Finite Group Databasebeta

← #6354483

Subgroups — #6354483

Order 512.

1741 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60846C211yesmaximal, characteristic, complemented
2#60558C211yesmaximal, characteristic
3C2×#3101C211yesmaximal, characteristic, complemented
4C2×#2975C211yesmaximal, characteristic, complemented
5#19431C214yesmaximal, complemented
6#19430C214yesmaximal, complemented
7#19429C212yesmaximal, complemented
8#19428C212yesmaximal, complemented
9#19427C212yesmaximal, complemented
10#19426C212yesmaximal, complemented
11#19142C214yesmaximal, complemented
12#19137C214yesmaximal, complemented
13#17803C211yesmaximal, characteristic, complemented
14#17718C211yesmaximal, characteristic, complemented
15#17700C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3233—22no—
20#3232—22no—
21#3231C2212yescomplemented
22#3230C2211yescharacteristic, complemented
23#3229C2211yescharacteristic, complemented
24#3228C2212yescomplemented
25#3227C2211yescharacteristic, complemented
26#3226C2211yescharacteristic, complemented
27#3160C2212yescomplemented
28#3131C2212yescomplemented
29#3114—22no—
30#3110C2212yes—
31#3109C2212yes—
32#3105—22no—
33#3101—22no—
34#3101—22no—
35#3101C2214yes—
36#3101C2214yes—
37#3101C2212yes—
38#3100C2212yes—
39#3092—21no—
40#3091—22no—
41C2×#483—21no—
42#3077C2212yes—
43#3076—22no—
44C2×#476C2212yes—
45#3036C2212yes—
46C2×#461C2212yes—
47#3003—21no—
48C2×#450—21no—
49#2975—22no—
50#2975—22no—
51#2975C2212yes—
52#2975C2212yes—
53#2975C2214yes—
54C4×AGL(1,ℤ/8)—22no—
55C2×C4×SD16—22no—
56C2×#436—22no—
57C2×#436C2214yes—
58#2920C2211yescharacteristic
59C2×#431C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C2×#431C2211yescharacteristic
62C2×#407—22no—
63#1703C2218yescomplemented
64#1701C2218yescomplemented
65#1699C2218yescomplemented
66#1697C2218yescomplemented
67#1695C2218yescomplemented
68#1693C2218yescomplemented
69#1691C2218yescomplemented
70#1691C2218yescomplemented
71#1572C2214yescomplemented
72#1571C2212yescomplemented
73#1570C2212yescomplemented
74#1564C2214yescomplemented
75#1554C2214yescomplemented
76#1548C2212yescomplemented
77#1546C2212yescomplemented
78#1544C2212yescomplemented
79#1543C2212yescomplemented
80#1542C2212yescomplemented
81#1542C2212yescomplemented
82#1542C2214yescomplemented
83#1540C2212yescomplemented
84#1540C2212yescomplemented
85C23×D4D411yescharacteristic
86C24×C4—21no—
87C2×AGL(1,ℤ/8)—22no—
88C22×SD16—22no—
89C22×M16—24no—
90#533—21no—
91#533—21no—
92#533—21no—
93#533—21no—
94#533—21no—
95#533—21no—
96#533C2311yescharacteristic
97#533C2312yes—
98#533C2311yescharacteristic
99#533C2311yescharacteristic
100#533C2312yes—
101#532C2311yescharacteristic
102#532C2311yescharacteristic
103#532C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106#529—21no—
107#529—21no—
108#529—21no—
109#529—21no—
110#529—21no—
111#529—21no—
112C2×#125—21no—
113#525—21no—
114#525—21no—
115#525—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C4×#41—21no—
119C4×#41—21no—
120C4×#41—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4D412yes—
130C2×C4×D4D412yes—
131C2×C4×D4D412yes—
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C22×C4⋊C4D411yescharacteristic
137C22×C22⋊C4—21no—
138C22×C42C2311yescharacteristic
139#500C2312yescomplemented
140#500C2312yescomplemented
141#499—24no—
142#498C2312yescomplemented
143#498C2312yescomplemented
144#488—24no—
145#487C2312yescomplemented
146#487C2312yescomplemented
147#486C2312yescomplemented
148#486C2312yescomplemented
