Finite Group Databasebeta

← #6355338

Subgroups — #6355338

Order 512.

1740 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60947C211yesmaximal, characteristic, complemented
2#60558C211yesmaximal, characteristic
3C2×#3110C211yesmaximal, characteristic, complemented
4C2×#2975C211yesmaximal, characteristic, complemented
5#19431C214yesmaximal, complemented
6#19428C212yesmaximal, complemented
7#19426C212yesmaximal, complemented
8#19424C214yesmaximal, complemented
9#19423C212yesmaximal, complemented
10#19421C212yesmaximal, complemented
11#19284C214yesmaximal, complemented
12#19282C214yesmaximal, complemented
13#17816C211yesmaximal, characteristic, complemented
14#17715C211yesmaximal, characteristic, complemented
15#17713C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3248—22no—
20#3231C2212yescomplemented
21#3229C2211yescharacteristic, complemented
22#3226C2211yescharacteristic, complemented
23#3210C2212yescomplemented
24#3209C2211yescharacteristic, complemented
25#3206C2211yescharacteristic, complemented
26#3190—22no—
27#3170C2212yescomplemented
28#3167C2212yescomplemented
29#3114—22no—
30#3110—22no—
31#3110—22no—
32#3110C2214yes—
33#3110C2212yes—
34#3110C2214yes—
35#3109C2212yes—
36#3108—22no—
37#3107—22no—
38#3105—22no—
39#3101C2212yes—
40#3100C2212yes—
41#3092—21no—
42C2×#484—21no—
43#3077C2212yes—
44C2×#474C2212yes—
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×#440—22no—
56C2×#436—22no—
57C2×#436C2214yes—
58#2920C2211yescharacteristic
59C2×#432C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C2×#431C2211yescharacteristic
62C2×#407—22no—
63#1703C2218yescomplemented
64#1701C2218yescomplemented
65#1699C2218yescomplemented
66#1699C2218yescomplemented
67#1693C2218yescomplemented
68#1691C2218yescomplemented
69#1689C2218yescomplemented
70#1685C2218yescomplemented
71#1572C2214yescomplemented
72#1571C2212yescomplemented
73#1570C2212yescomplemented
74#1568C2214yescomplemented
75#1554C2212yescomplemented
76#1553C2212yescomplemented
77#1552C2212yescomplemented
78#1552C2214yescomplemented
79#1551C2212yescomplemented
80#1549C2212yescomplemented
81#1546C2212yescomplemented
82#1544C2214yescomplemented
83#1542C2212yescomplemented
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#533C2312yes—
99#533C2311yescharacteristic
100#533C2311yescharacteristic
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#498C2312yescomplemented
141#494C2312yescomplemented
142#492C2312yescomplemented
143#487C2312yescomplemented
144#487C2312yescomplemented
145#487C2312yescomplemented
146#486C2312yescomplemented
147#485—24no—
148#484—24no—
149#484—24no—
150#484—22no—
151#483—22no—
152#483—22no—
153#481—22no—
154#480—22no—
155#480—22no—
156#479—24no—
157#478—24no—
158#476C2314yes—
159#475—22no—
160#474—24no—
161#474—22no—
162#474C2314yes—
163#474C2318yes—
164#467—24no—
165#465—24no—
166#464—24no—
167#463—24no—
168#462—22no—
169#461—24no—
170#461—22no—
171#461C2314yes—
172#461C2318yes—
173#459C2314yes—
174#452—24no—
175#451—22no—
176#450—22no—
177#450—22no—
178#450—24no—
179#450—24no—
180#448—22no—
181#447—22no—
182#447—22no—
183#442—24no—
184#440—28no—
185#440—28no—
186#440—24no—
187C4×Q16—24no—
188C4×SD16—24no—
189C4×SD16—24no—
190C4×D8—24no—
191#436—24no—
192#436—24no—
193#436—28no—
194#436—24no—
195#436—24no—
196#436C2318yes—
197#436C2318yes—
198#432—21no—
199#432—21no—
200#432—21no—
201#432C2312yes—
202#432C2312yes—
203#432C2312yes—
204#431—21no—
205#431—21no—
206#431—21no—
207#431D412yes—
208#431D412yes—
209#431C2312yes—
210#431C2312yes—
211#431C2312yes—
212#431C2311yescharacteristic
213#431C2311yescharacteristic
214#431C2312yes—
215#428—22no—
216C2×#107—22no—
217#423C2312yes—
218C2×C4⋊C8D412yes—
219C2×C4⋊C8C2312yes—
220C2×C4⋊C8C2312yes—
221#418—22no—
222#417—22no—
