Finite Group Databasebeta

← #6345461

Subgroups — #6345461

Order 512.

1581 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic
2#60211C211yesmaximal, characteristic, complemented
3#60210C211yesmaximal, characteristic, complemented
4#18073C212yesmaximal, complemented
5#18072C212yesmaximal, complemented
6#18071C212yesmaximal, complemented
7#18071C212yesmaximal, complemented
8#18005C214yesmaximal, complemented
9#18002C214yesmaximal, complemented
10#17937C214yesmaximal, complemented
11#17909C214yesmaximal, complemented
12C2×#1572C211yesmaximal, characteristic, complemented
13#17488C211yesmaximal, characteristic, complemented
14#17458C211yesmaximal, characteristic, complemented
15#17319C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#3444C2211yescharacteristic
18#3250C414yes—
19#3231C411yescharacteristic
20#3231C411yescharacteristic
21#3228C411yescharacteristic
22#3228C411yescharacteristic
23#3210C411yescharacteristic
24#3210C411yescharacteristic
25#3207C411yescharacteristic
26#3207C411yescharacteristic
27#3191C414yes—
28#3170C2211yescharacteristic
29#3169C2211yescharacteristic
30#3167C2211yescharacteristic
31#3167C2211yescharacteristic
32#3102C2214yes—
33#3102C2214yes—
34#2971C2212yes—
35#2970—22no—
36#2970C2212yes—
37#2969C2214yes—
38#2968—22no—
39#2967—22no—
40#2966C2214yes—
41#2965—22no—
42#2965C2212yes—
43C8×#41C2212yes—
44#2962C2212yes—
45#2962C2212yes—
46#2935C2212yes—
47#2935C2212yes—
48#2933—22no—
49#2933—22no—
50#2923—21no—
51#2922—21no—
52C2×#432C2211yescharacteristic
53C2×#431C2211yescharacteristic
54C2×#431C2211yescharacteristic
55C2×#431C2211yescharacteristic
56#2875—21no—
57C4×#131—21no—
58C2×C4×M16C2211yescharacteristic
59#1594C2214yescomplemented
60#1593C2214yescomplemented
61#1592C2214yescomplemented
62#1592C2214yescomplemented
63#1588C2214yescomplemented
64#1588C2214yescomplemented
65#1587C2214yescomplemented
66#1587C2214yescomplemented
67#1586C2214yescomplemented
68#1586C2214yescomplemented
69#1585C2214yescomplemented
70#1585C2214yescomplemented
71#1580C2214yescomplemented
72#1579C2214yescomplemented
73SD16⋊C8C2214yescomplemented
74SD16⋊C8C2214yescomplemented
75#1572C2212yes—
76#1572C2212yes—
77#1572C2212yes—
78#1572C2212yes—
79C2×#335C2212yes—
80C2×#334C2212yes—
81#1494C2212yes—
82#1494C2212yes—
83#1488C2214yes—
84#1482C2212yes—
85#1482C2212yes—
86#1481C2212yes—
87#1481C2212yes—
88#1480C2212yes—
89#1480C2212yes—
90#1476C2214yes—
91#1452C2214yes—
92#1451C2214yes—
93#1451C2212yes—
94#1451C2212yes—
95C2×#131—22no—
96C2×#131—22no—
97C22×M16D412yes—
98#533C2×C411yescharacteristic
99#533C2×C411yescharacteristic
100#533C2×C411yescharacteristic
101#533C2×C411yescharacteristic
102#533C2311yescharacteristic
103#533C2311yescharacteristic
104#532C2×C411yescharacteristic
105#532C2×C411yescharacteristic
106#532C2×C411yescharacteristic
107#532C2×C411yescharacteristic
108#532C2311yescharacteristic
109#532C2311yescharacteristic
110#529—21no—
111#529—21no—
112#529—21no—
113#529—21no—
114#529—21no—
