Finite Group Databasebeta

← #6344748

Subgroups — #6344748

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60561C211yesmaximal, characteristic
2#60277C211yesmaximal, characteristic, complemented
3#60236C211yesmaximal, characteristic, complemented
4#17951C214yesmaximal, complemented
5#17950C214yesmaximal, complemented
6#17945C214yesmaximal, complemented
7#17940C214yesmaximal, complemented
8#17893C214yesmaximal, complemented
9#17873C214yesmaximal, complemented
10#17808C211yesmaximal, characteristic, complemented
11C2×#1484C211yesmaximal, characteristic, complemented
12#17447C211yesmaximal, characteristic, complemented
13#17327C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C2×C4×#41C2211yescharacteristic
16#3115C414yes—
17#3112C412yes—
18#3109C412yes—
19#3106C2214yes—
20#3103C2212yes—
21#3100C2212yes—
22#3094C412yes—
23#3079C412yes—
24#3038C412yes—
25#3006C412yes—
26#2989—22no—
27#2989C2212yes—
28#2989C2212yes—
29#2983C2212yescomplemented
30#2981C2214yes—
31#2979—22no—
32#2975C2212yescomplemented
33#2975C2212yescomplemented
34#2974—22no—
35#2973—22no—
36#2972C2212yescomplemented
37#2965—22no—
38#2965C2214yescomplemented
39C4×AGL(1,ℤ/8)C2214yes—
40#2935—22no—
41#2931C2214yescomplemented
42#2920—22no—
43#2920C2211yescharacteristic
44#2920C2211yescharacteristic
45C2×#431C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2880—22no—
48C2×C4×M16C2211yescharacteristic
49#1594C2218yescomplemented
50#1592C2218yescomplemented
51#1590C2218yescomplemented
52#1589C2218yescomplemented
53#1586C2218yescomplemented
54#1585C2218yescomplemented
55#1582C2218yescomplemented
56#1581C2218yescomplemented
57#1574C2214yescomplemented
58#1572C2214yescomplemented
59#1565C2214yescomplemented
60#1491C2212yescomplemented
61#1490C2212yescomplemented
62#1489C2212yescomplemented
63#1489C2212yescomplemented
64#1484C2214yes—
65#1484C2212yescomplemented
66#1484C2212yescomplemented
67#1476C2214yes—
68C2×D4⋊C8C2214yescomplemented
69#1453C2214yescomplemented
70#1451C2212yescomplemented
71#1451C2212yescomplemented
72C8×M16C2212yescomplemented
73C8×M16C2212yescomplemented
74C22×#41—21no—
75C22×#41C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C414yes—
77C2×#131—24no—
78C22×M16—21no—
79#532—21no—
80#532—21no—
81#532—21no—
82#532—21no—
83#532C2×C412yes—
84#532C2×C412yes—
85#532C2312yes—
86#532C2312yes—
87C2×C4⋊Q8C2×C411yescharacteristic
88C2×#127C2×C411yescharacteristic
89C2×#124—21no—
90C2×#121—21no—
91C2×#121C2×C412yes—
92C4×#41—21no—
93C4×#41—21no—
94C4×#41—21no—
95C4×#41—21no—
96C4×#41C2×C412yes—
97C4×#41C2×C412yes—
98C4×#41C2312yes—
99C4×#41C2312yes—
100C2×C4×Q8C2×C411yescharacteristic
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4C2×C411yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117C2×C411yescharacteristic
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112#482C2×C418yes—
113#481C2×C414yes—
114#480C2×C414yes—
115#477C2×C414yes—
116#476C2×C414yes—
117#475C2×C414yes—
118#474C2×C414yes—
119#460C2×C414yes—
120#460C2×C414yes—
121#459C2×C414yes—
122#459C2×C414yes—
123#449C2×C418yes—
124#448C2×C414yes—
125#447C2×C414yes—
126#446—24no—
127#446C2314yes—
128#446C2314yes—
129#442C2×C418yes—
130#440C2×C418yes—
131C4×SD16C2318yes—
132C4×D8C2318yes—
133#436—24no—
134#436—24no—
135#436—24no—
136#436C2314yes—
137#436C2314yes—
138#436C2318yes—
139#435—24no—
140#435—24no—
141#435—28no—
142#435—24no—
143#435—24no—
144C8×D4—24no—
145C8×D4—24no—
146C8×D4C2318yes—
147#432—21no—
148#432—22no—
149#432—22no—
150#432—24no—
151#432C2×C412yes—
152#432C2×C412yes—
153#432C2312yes—
154#431—21no—
155#431—21no—
156#431—21no—
157#431—24no—
158#431D412yes—
159#431D412yes—
160#431C2312yes—
161#431C2312yes—
162#431C2312yes—
163#431C2312yes—
164#431C2312yes—
165#431C2311yescharacteristic
166#431C2311yescharacteristic
167#431C2312yes—
168#428C2×C414yes—
169#423—22no—
170#423—22no—
171#423C2×C412yes—
172#423C2312yes—
