Finite Group Databasebeta

← #6353739

Subgroups — #6353739

Order 512.

1851 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60941C211yesmaximal, characteristic, complemented
2#60940C211yesmaximal, characteristic
3#60618C211yesmaximal, characteristic, complemented
4C2×#2935C211yesmaximal, characteristic, complemented
5#19237C214yesmaximal, complemented
6#19236C214yesmaximal, complemented
7#19235C214yesmaximal, complemented
8#19234C214yesmaximal, complemented
9#19142C214yesmaximal, complemented
10#19110C214yesmaximal, complemented
11#17787C211yesmaximal, characteristic, complemented
12C2×#1542C211yesmaximal, characteristic, complemented
13C2×#1540C211yesmaximal, characteristic, complemented
14C22×#131—21no—
15C2×#533C2211yescharacteristic
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3246C2214yes—
19#3245C2214yes—
20#3229—21no—
21#3227C2212yescomplemented
22#3226—21no—
23#3226C2212yes—
24#3208—21no—
25#3206C2212yescomplemented
26#3205—21no—
27#3205C2212yes—
28#3187—22no—
29#3186—22no—
30#3169—22no—
31#3167—22no—
32#3142—22no—
33#3141C2212yescomplemented
34#3141C2212yescomplemented
35#3140—22no—
36#3139C2212yescomplemented
37#3138C2212yescomplemented
38#3135—22no—
39#3131C2214yes—
40#3103—22no—
41#3101C2214yescomplemented
42#3100—22no—
43#3100C2214yes—
44#2935—22no—
45#2935—22no—
46#2935C2214yes—
47#2935C2214yescomplemented
48C2×#436C2212yescomplemented
49C2×#435—22no—
50C2×C8×D4C2212yescomplemented
51C2×#431C2211yescharacteristic
52C2×#431C2211yescharacteristic
53C22×C4⋊C8C2211yescharacteristic
54#2896—21no—
55#2896—21no—
56C2×#408—21no—
57C2×C4×M16C2211yescharacteristic
58#1703C2218yescomplemented
59#1701C2218yescomplemented
60#1699C2218yescomplemented
61#1697C2218yescomplemented
62#1683C2218yescomplemented
63#1681C2218yescomplemented
64#1679C2218yescomplemented
65#1677C2218yescomplemented
66#1565C2214yescomplemented
67#1565C2214yescomplemented
68#1564C2214yescomplemented
69#1542C2214yescomplemented
70#1542C2212yescomplemented
71#1542C2212yescomplemented
72C2×#333C2212yescomplemented
73#1540C2212yes—
74#1540C2212yes—
75#1540C2214yescomplemented
76C2×#332C2212yescomplemented
77C2×#332C2212yescomplemented
78C2×#331C2212yescomplemented
79C22×#41—21no—
80C22×#41—21no—
81C2×#131—24no—
82C2×#131—24no—
83C2×#131—48no—
84C22×M16—22no—
85C22×M16D411yescharacteristic
86C22×M16D411yescharacteristic
87C23×C8—21no—
88C23×C8—22no—
89#533—21no—
90#533—21no—
91#533—21no—
92#533—21no—
93#533—21no—
94#533—21no—
95#533C2312yes—
96#533C2312yes—
97#532—21no—
98#532—21no—
99#532—21no—
100#532—21no—
101#532—21no—
102#532—21no—
103#532—21no—
104#532—21no—
105#532—21no—
106#532—21no—
107#532D412yes—
108#532D412yes—
109#532C2312yes—
110#532C2312yes—
111#532C2312yes—
112#532C2312yes—
113C2×C4⋊Q8D411yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#127D411yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#125—21no—
120C2×#124—21no—
121C2×#124—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×#121—21no—
125C2×#121—21no—
126C2×#121—21no—
127C2×#121D412yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C22×C42C2311yescharacteristic
136#500—22no—
137#498—22no—
138#493—22no—
139#492—22no—
140#491—24no—
141#490—24no—
142#489—24no—
143#489—24no—
144#488—24no—
145#488—24no—
146#487—22no—