115#529—21no—
116#529—21no—
117#529—21no—
118#529C2×C412yes—
119#529C2×C412yes—
120#529C2312yes—
121#529C2312yes—
122#525—21no—
123#525—21no—
124#525—21no—
125#525—21no—
126#525C2×C412yes—
127#525C2×C412yes—
128C4×#41—21no—
129C4×#41—21no—
130C4×#41—21no—
131C4×#41—21no—
132C4×#41C2312yes—
133C4×#41C2312yes—
134C22×C42C2311yescharacteristic
135#500C2×C411yescharacteristic
136#500C2×C411yescharacteristic
137#498C2×C411yescharacteristic
138#498C2×C411yescharacteristic
139#494C2×C411yescharacteristic
140#493C2×C411yescharacteristic
141#492C2×C411yescharacteristic
142#492C2×C411yescharacteristic
143#487C2×C411yescharacteristic
144#487C2×C411yescharacteristic
145#487C2×C411yescharacteristic
146#487C2×C411yescharacteristic
147#486C2×C411yescharacteristic
148#486C2×C411yescharacteristic
149#486C2×C411yescharacteristic
150#486C2×C411yescharacteristic
151#485C2×C414yes—
152#485C2×C414yes—
153#482C2×C414yes—
154#482C2×C414yes—
155#479C2×C414yes—
156#479C2×C414yes—
157#478C2×C414yes—
158#478C2×C414yes—
159#467C2×C414yes—
160#466C2×C414yes—
161#465C2×C414yes—
162#465C2×C414yes—
163#464C2×C414yes—
164#464C2×C414yes—
165#463C2×C414yes—
166#463C2×C414yes—
167#452C2×C414yes—
168#452C2×C414yes—
169#449C2×C414yes—
170#449C2×C414yes—
171#446—24no—
172#446—24no—
173#446C2314yes—
174#446C2314yes—
175C8×Q8C2314yes—
176C8×Q8C2314yes—
177C4×Q16C2314yes—
178C4×SD16C2314yes—
179C4×SD16C2314yes—
180C4×D8C2314yes—
181#436—24no—
182#436—24no—
183#436C2314yes—
184#436C2314yes—
185#435—24no—
186#435—24no—
187#435—24no—
188#435—24no—
189#435—24no—
190#435—24no—
191C8×D4—24no—
192C8×D4—24no—
193C8×D4C2314yes—
194C8×D4C2314yes—
195#433—24no—
196#433—24no—
197#432—21no—
198#432—21no—
199#432—22no—
200#432—22no—
201#432C2312yes—
202#432C2311yescharacteristic
203#432C2311yescharacteristic
204#431—21no—
205#431—21no—
206#431—21no—
207#431—21no—
208#431—22no—
209#431—22no—
210#431C2×C411yescharacteristic
211#431C2×C411yescharacteristic
212#431C2×C411yescharacteristic
213#431C2×C411yescharacteristic
214#431C2311yescharacteristic
215#431C2311yescharacteristic
216#431C2312yes—
217#431C2312yes—
218#431C2312yes—
219#431C2312yes—
220#431C2312yes—
221#424—22no—
222#424—22no—
223C2×C4⋊C8—21no—
224C2×C4⋊C8C2×C412yes—
225C2×C4⋊C8C2312yes—
226C2×C4⋊C8C2312yes—
227#408—22no—
228#408—22no—
229C2×C22⋊C8—21no—
230C2×C22⋊C8—21no—
231C2×C22⋊C8—21no—
232C2×C22⋊C8C2×C412yes—
233#405—22no—
234#405—22no—
235#405—24no—
236#405—24no—
237C4×M16—22no—
238C4×M16—22no—
239C4×M16D411yescharacteristic
240C4×M16D411yescharacteristic
241C4×M16D412yes—
242C4×M16C2312yes—
243C4×M16C2312yes—
244C2×#97—21no—
245C2×#97—22no—
246C2×#97D412yes—
247C2×C4×C8—21no—
248C2×C4×C8—22no—
249C2×C4×C8C2312yes—
250C2×C4×C8C2311yescharacteristic
251C2×C4×C8C2311yescharacteristic
252C2×C4×C8C2312yes—
253#335C2312yes—
254#335C2312yes—
255#335C2312yes—
256#335C2312yes—
257#334C2312yes—
258#334C2312yes—
259#334C2312yes—
260#334C2312yes—
261Q8⋊C8C2314yes—
262Q8⋊C8C2314yes—
263Q8⋊C8C2314yes—
264Q8⋊C8C2314yes—