173C2×C4⋊C8D412yes—
174C2×C4⋊C8C2312yes—
175#418C2×C414yes—
176#417C2×C414yes—
177#408—24no—
178#408—28no—
179#407—21no—
180#407—24no—
181#407—24no—
182#407C2×C412yes—
183C2×C22⋊C8—21no—
184C2×C22⋊C8—24no—
185C2×C22⋊C8D412yes—
186C4×M16—21no—
187C4×M16—21no—
188C4×M16—21no—
189C4×M16—24no—
190C4×M16C2311yescharacteristic
191C4×M16C2311yescharacteristic
192C4×M16C2312yes—
193C4×M16C2312yes—
194C2×#97—21no—
195C2×C4×C8—21no—
196C2×C4×C8C2312yes—
197C2×C4×C8C2312yes—
198#335C2318yes—
199#334C2318yes—
200Q8⋊C8C2314yes—
201Q8⋊C8C2314yes—
202D4⋊C8C2314yes—
203D4⋊C8C2318yes—
204D4⋊C8C2318yes—
205D4⋊C8C2314yes—
206#322C2314yes—
207#322C2314yes—
208#322C2314yes—
209C82C2314yes—
210C2×#41—21no—
211C2×#41—21no—
212C2×#41—22no—
213C2×#41—22no—
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—44no—
217C2×#41—84no—
218C2×#41C22×C412yes—
219C2×#41C22×C412yes—
220C22×Q8—21no—
221C22×Q8C22×C411yescharacteristic
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C22×C412yes—
227C22×D4C22×C411yescharacteristic
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4C22×C411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4#4111yescharacteristic
235AGL(1,ℤ/8)—48no—
236C2×SD16C22×C418yes—
237C2×D8C22×C418yes—
238#131—88no—
239C2×M16—21no—
240C2×M16—22no—
241C2×M16—22no—
242C2×M16—21no—
243C2×M16—28no—
244C2×M16—24no—
245C2×M16—24no—
246C2×M16—44no—
247C2×M16C22×C414yes—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—28no—
251C22×C8C2×D414yes—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8C22×C412yes—
255C4⋊Q8C22×C412yes—
256#127—21no—
257#127—21no—
258#127C22×C412yes—
259#127C22×C412yes—
260#124—22no—
261#124—22no—
262#124—22no—
263#124—22no—
264#121—22no—
265#121—22no—
266#121—22no—
267#121—22no—
268#121—22no—
269#121—22no—
270#121C22×C414yes—
271#121C22×C414yes—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8C22×C412yes—
277C4×Q8C22×C412yes—
278C4×Q8C22×C412yes—
279C4×Q8C2412yes—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—21no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—22no—
298C4×D4C22×C412yes—
299C4×D4C22×C412yes—
300C4×D4C22×C412yes—
301C4×D4C22×C414yes—
302C4×D4C2412yes—
303C4×D4C2414yes—
304#117—21no—
305#117—21no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117C22×C412yes—
311#117C22×C412yes—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C22×C411yescharacteristic
316C2×C4⋊C4C22×C411yescharacteristic
317C2×C4⋊C4C22×C412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42C22×C411yescharacteristic
331C2×C42C22×C411yescharacteristic
332C2×C42C22×C411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4111yescharacteristic
337C2×C42#4111yescharacteristic
338C2×C42#4111yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107C22×C418yes—
343#106C22×C418yes—
344C4⋊C8—24no—
345C4⋊C8—24no—
346C4⋊C8—22no—
347C4⋊C8—22no—
348C4⋊C8C22×C414yes—
349C4⋊C8C22×C414yes—
350C4⋊C8C2×D412yes—
351C4⋊C8C2×D412yes—
352C4⋊C8C2×D414yes—
353C4⋊C8C2414yes—
354C4⋊C8C2414yes—
355#103C22×C418yes—
356#102C22×C418yes—
357#102C22×C418yes—
358#102C22×C418yes—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8—28no—
364C22⋊C8—28no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8C22×C414yes—
368C22⋊C8C22×C414yes—
369C22⋊C8C2×D412yes—
370C22⋊C8C2×D412yes—
371C22⋊C8C2×D414yes—
372#97—24no—
373#97—24no—
374#97—22no—
375#97—21no—
376#97—21no—
377#97—22no—
378#97C22×C414yes—
379C4×C8—22no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8C2×D414yes—
384C4×C8C2412yes—
385C4×C8C2412yes—
386C4×C8C2414yes—
387C4×C8C2412yes—
388C4×C8C2412yes—
389C24—21no—
390C24—22no—
391C24C4×D411yescharacteristic
392#41—24no—
393#41—24no—
394#41—42no—
395#41—42no—
396#41—48no—
397#41—44no—
398#41—44no—
399#41—84no—
400#41—84no—
401#41—84no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—82no—
407C2×Q8C23×C412yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—28no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—84no—