149#485—24no—
150#484—22no—
151#484—22no—
152#483—24no—
153#483—24no—
154#483—22no—
155#483—22no—
156#482—24no—
157#481—22no—
158#481—22no—
159#480—22no—
160#480—22no—
161#479—24no—
162#478—24no—
163#478—24no—
164#477—22no—
165#476—24no—
166#476—22no—
167#476C2314yes—
168#476C2318yes—
169#475—22no—
170#474—22no—
171#474C2314yes—
172#463—24no—
173#461—24no—
174#461C2318yes—
175#461C2314yes—
176#459C2314yes—
177#450—24no—
178#450—24no—
179#450—22no—
180#450—22no—
181#447—22no—
182#447—22no—
183#442—24no—
184#440—24no—
185C4×SD16—28no—
186C4×SD16—28no—
187C4×SD16—24no—
188C4×D8—24no—
189#436—24no—
190#436—24no—
191#436—28no—
192#436—24no—
193#436—24no—
194#436C2318yes—
195#436C2318yes—
196#432—21no—
197#432C2312yes—
198#431—21no—
199#431—21no—
200#431—21no—
201#431—21no—
202#431—21no—
203#431D412yes—
204#431D412yes—
205#431C2312yes—
206#431C2312yes—
207#431C2312yes—
208#431C2312yes—
209#431C2312yes—
210#431C2311yescharacteristic
211#431C2311yescharacteristic
212#431C2312yes—
213#428—22no—
214#427—24no—
215C2×#106—22no—
216#423C2312yes—
217C2×C4⋊C8D412yes—
218C2×C4⋊C8C2312yes—
219C2×C4⋊C8C2312yes—
220#418—22no—
221#417—22no—
222#416—24no—
223C2×#103—22no—
224C2×#102—22no—
225#407—21no—
226#407—24no—
227#407—24no—
228#407—28no—
229C2×C22⋊C8—21no—
230C2×C22⋊C8—21no—
231C2×C22⋊C8—24no—
232C2×C22⋊C8D412yes—
233C4×M16—21no—
234C2×C4×C8—21no—
235C2×C4×C8—21no—
236C2×C4×C8C2312yes—
237#335C2318yescomplemented
238#334C2318yescomplemented
239#333C2314yescomplemented
240#333C2318yescomplemented
241#332C2314yescomplemented
242#332C2314yescomplemented
243#332C2318yescomplemented
244#331C2314yescomplemented
245C25—21no—
246C2×#41—21no—
247C2×#41—21no—
248C2×#41—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—44no—
255C22×D4C2×D412yes—
256C22×D4C2×D412yes—
257C22×D4C2×D412yes—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—84no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4#4111yescharacteristic
269AGL(1,ℤ/8)—48no—
270C2×SD16—28no—
271C2×SD16—24no—
272C2×SD16—48no—
273C2×D8—24no—
274C2×M16—22no—
275C2×M16—28no—
276C2×M16—28no—
277C2×M16—88no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8C2×D414yes—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8C2412yes—
288C4⋊Q8C2412yes—
289C4⋊Q8C2411yescharacteristic
290C4⋊Q8C2411yescharacteristic
291C4⋊Q8C2411yescharacteristic
292#127C2411yescharacteristic
293#126—41no—
294#126—41no—
295#125—22no—
296#125—22no—
297#125—21no—
298#125—21no—
299#125—21no—
300#125—41no—
301#125—41no—
302#124—21no—
303#124—21no—
304#124—21no—
305#123—41no—
306#123—41no—
307#123—41no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—41no—
319C22⋊Q8—41no—
320C22⋊Q8—41no—
321C22⋊Q8—41no—
322#121—21no—
323#121—21no—
324#121—21no—
325#121—21no—
326#121—21no—
327#121—21no—
328#120—41no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—21no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—41no—
340C4×Q8—41no—
341C4×Q8—41no—
342C4×Q8—41no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—41no—
362C4×D4—41no—
363C4×D4—41no—
364C4×D4—41no—
365C4×D4—41no—
366C4×D4—41no—
367C4×D4C2×D414yes—
368C4×D4C2×D414yes—
369C4×D4C2×D414yes—
370C4×D4C2414yes—
371#117—22no—
372#117—22no—
373#117—21no—
374#117—21no—
375#117—21no—
376#117—41no—
377#117—41no—
378#117—41no—
379#117—41no—
380#117—41no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—41no—
390C2×C4⋊C4C2×D412yes—
391C2×C4⋊C4C2×D412yes—
392C2×C4⋊C4C2×D412yes—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—41no—