265D4⋊C8C2314yes—
266D4⋊C8C2314yes—
267D4⋊C8C2314yes—
268D4⋊C8C2314yes—
269#322C2314yes—
270#322C2314yes—
271#322C2314yes—
272#322C2314yes—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41C22×C412yes—
278C2×#41C22×C412yes—
279C23×C4C22×C411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4#4111yescharacteristic
282C2×Q16C22×C414yes—
283C2×SD16C22×C414yes—
284C2×SD16C22×C414yes—
285C2×D8C22×C414yes—
286#131—48no—
287#131—48no—
288C2×M16—24no—
289C2×M16—24no—
290C2×M16—44no—
291C2×M16C2×D412yes—
292C2×M16C2×D412yes—
293C2×M16C2×D414yes—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—24no—
298C22×C8—24no—
299C22×C8C22×C412yes—
300C22×C8C22×C412yes—
301C22×C8C2×D414yes—
302C4⋊Q8C22×C411yescharacteristic
303C4⋊Q8C22×C411yescharacteristic
304C4⋊Q8C22×C411yescharacteristic
305C4⋊Q8C22×C411yescharacteristic
306C4⋊Q8C22×C411yescharacteristic
307C4⋊Q8C22×C411yescharacteristic
308#127C22×C411yescharacteristic
309#127C22×C411yescharacteristic
310#126—22no—
311#126—22no—
312#126—22no—
313#126—22no—
314#125—21no—
315#125—21no—
316#125—21no—
317#125—21no—
318#125C22×C412yes—
319#125C22×C412yes—
320#124—21no—
321#124—21no—
322#124—21no—
323#124—21no—
324#124C22×C412yes—
325#124C22×C412yes—
326#123—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—22no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8C22×C412yes—
341C22⋊Q8C22×C412yes—
342C22⋊Q8C22×C412yes—
343C22⋊Q8C22×C412yes—
344#121—21no—
345#121—21no—
346#121—21no—
347#121—21no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—21no—
352#121C22×C412yes—
353#121C22×C412yes—
354#121C22×C412yes—
355#121C22×C412yes—
356#120—22no—
357#120—22no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8C22×C412yes—
367C4×Q8C22×C412yes—
368C4×Q8C2412yes—
369C4×Q8C2412yes—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4C22×C412yes—
391C4×D4C22×C412yes—
392C4×D4C2412yes—
393C4×D4C2412yes—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—22no—
399#117—22no—
400#117—22no—
401#117—22no—
402#117—22no—
403#117—22no—
404#117C22×C412yes—
405#117C22×C412yes—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42C22×C411yescharacteristic
409C2×C42C22×C411yescharacteristic
410C2×C42C22×C411yescharacteristic
411C2×C42C2×D411yescharacteristic
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42#4111yescharacteristic
415C2×C42#4111yescharacteristic
416C2×C42#4111yescharacteristic
417C2×C42C2411yescharacteristic
418C2×C42C2411yescharacteristic
419C2×C42C2411yescharacteristic
420#107C22×C414yes—
421#107C22×C414yes—
422#106C22×C414yes—
423#106C22×C414yes—
424C4⋊C8—24no—
425C4⋊C8—24no—
426C4⋊C8—24no—
427C4⋊C8—24no—
428C4⋊C8—22no—
429C4⋊C8—22no—
430C4⋊C8—22no—
431C4⋊C8—22no—
432C4⋊C8C22×C412yes—
433C4⋊C8C22×C412yes—
434C4⋊C8C22×C412yes—
435C4⋊C8C22×C412yes—
436C4⋊C8C2412yes—
437C4⋊C8C2412yes—
438C4⋊C8C2412yes—
439C4⋊C8C2412yes—
440#103C22×C414yes—
441#103C22×C414yes—
442#103C22×C414yes—
443#103C22×C414yes—
444#102C22×C414yes—
445#102C22×C414yes—
446#102C22×C414yes—