147#487C2314yes—
148#487C2314yes—
149#486—22no—
150#486—22no—
151#486—22no—
152#486C2314yes—
153#486C2314yes—
154#483—24no—
155#481—24no—
156#480—24no—
157#480—24no—
158#476C2318yes—
159#475—24no—
160#474—24no—
161#474C2318yes—
162#466—28no—
163#465—28no—
164#461C2318yes—
165#460—24no—
166#459—24no—
167#459C2318yes—
168#450—24no—
169#448—24no—
170#447—24no—
171#447—24no—
172C4×SD16—28no—
173C4×D8—28no—
174#436—24no—
175#436C2318yes—
176#435—28no—
177#435—28no—
178C8×D4—24no—
179C8×D4C2318yes—
180#431—21no—
181#431—21no—
182#431—21no—
183#431—21no—
184#431C2312yes—
185#431C2312yes—
186#431C2312yes—
187#431C2312yes—
188C2×C4⋊C8—21no—
189C2×C4⋊C8—21no—
190C2×C4⋊C8—21no—
191C2×C4⋊C8C2312yes—
192C2×C4⋊C8C2312yes—
193C2×C4⋊C8C2312yes—
194C2×C4⋊C8C2312yes—
195C2×C4⋊C8C2311yescharacteristic
196C2×C4⋊C8C2311yescharacteristic
197#419—22no—
198#419—22no—
199#418—22no—
200#418—22no—
201#417—22no—
202#417—22no—
203#417—22no—
204#417—22no—
205#416—24no—
206#416—24no—
207#408—24no—
208#408—24no—
209C2×C22⋊C8—21no—
210C2×C22⋊C8—21no—
211C2×C22⋊C8—22no—
212C2×C22⋊C8—22no—
213C4×M16D412yes—
214C4×M16D412yes—
215C4×M16C2312yes—
216C4×M16C2312yes—
217C2×#97D412yes—
218C2×C4×C8—21no—
219C2×C4×C8—21no—
220C2×C4×C8C2312yes—
221#335C2318yes—
222#334C2318yes—
223#333C2314yes—
224#333C2314yes—
225#332C2314yes—
226#332C2314yes—
227#332C2314yes—
228#332C2314yes—
229#331C2314yes—
230#331C2314yes—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—41no—
238C2×#41—41no—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—44no—
242C2×#41—44no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—41no—
254C22×D4C2×D412yes—
255C23×C4—22no—
256C23×C4—41no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4#4111yescharacteristic
260C2×SD16—28no—
261C2×D8—28no—
262#131—416no—
263#131—416no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—28no—
267C2×M16—48no—
268C2×M16C2×D412yes—
269C2×M16C2×D412yes—
270C2×M16C2×D412yes—
271C2×M16C2×D412yes—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—28no—
279C22×C8—48no—
280C22×C8—44no—
281C22×C8C2×D412yes—
282C22×C8C2×D412yes—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C2×D412yes—
295C4⋊Q8C2×D412yes—
296C4⋊Q8C2412yes—
297C4⋊Q8C2412yes—
298C4⋊Q8C2412yes—
299#127—21no—
300#127—21no—
301#127—21no—
302#127—21no—
303#127—21no—
304#127C2×D412yes—
305#127C2×D412yes—
306#127C2412yes—
307#125—22no—
308#125—22no—
309#125—41no—
310#125—41no—
311#124—22no—
312#124—22no—
313#124—22no—
314#124—22no—
315#124—22no—
316#124—22no—
317#124—41no—
318#124—41no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—41no—
324C22⋊Q8—41no—
325C22⋊Q8—41no—
326C22⋊Q8—41no—
327#121—22no—
328#121—22no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—22no—
333#121—22no—
334#121—22no—
335#121—22no—
336#121—22no—
337#121—41no—
338#121—41no—
339#121—41no—
340#121—41no—
341#121C2×D414yes—
342#121C2×D414yes—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—41no—
348C4×Q8—41no—
349C4×Q8—41no—
350C4×Q8—41no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—41no—
362C4×D4—41no—
363C4×D4—41no—
364C4×D4—41no—
365C4×D4C2×D414yes—
366C4×D4C2414yes—
367#117—22no—
368#117—22no—
369#117—41no—