400C2×C42—41no—
401C2×C42—41no—
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42#4111yescharacteristic
409C2×C42#4111yescharacteristic
410C2×C42#4111yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414#107—24no—
415#107—24no—
416#107—24no—
417#106—24no—
418#106—24no—
419#106—28no—
420#106—24no—
421C4⋊C8—22no—
422C4⋊C8—22no—
423C4⋊C8—22no—
424C4⋊C8C2×D412yes—
425C4⋊C8C2×D412yes—
426C4⋊C8C2×D414yes—
427C4⋊C8C2414yes—
428C4⋊C8C2414yes—
429C4⋊C8C2414yes—
430#103—24no—
431#103—24no—
432#103—28no—
433#103—24no—
434#102—28no—
435#102—24no—
436#102—24no—
437#102—24no—
438#102—24no—
439#102—24no—
440C22⋊C8—22no—
441C22⋊C8—22no—
442C22⋊C8—22no—
443C22⋊C8—22no—
444C22⋊C8—22no—
445C22⋊C8—22no—
446C22⋊C8—28no—
447C22⋊C8—28no—
448C22⋊C8C2×D412yes—
449C22⋊C8C2×D412yes—
450C22⋊C8C2×D414yes—
451#97—22no—
452C4×C8—22no—
453C4×C8—22no—
454C4×C8—22no—
455C4×C8—22no—
456C4×C8—22no—
457C4×C8C2×D414yes—
458C4×C8C2412yes—
459C4×C8C2412yes—
460C24—22no—
461C24—22no—
462C24—22no—
463C24—44no—
464C24—82no—
465C24—82no—
466C24#12111yescharacteristic
467#41—44no—
468#41—44no—
469#41—82no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—41no—
476C2×Q8—41no—
477C2×Q8—44no—
478C2×Q8—82no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—44no—
488C2×D4—44no—
489C2×D4—44no—
490C2×D4—44no—
491C2×D4—44no—
492C2×D4—41no—
493C2×D4—41no—
494C2×D4—41no—
495C2×D4C22×D414yes—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—48no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—41no—
510C22×C4—41no—
511C22×C4—84no—
512C22×C4—84no—
513C22×C4—84no—
514C22×C4—84no—
515C22×C4—84no—
516C22×C4—81no—
517C22×C4—82no—
518C22×C4#12111yescharacteristic
519C22×C4#12111yescharacteristic
520C22×C4#12111yescharacteristic
521C22×C4#12111yescharacteristic
522C22×C4#12111yescharacteristic
523C22×C4#12111yescharacteristic
524C22×C4#12111yescharacteristic
525C22×C4#12111yescharacteristic
526C22×C4#12111yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C2×#4111yescharacteristic
534C22×C4C2×#4111yescharacteristic
535C22×C4C2×#4111yescharacteristic
536SD16—416no—
537SD16—48no—
538D8—48no—
539M16—44no—
540M16—88no—
541M16—88no—
542C2×C8—28no—
543C2×C8—24no—
544C2×C8—24no—
545C2×C8—24no—
546C2×C8—44no—
547C2×C8—44no—
548C2×C8C22×D414yes—
549C2×C8C22×D414yes—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—41no—
564C4⋊C4—41no—
565C4⋊C4—41no—
566C4⋊C4—41no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—41no—
572C4⋊C4—41no—
573C4⋊C4—41no—
574C4⋊C4—41no—
575C4⋊C4—41no—
576C4⋊C4—41no—
577C4⋊C4C22×D414yes—
578C22⋊C4—24no—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—42no—
583C22⋊C4—42no—
584C22⋊C4—41no—
585C22⋊C4—41no—
586C22⋊C4—41no—
587C22⋊C4—41no—
588C22⋊C4—41no—
589C22⋊C4—41no—
590C22⋊C4—41no—
591C22⋊C4—41no—
592C22⋊C4—41no—
593C42—41no—
594C42—41no—
595C42—42no—
596C42—41no—
597C42—42no—
598C42#12111yescharacteristic
599C42#12111yescharacteristic
600C42#12111yescharacteristic
601C42#12111yescharacteristic
602C42#12111yescharacteristic
603C42#12111yescharacteristic
604C42C22×D411yescharacteristic
605C42C22×D411yescharacteristic
606C42C22×D411yescharacteristic
607C42C22×D411yescharacteristic
608C42C22×D411yescharacteristic
609C42C22×D411yescharacteristic
610C42C2×#4111yescharacteristic
611C42C2×#4111yescharacteristic
612C42C2×#4111yescharacteristic
613C42C2511yescharacteristic
614C23—22no—
615C23—22no—
616C23—24no—
617C23—41no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—41no—
624C23—82no—
625C23—82no—
626C23—82no—