447#102C22×C414yes—
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—22no—
451C22⋊C8—22no—
452C22⋊C8—21no—
453C22⋊C8—21no—
454C22⋊C8—22no—
455C22⋊C8—22no—
456C22⋊C8—24no—
457C22⋊C8—24no—
458C22⋊C8—24no—
459C22⋊C8—24no—
460C22⋊C8C22×C412yes—
461C22⋊C8C22×C412yes—
462C22⋊C8C22×C412yes—
463C22⋊C8C22×C412yes—
464#97—24no—
465#97—24no—
466#97—21no—
467#97—21no—
468#97—22no—
469#97C2×D412yes—
470#97C2×D412yes—
471#97C2×D414yes—
472C4×C8—24no—
473C4×C8—24no—
474C4×C8—21no—
475C4×C8—21no—
476C4×C8—22no—
477C4×C8—22no—
478C4×C8C22×C411yescharacteristic
479C4×C8C22×C411yescharacteristic
480C4×C8C22×C411yescharacteristic
481C4×C8C22×C411yescharacteristic
482C4×C8C2×D412yes—
483C4×C8C2×D412yes—
484C4×C8C2414yes—
485C4×C8C2412yes—
486C4×C8C2412yes—
487C4×C8C2414yes—
488C24C4×D411yescharacteristic
489#41—44no—
490#41—44no—
491#41—44no—
492#41—44no—
493#41—82no—
494#41—82no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8C23×C412yes—
500C2×Q8C23×C412yes—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—22no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4C23×C412yes—
512C2×D4C23×C412yes—
513C22×C4—24no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—42no—
521C22×C4—81no—
522C22×C4C4×D411yescharacteristic
523C22×C4C4×D411yescharacteristic
524C22×C4C4×D411yescharacteristic
525C22×C4C4×D411yescharacteristic
526C22×C4C4×D411yescharacteristic
527C22×C4C4×D411yescharacteristic
528C22×C4C4×D411yescharacteristic
529C22×C4C4×D411yescharacteristic
530C22×C4C4×D411yescharacteristic
531C22×C4C23×C411yescharacteristic
532C22×C4C23×C411yescharacteristic
533C22×C4C23×C411yescharacteristic
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C22×D411yescharacteristic
537C22×C4C2×#4111yescharacteristic
538C22×C4C2×#4111yescharacteristic
539C22×C4C2×#4111yescharacteristic
540Q16—44no—
541SD16—44no—
542SD16—44no—
543D8—44no—
544M16—24no—
545M16—24no—
546M16—44no—
547M16—48no—
548M16—48no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—24no—
554C2×C8—42no—
555C2×C8—42no—
556C2×C8—44no—
557C2×C8—48no—
558C2×C8—48no—
559C2×C8—44no—
560C2×C8—44no—
561C2×C8—82no—
562C2×C8C23×C412yes—
563C2×C8C23×C412yes—
564C2×C8C23×C412yes—
565C2×C8C23×C412yes—
566C2×C8C22×D414yes—
567C2×C8C22×D414yes—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—22no—
577C4⋊C4—22no—
578C4⋊C4—22no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—22no—
582C4⋊C4—22no—
583C4⋊C4—22no—
584C4⋊C4—22no—
585C4⋊C4—22no—
586C4⋊C4—22no—
587C4⋊C4—22no—
588C4⋊C4C23×C412yes—
589C4⋊C4C23×C412yes—
590C4⋊C4C23×C412yes—
591C4⋊C4C23×C412yes—
592C22⋊C4—22no—
593C22⋊C4—22no—
594C22⋊C4—22no—
595C22⋊C4—22no—
596C22⋊C4—22no—
597C22⋊C4—22no—
598C22⋊C4—22no—
599C22⋊C4—22no—
600C22⋊C4—22no—
601C22⋊C4—22no—
602C22⋊C4—22no—
603C22⋊C4—22no—
604C22⋊C4—22no—
605C22⋊C4—22no—
606C22⋊C4—22no—
607C22⋊C4—22no—
608C22⋊C4—22no—
609C22⋊C4—22no—
610C42—22no—
611C42—22no—
612C42—22no—
613C42—22no—
614C42—22no—
615C42—22no—
616C42C4×D411yescharacteristic
617C42C4×D411yescharacteristic