370#117—41no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4C2×D412yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—41no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4—41no—
386C2×C42—41no—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42#4111yescharacteristic
394C2×C42#4111yescharacteristic
395C2×C42#4111yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—28no—
400#106—28no—
401C4⋊C8—21no—
402C4⋊C8—22no—
403C4⋊C8—22no—
404C4⋊C8—21no—
405C4⋊C8—22no—
406C4⋊C8—22no—
407C4⋊C8C2412yes—
408C4⋊C8C2412yes—
409C4⋊C8C2414yes—
410C4⋊C8C2414yes—
411#103—24no—
412#103—24no—
413#103—24no—
414#103—24no—
415#103—28no—
416#102—28no—
417#102—28no—
418#102—24no—
419#102—24no—
420#102—24no—
421#102—24no—
422#102—28no—
423C22⋊C8—22no—
424C22⋊C8—22no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8—28no—
428C22⋊C8—28no—
429#97C2×D414yes—
430#97C2×D414yes—
431C4×C8—22no—
432C4×C8—22no—
433C4×C8—22no—
434C4×C8—22no—
435C4×C8C2×D414yes—
436C4×C8C2414yes—
437C24—22no—
438C24—41no—
439C24#12111yescharacteristic
440#41—44no—
441#41—44no—
442#41—44no—
443#41—44no—
444#41—48no—
445#41—48no—
446#41—44no—
447#41—44no—
448#41—44no—
449#41—44no—
450#41—84no—
451#41—84no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—41no—
460C2×Q8—44no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—82no—
464C2×Q8—82no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—44no—
476C2×D4—48no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—44no—
485C2×D4—82no—
486C2×D4—82no—
487C2×D4—82no—
488C2×D4—84no—
489C2×D4C22×D414yes—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—81no—
502C22×C4—84no—
503C22×C4#12111yescharacteristic
504C22×C4#12111yescharacteristic
505C22×C4#12111yescharacteristic
506C22×C4#12111yescharacteristic
507C22×C4#12111yescharacteristic
508C22×C4#12111yescharacteristic
509C22×C4#12111yescharacteristic
510C22×C4#12111yescharacteristic
511C22×C4#12111yescharacteristic
512C22×C4#13514yes—
513C22×C4AGL(1,ℤ/8)14yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C2×#4111yescharacteristic
521C22×C4C2×#4111yescharacteristic
522C22×C4C2×#4111yescharacteristic
523SD16—88no—
524D8—88no—
525M16—24no—
526M16—24no—
527M16—24no—
528M16—24no—
529M16—416no—
530C2×C8—24no—
531C2×C8—24no—
532C2×C8—24no—
533C2×C8—24no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—416no—
537C2×C8—42no—
538C2×C8—42no—
539C2×C8—44no—
540C2×C8—44no—
541C2×C8—82no—
542C2×C8—82no—
543C2×C8C22×D414yes—
544C2×C8C22×D414yes—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4—41no—
558C4⋊C4—41no—
559C4⋊C4—41no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4C22×D414yes—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—42no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C42—42no—
576C42—42no—
577C42#12111yescharacteristic
578C42#12111yescharacteristic
579C42#12111yescharacteristic
580C42#12111yescharacteristic
581C42#12111yescharacteristic
582C42#12111yescharacteristic
583C42C22×D411yescharacteristic
584C42C22×D411yescharacteristic
585C42C22×D411yescharacteristic
586C42C22×D411yescharacteristic
587C42C22×D411yescharacteristic
588C42C22×D411yescharacteristic
589C42C2×#4111yescharacteristic
590C42C2×#4111yescharacteristic
591C42C2×#4111yescharacteristic
592C42C2511yescharacteristic
593C23—24no—
594C23—41no—
595C23—41no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—82no—