627C23—82no—
628C23—816no—
629C23C2×#12111yescharacteristic
630C23C2×#12111yescharacteristic
631C23C2×#12111yescharacteristic
632Q8—44no—
633Q8—42no—
634Q8—42no—
635Q8—82no—
636Q8—81no—
637Q8—82no—
638Q8—82no—
639D4—24no—
640D4—24no—
641D4—44no—
642D4—44no—
643D4—44no—
644D4—42no—
645D4—42no—
646D4—81no—
647D4—82no—
648D4—82no—
649D4—82no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—24no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—44no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—82no—
678C2×C4—82no—
679C2×C4—82no—
680C2×C4—82no—
681C2×C4—82no—
682C2×C4—82no—
683C2×C4—82no—
684C2×C4C2×#12111yescharacteristic
685C2×C4C2×#12111yescharacteristic
686C2×C4C2×#12111yescharacteristic
687C2×C4C2×#12111yescharacteristic
688C2×C4C2×#12111yescharacteristic
689C2×C4C2×#12111yescharacteristic
690C2×C4C2×#12111yescharacteristic
691C2×C4C2×#12111yescharacteristic
692C2×C4C2×#12111yescharacteristic
693C2×C4#57514yes—
694C2×C4#57714yes—
695C2×C4C23×D411yescharacteristic
696C2×C4C23×D411yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C8—24no—
699C8—24no—
700C8—44no—
701C8—44no—
702C8—44no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—88no—
715C22—88no—
716C22—88no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22#304212yes—
721C22#305412yes—
722C22C22×#12111yescharacteristic
723C4—21no—
724C4—21no—
725C4—21no—
726C4—21no—
727C4—21no—
728C4—21no—
729C4—21no—
730C4—21no—
731C4—44no—
732C4—81no—
733C4—81no—
734C4—82no—
735C4—81no—
736C4—82no—
737C4#304212yes—
738C4#305412yes—
739C4C2×#57512yes—
740C4C2×#57712yes—
741C2—22no—
742C2—22no—
743C2—41no—
744C2—84no—
745C2—161no—
746C2C2×#304211yescharacteristic
747C2C2×#305411yescharacteristic
748C2#6046911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutator, Frattini, cosocleC42C25
socleC22C22×#121
prime radical#60558C2
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. #60846C2semidirect
3. C2×#3101C2semidirect
4. C2×#2975C2semidirect
5. #19431C2semidirect
6. #19430C2semidirect
7. #19429C2semidirect
8. #19428C2semidirect
9. #19427C2semidirect
10. #19426C2semidirect
11. #19142C2semidirect
12. #19137C2semidirect
13. #17803C2semidirect
14. #17718C2semidirect
15. #17700C2semidirect
21. #3231C22semidirect
22. #3230C22semidirect
23. #3229C22semidirect
24. #3228C22semidirect
25. #3227C22semidirect
26. #3226C22semidirect
27. #3160C22semidirect
28. #3131C22semidirect
63. #1703C22semidirect
64. #1701C22semidirect
65. #1699C22semidirect
66. #1697C22semidirect
67. #1695C22semidirect
68. #1693C22semidirect
69. #1691C22semidirect
70. #1691C22semidirect
71. #1572C22semidirect
72. #1571C22semidirect
73. #1570C22semidirect
74. #1564C22semidirect
75. #1554C22semidirect
76. #1548C22semidirect
77. #1546C22semidirect
78. #1544C22semidirect
79. #1543C22semidirect
80. #1542C22semidirect
81. #1542C22semidirect
82. #1542C22semidirect
83. #1540C22semidirect
84. #1540C22semidirect
139. #500C23semidirect
140. #500C23semidirect
142. #498C23semidirect
143. #498C23semidirect
145. #487C23semidirect
146. #487C23semidirect
147. #486C23semidirect
148. #486C23semidirect
237. #335C23semidirect
238. #334C23semidirect
239. #333C23semidirect
240. #333C23semidirect
241. #332C23semidirect
242. #332C23semidirect
243. #332C23semidirect
244. #331C23semidirect

Maximal subgroups

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

subgroupindex
#608462
#605582
C2×#31012
C2×#29752
#194312
#194302
#194292
#194282
#194272
#194262
#191422
#191372
#178032
#177182
#177002