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×#97—21no—
235C2×C4×C8—21no—
236C2×C4×C8C2312yes—
237#335C2318yescomplemented
238#334C2318yescomplemented
239#333C2314yescomplemented
240#332C2314yescomplemented
241#332C2314yescomplemented
242#332C2318yescomplemented
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×Q16—24no—
271C2×SD16—28no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—48no—
275C2×D8—24no—
276C2×M16—22no—
277C2×M16—28no—
278C2×M16—28no—
279C2×M16—88no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8C2×D414yes—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8C2412yes—
290C4⋊Q8C2412yes—
291C4⋊Q8C2411yescharacteristic
292C4⋊Q8C2411yescharacteristic
293C4⋊Q8C2411yescharacteristic
294#127C2411yescharacteristic
295#126—41no—
296#126—41no—
297#125—21no—
298#125—21no—
299#125—22no—
300#125—22no—
301#125—21no—
302#125—41no—
303#125—41no—
304#124—21no—
305#124—21no—
306#124—21no—
307#123—41no—
308#123—41no—
309#123—41no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C22⋊Q8—41no—
321C22⋊Q8—41no—
322C22⋊Q8—41no—
323C22⋊Q8—41no—
324#121—21no—
325#121—21no—
326#121—21no—
327#121—21no—
328#121—21no—
329#121—21no—
330#120—41no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8—41no—
342C4×Q8—41no—
343C4×Q8—41no—
344C4×Q8—41no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—41no—
364C4×D4—41no—
365C4×D4—41no—
366C4×D4—41no—
367C4×D4—41no—
368C4×D4—41no—
369C4×D4C2×D414yes—
370C4×D4C2×D414yes—
371C4×D4C2×D414yes—
372C4×D4C2414yes—
373#117—21no—
374#117—21no—
375#117—22no—
376#117—22no—
377#117—21no—
378#117—41no—
379#117—41no—
380#117—41no—
381#117—41no—
382#117—41no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—41no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2×D412yes—
394C2×C4⋊C4C2×D412yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—41no—
402C2×C42—41no—
403C2×C42—41no—
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42#4111yescharacteristic
411C2×C42#4111yescharacteristic
412C2×C42#4111yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416#107—24no—
417#107—28no—
418#107—24no—
419#106—24no—
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—24no—
433#103—28no—
434#103—24no—
435#102—28no—
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—
452#97—22no—
453#97—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—82no—
469#41—82no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—21no—
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—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—48no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—41no—
510C22×C4—41no—
511C22×C4—41no—
512C22×C4—84no—
513C22×C4—84no—
514C22×C4—84no—
515C22×C4—84no—
516C22×C4—84no—
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
536Q16—84no—
537SD16—416no—
538SD16—48no—
539SD16—84no—
540D8—48no—
541M16—44no—
542M16—88no—
543M16—88no—
544C2×C8—24no—
545C2×C8—24no—
546C2×C8—24no—
547C2×C8—44no—
548C2×C8—44no—
549C2×C8—44no—
550C2×C8C22×D414yes—
551C2×C8C22×D414yes—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—41no—
566C4⋊C4—41no—
567C4⋊C4—41no—
568C4⋊C4—41no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—41no—
574C4⋊C4—41no—
575C4⋊C4—41no—
576C4⋊C4—41no—
577C4⋊C4—41no—
578C4⋊C4—41no—
579C4⋊C4C22×D414yes—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—24no—
583C22⋊C4—24no—
584C22⋊C4—42no—
585C22⋊C4—42no—
586C22⋊C4—41no—
587C22⋊C4—41no—
588C22⋊C4—41no—
589C22⋊C4—41no—
590C22⋊C4—41no—
591C22⋊C4—41no—
592C22⋊C4—41no—
593C22⋊C4—41no—
594C22⋊C4—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—42no—
599C42—42no—
600C42#12111yescharacteristic
601C42#12111yescharacteristic
602C42#12111yescharacteristic
603C42#12111yescharacteristic
604C42#12111yescharacteristic
605C42#12111yescharacteristic
606C42C22×D411yescharacteristic