424C2×D4C23×C412yes—
425C2×D4C23×C414yes—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—84no—
445C22×C4C4×D411yescharacteristic
446C22×C4C4×D411yescharacteristic
447C22×C4C4×D411yescharacteristic
448C22×C4C4×D411yescharacteristic
449C22×C4C4×D411yescharacteristic
450C22×C4C4×D411yescharacteristic
451C22×C4C4×D411yescharacteristic
452C22×C4C4×D411yescharacteristic
453C22×C4C4×D411yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C411yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463SD16—48no—
464D8—48no—
465M16—24no—
466M16—28no—
467M16—44no—
468M16—42no—
469M16—44no—
470M16—84no—
471M16—84no—
472M16—88no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8—88no—
480C2×C8—82no—
481C2×C8C23×C418yes—
482C2×C8C22×D414yes—
483C2×C8C22×D414yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4C23×C412yes—
494C4⋊C4C23×C412yes—
495C4⋊C4C23×C414yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42C4×D411yescharacteristic
513C42C4×D411yescharacteristic
514C42C4×D411yescharacteristic
515C42C4×D411yescharacteristic
516C42C4×D411yescharacteristic
517C42C4×D411yescharacteristic
518C42C23×C411yescharacteristic
519C42C23×C411yescharacteristic
520C42C23×C411yescharacteristic
521C42C22×D411yescharacteristic
522C42C22×D411yescharacteristic
523C42C22×D411yescharacteristic
524C42C2×#4111yescharacteristic
525C42C2×#4111yescharacteristic
526C42C2×#4111yescharacteristic
527C42C2511yescharacteristic
528C23—22no—
529C23—22no—
530C23—24no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—44no—
535C23—81no—
536C23—81no—
537C23—82no—
538C23—82no—
539C23C2×C4×D411yescharacteristic
540C23C2×C4×D411yescharacteristic
541C23C2×C4×D411yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—42no—
545Q8—44no—
546Q8—82no—
547D4—22no—
548D4—28no—
549D4—22no—
550D4—42no—
551D4—44no—
552D4—44no—
553D4—48no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—82no—
558D4—82no—
559D4—84no—
560C2×C4—22no—
561C2×C4—22no—
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—24no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—84no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×C4×D411yescharacteristic
603C2×C4#56814yes—
604C2×C4C24×C411yescharacteristic
605C2×C4C23×D411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C8—24no—
608C8—24no—
609C8—24no—
610C8—28no—
611C8—42no—
612C8—44no—
613C8—44no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—82no—
629C22—84no—
630C22—164no—
631C22#293012yes—
632C22C22×C4×D411yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4#293012yes—
646C4#295612yes—
647C4C2×#56812yes—
648C4#358512yes—
649C2—22no—
650C2—41no—
651C2—41no—
652C2—82no—
653C2—84no—
654C2C2×#293011yescharacteristic
655C2#6016811yescharacteristic
656C2C2×#295611yescharacteristic
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. #60236C2semidirect
4. #17951C2semidirect
5. #17950C2semidirect
6. #17945C2semidirect
7. #17940C2semidirect
8. #17893C2semidirect
9. #17873C2semidirect
10. #17808C2semidirect
11. C2×#1484C2semidirect
12. #17447C2semidirect
13. #17327C2semidirect
29. #2983C22semidirect
32. #2975C22semidirect
33. #2975C22semidirect
36. #2972C22semidirect
38. #2965C22semidirect
41. #2931C22semidirect
49. #1594C22semidirect
50. #1592C22semidirect
51. #1590C22semidirect
52. #1589C22semidirect
53. #1586C22semidirect
54. #1585C22semidirect
55. #1582C22semidirect
56. #1581C22semidirect
57. #1574C22semidirect
58. #1572C22semidirect
59. #1565C22semidirect
60. #1491C22semidirect
61. #1490C22semidirect
62. #1489C22semidirect
63. #1489C22semidirect
65. #1484C22semidirect
66. #1484C22semidirect
68. C2×D4⋊C8C22semidirect
69. #1453C22semidirect
70. #1451C22semidirect
71. #1451C22semidirect
72. C8×M16C22semidirect
73. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#605612
#602772
#602362
#179512
#179502
#179452
#179402
#178932
#178732
#178082
C2×#14842
#174472
#173272