618C42C4×D411yescharacteristic
619C42C4×D411yescharacteristic
620C42C4×D411yescharacteristic
621C42C4×D411yescharacteristic
622C42C23×C411yescharacteristic
623C42C23×C411yescharacteristic
624C42C23×C411yescharacteristic
625C42C22×D411yescharacteristic
626C42C22×D411yescharacteristic
627C42C22×D411yescharacteristic
628C42C2×#4111yescharacteristic
629C42C2×#4111yescharacteristic
630C42C2×#4111yescharacteristic
631C42C2511yescharacteristic
632C23—22no—
633C23—22no—
634C23—22no—
635C23—41no—
636C23—41no—
637C23C2×C4×D411yescharacteristic
638C23C2×C4×D411yescharacteristic
639C23C2×C4×D411yescharacteristic
640Q8—24no—
641Q8—24no—
642Q8—42no—
643Q8—42no—
644Q8—81no—
645Q8—81no—
646D4—24no—
647D4—24no—
648D4—44no—
649D4—44no—
650D4—44no—
651D4—44no—
652D4—42no—
653D4—42no—
654D4—81no—
655D4—81no—
656D4—82no—
657D4—82no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—82no—
686C2×C4—82no—
687C2×C4C2×C4×D411yescharacteristic
688C2×C4C2×C4×D411yescharacteristic
689C2×C4C2×C4×D411yescharacteristic
690C2×C4C2×C4×D411yescharacteristic
691C2×C4C2×C4×D411yescharacteristic
692C2×C4C2×C4×D411yescharacteristic
693C2×C4C2×C4×D411yescharacteristic
694C2×C4C2×C4×D411yescharacteristic
695C2×C4C2×C4×D411yescharacteristic
696C2×C4#57514yes—
697C2×C4C24×C411yescharacteristic
698C2×C4C23×D411yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C8—22no—
701C8—22no—
702C8—22no—
703C8—22no—
704C8—24no—
705C8—24no—
706C8—44no—
707C8—42no—
708C8—42no—
709C8—42no—
710C8—42no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—21no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—42no—
723C22—42no—
724C22—42no—
725C22—82no—
726C22#294312yes—
727C22C22×C4×D411yescharacteristic
728C4—21no—
729C4—21no—
730C4—21no—
731C4—21no—
732C4—21no—
733C4—21no—
734C4—21no—
735C4—21no—
736C4—21no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4#294312yes—
744C4#357312yes—
745C4C2×#57512yes—
746C2—22no—
747C2—41no—
748C2—41no—
749C2—82no—
750C2—82no—
751C2#6014211yescharacteristic
752C2C2×#294311yescharacteristic
753C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60944C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60211C2semidirect
3. #60210C2semidirect
4. #18073C2semidirect
5. #18072C2semidirect
6. #18071C2semidirect
7. #18071C2semidirect
8. #18005C2semidirect
9. #18002C2semidirect
10. #17937C2semidirect
11. #17909C2semidirect
12. C2×#1572C2semidirect
13. #17488C2semidirect
14. #17458C2semidirect
15. #17319C2semidirect
59. #1594C22semidirect
60. #1593C22semidirect
61. #1592C22semidirect
62. #1592C22semidirect
63. #1588C22semidirect
64. #1588C22semidirect
65. #1587C22semidirect
66. #1587C22semidirect
67. #1586C22semidirect
68. #1586C22semidirect
69. #1585C22semidirect
70. #1585C22semidirect
71. #1580C22semidirect
72. #1579C22semidirect
73. SD16⋊C8C22semidirect
74. SD16⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#609442
#602112
#602102
#180732
#180722
#180712
#180052
#180022
#179372
#179092
C2×#15722
#174882
#174582
#173192