607C42C22×D411yescharacteristic
608C42C22×D411yescharacteristic
609C42C22×D411yescharacteristic
610C42C22×D411yescharacteristic
611C42C22×D411yescharacteristic
612C42C2×#4111yescharacteristic
613C42C2×#4111yescharacteristic
614C42C2×#4111yescharacteristic
615C42C2511yescharacteristic
616C23—22no—
617C23—22no—
618C23—24no—
619C23—41no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—41no—
626C23—82no—
627C23—82no—
628C23—82no—
629C23—82no—
630C23—816no—
631C23C2×#12111yescharacteristic
632C23C2×#12111yescharacteristic
633C23C2×#12111yescharacteristic
634Q8—44no—
635Q8—42no—
636Q8—44no—
637Q8—42no—
638Q8—81no—
639Q8—82no—
640Q8—82no—
641D4—24no—
642D4—24no—
643D4—44no—
644D4—44no—
645D4—42no—
646D4—44no—
647D4—42no—
648D4—81no—
649D4—82no—
650D4—82no—
651D4—82no—
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—22no—
661C2×C4—22no—
662C2×C4—24no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—48no—
671C2×C4—44no—
672C2×C4—48no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—82no—
681C2×C4—82no—
682C2×C4—82no—
683C2×C4—82no—
684C2×C4—82no—
685C2×C4—82no—
686C2×C4—82no—
687C2×C4C2×#12111yescharacteristic
688C2×C4C2×#12111yescharacteristic
689C2×C4C2×#12111yescharacteristic
690C2×C4C2×#12111yescharacteristic
691C2×C4C2×#12111yescharacteristic
692C2×C4C2×#12111yescharacteristic
693C2×C4C2×#12111yescharacteristic
694C2×C4C2×#12111yescharacteristic
695C2×C4C2×#12111yescharacteristic
696C2×C4#57514yes—
697C2×C4C23×D411yescharacteristic
698C2×C4C23×D411yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C8—24no—
701C8—24no—
702C8—44no—
703C8—44no—
704C8—44no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—88no—
717C22—88no—
718C22—88no—
719C22—81no—
720C22—81no—
721C22—81no—
722C22#305412yes—
723C22#306912yes—
724C22C22×#12111yescharacteristic
725C4—21no—
726C4—21no—
727C4—21no—
728C4—21no—
729C4—21no—
730C4—21no—
731C4—21no—
732C4—21no—
733C4—21no—
734C4—44no—
735C4—81no—
736C4—82no—
737C4—81no—
738C4—82no—
739C4—81no—
740C4#305412yes—
741C4C2×#57512yes—
742C4#358512yes—
743C2—22no—
744C2—22no—
745C2—41no—
746C2—84no—
747C2—161no—
748C2#6042311yescharacteristic
749C2C2×#305411yescharacteristic
750C2C2×#306911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60558C2
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. #60947C2semidirect
3. C2×#3110C2semidirect
4. C2×#2975C2semidirect
5. #19431C2semidirect
6. #19428C2semidirect
7. #19426C2semidirect
8. #19424C2semidirect
9. #19423C2semidirect
10. #19421C2semidirect
11. #19284C2semidirect
12. #19282C2semidirect
13. #17816C2semidirect
14. #17715C2semidirect
15. #17713C2semidirect
20. #3231C22semidirect
21. #3229C22semidirect
22. #3226C22semidirect
23. #3210C22semidirect
24. #3209C22semidirect
25. #3206C22semidirect
27. #3170C22semidirect
28. #3167C22semidirect
63. #1703C22semidirect
64. #1701C22semidirect
65. #1699C22semidirect
66. #1699C22semidirect
67. #1693C22semidirect
68. #1691C22semidirect
69. #1689C22semidirect
70. #1685C22semidirect
71. #1572C22semidirect
72. #1571C22semidirect
73. #1570C22semidirect
74. #1568C22semidirect
75. #1554C22semidirect
76. #1553C22semidirect
77. #1552C22semidirect
78. #1552C22semidirect
79. #1551C22semidirect
80. #1549C22semidirect
81. #1546C22semidirect
82. #1544C22semidirect
83. #1542C22semidirect
84. #1540C22semidirect
139. #500C23semidirect
140. #498C23semidirect
141. #494C23semidirect
142. #492C23semidirect
143. #487C23semidirect
144. #487C23semidirect
145. #487C23semidirect
146. #486C23semidirect
237. #335C23semidirect
238. #334C23semidirect
239. #333C23semidirect
240. #332C23semidirect
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
#609472
#605582
C2×#31102
C2×#29752
#194312
#194282
#194262
#194242
#194232
#194212
#192842
#192822
#178162
#177152
#177132