteristic
642C2#6016811yescharacteristic
643C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60561C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60277C2semidirect
3. C2×#2981C2semidirect
4. #17951C2semidirect
5. #17950C2semidirect
6. #17948C2semidirect
7. #17891C2semidirect
8. #17869C2semidirect
9. #17810C2semidirect
10. #17463C2semidirect
11. #17452C2semidirect
12. #17324C2semidirect
26. #2989C22semidirect
27. #2983C22semidirect
31. #2981C22semidirect
32. #2975C22semidirect
38. C2×#436C22semidirect
40. C2×C8×D4C22semidirect
48. #1594C22semidirect
49. #1592C22semidirect
50. #1590C22semidirect
51. #1589C22semidirect
52. #1586C22semidirect
53. #1585C22semidirect
54. #1582C22semidirect
55. #1581C22semidirect
56. #1573C22semidirect
57. #1572C22semidirect
58. #1565C22semidirect
59. #1491C22semidirect
60. #1490C22semidirect
61. #1488C22semidirect
63. #1480C22semidirect
64. #1478C22semidirect
66. #1454C22semidirect
67. #1453C22semidirect
68. #1451C22semidirect
69. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#605612
#602772
C2×#29812
#179512
#179502
#179482
#178912
#178692
#178102
#174632
#174522
#173242