601C23—82no—
602C23—82no—
603C23#46314yes—
604C23C2×#12111yescharacteristic
605C23C2×#12111yescharacteristic
606C23C2×#12111yescharacteristic
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—42no—
611Q8—42no—
612Q8—82no—
613Q8—82no—
614Q8—82no—
615Q8—82no—
616D4—28no—
617D4—44no—
618D4—48no—
619D4—48no—
620D4—44no—
621D4—44no—
622D4—44no—
623D4—42no—
624D4—42no—
625D4—82no—
626D4—82no—
627D4—82no—
628D4—82no—
629D4—82no—
630D4—84no—
631D4—84no—
632D4—82no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—48no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4#46314yes—
659C2×C4#46314yes—
660C2×C4#46314yes—
661C2×C4C2×#12111yescharacteristic
662C2×C4C2×#12111yescharacteristic
663C2×C4C2×#12111yescharacteristic
664C2×C4C2×#12111yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4C2×#12111yescharacteristic
667C2×C4C2×#12111yescharacteristic
668C2×C4C2×#12111yescharacteristic
669C2×C4C2×#12111yescharacteristic
670C2×C4C2×#13512yes—
671C2×C4C2×#13512yes—
672C2×C4C2×#13512yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4C2×AGL(1,ℤ/8)12yes—
675C2×C4C2×AGL(1,ℤ/8)12yes—
676C2×C4C23×D411yescharacteristic
677C2×C4C23×D411yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C8—24no—
680C8—24no—
681C8—44no—
682C8—44no—
683C8—42no—
684C8—42no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—88no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—164no—
700C22C2×#46312yes—
701C22C2×#46312yes—
702C22C2×#46312yes—
703C22C22×#12111yescharacteristic
704C4—21no—
705C4—21no—
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—21no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4C2×#46312yes—
716C4C2×#46312yes—
717C4C2×#46312yes—
718C4C22×#13511yescharacteristic
719C4C22×AGL(1,ℤ/8)11yescharacteristic
720C4#358412yes—
721C2—22no—
722C2—22no—
723C2—22no—
724C2—84no—
725C2—162no—
726C2C22×#46311yescharacteristic
727C2#6045311yescharacteristic
728C2#6046911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutator, Frattini, cosocleC42C25
socleC22C22×#121
prime radical#60940C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60941C2semidirect
3. #60618C2semidirect
4. C2×#2935C2semidirect
5. #19237C2semidirect
6. #19236C2semidirect
7. #19235C2semidirect
8. #19234C2semidirect
9. #19142C2semidirect
10. #19110C2semidirect
11. #17787C2semidirect
12. C2×#1542C2semidirect
13. C2×#1540C2semidirect
21. #3227C22semidirect
25. #3206C22semidirect
33. #3141C22semidirect
34. #3141C22semidirect
36. #3139C22semidirect
37. #3138C22semidirect
41. #3101C22semidirect
47. #2935C22semidirect
48. C2×#436C22semidirect
50. C2×C8×D4C22semidirect
58. #1703C22semidirect
59. #1701C22semidirect
60. #1699C22semidirect
61. #1697C22semidirect
62. #1683C22semidirect
63. #1681C22semidirect
64. #1679C22semidirect
65. #1677C22semidirect
66. #1565C22semidirect
67. #1565C22semidirect
68. #1564C22semidirect
69. #1542C22semidirect
70. #1542C22semidirect
71. #1542C22semidirect
72. C2×#333C22semidirect
75. #1540C22semidirect
76. C2×#332C22semidirect
77. C2×#332C22semidirect
78. C2×#331C22semidirect

Maximal subgroups

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

subgroupindex
#609412
#609402
#606182
C2×#29352
#192372
#192362
#192352
#192342
#191422
#191102
#177872
C2×#15422
C2×#15402