Finite Group Databasebeta

← #6354839

Subgroups — #6354839

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60969C211yesmaximal, characteristic, complemented
2#60940C211yesmaximal, characteristic
3C2×#3210C211yesmaximal, characteristic, complemented
4#60236C211yesmaximal, characteristic, complemented
5#19511C214yesmaximal, complemented
6#19508C214yesmaximal, complemented
7#19424C214yesmaximal, complemented
8#19388C214yesmaximal, complemented
9#19198C212yesmaximal, complemented
10#19197C212yesmaximal, complemented
11#19188C212yesmaximal, complemented
12#19186C212yesmaximal, complemented
13#17803C211yesmaximal, characteristic, complemented
14#17743C211yesmaximal, characteristic, complemented
15#17715C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#532C2211yescharacteristic
19#3254C2214yescomplemented
20#3245C2214yes—
21#3230—21no—
22#3226—21no—
23#3226C2212yes—
24#3226C2212yes—
25#3210—22no—
26#3210—22no—
27#3210C2212yes—
28#3210C2212yes—
29#3210C2212yescomplemented
30#3210C2212yescomplemented
31#3208—21no—
32#3206C2212yes—
33#3205—21no—
34#3205C2212yes—
35#3193—22no—
36#3186—22no—
37#3182—22no—
38#3169—22no—
39C2×#494C2211yescharacteristic, complemented
40C2×#492C2211yescharacteristic, complemented
41C2×#487C2211yescharacteristic, complemented
42C2×#486C2211yescharacteristic, complemented
43#3112—22no—
44#3110C2214yescomplemented
45#3100—22no—
46#3100C2214yes—
47C2×#464—22no—
48C2×#452—22no—
49#2981C2212yescomplemented
50#2981C2212yescomplemented
51#2979—22no—
52#2975C2212yescomplemented
53#2974—22no—
54#2972C2212yescomplemented
55#2935—22no—
56#2931C2212yes—
57#2931C2212yes—
58#2920C2211yescharacteristic
59#2920C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C2×#431C2211yescharacteristic
62#2880—22no—
63#1701C2218yescomplemented
64#1699C2218yescomplemented
65#1689C2218yescomplemented
66#1685C2218yescomplemented
67#1681C2218yescomplemented
68#1679C2218yescomplemented
69#1669C2218yescomplemented
70#1665C2218yescomplemented
71#1572C2212yescomplemented
72#1572C2212yescomplemented
73#1571C2212yescomplemented
74#1570C2212yescomplemented
75#1564C2212yescomplemented
76#1564C2212yescomplemented
77#1557C2214yescomplemented
78#1553C2214yes—
79#1552C2214yescomplemented
80#1549C2212yescomplemented
81#1546C2212yescomplemented
82#1546C2212yescomplemented
83#1545C2212yescomplemented
84#1544C2214yescomplemented
85C22×#41—21no—
86C22×#41—21no—
87C22×Q16—22no—
88C22×SD16—22no—
89C2×#131—22no—
90C2×#131—22no—
91#533—21no—
92#533—21no—
93#533—21no—
94#533—21no—
95#533—21no—
96#533—21no—
97#533C2312yes—
98#533C2312yes—
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#532—21no—
108#532—21no—
109#532D412yes—
110#532D412yes—
111#532C2312yes—
112#532C2312yes—
113#532C2312yes—
114#532C2312yes—
115C2×C4⋊Q8D411yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#127D411yescharacteristic
120C2×#127C2311yescharacteristic
121C2×#125—21no—
122C2×#124—21no—
123C2×#124—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×#121—21no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121D412yes—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C22×C42C2311yescharacteristic
138#500—22no—
139#500—22no—
140#494—22no—
141#494C2312yes—
142#494C2312yes—
143#493—22no—
144#492—22no—
145#492—22no—
146#492C2312yes—
147#492C2312yes—
148#487—22no—
149#487—22no—
150#487C2312yes—
151#487C2312yes—
152#486—22no—
153#486—22no—
154#486—22no—
155#486—22no—
156#486C2312yes—
157#486C2312yes—
158#484—24no—
159#480—24no—
160#480—24no—
161#480—24no—
162#477—24no—
163#474—24no—
164#474C2318yes—
165#474C2318yes—
166#468—28no—
167#466—28no—
168#464—24no—
169#464—24no—
170#464—24no—
171#464—24no—
172#461C2318yes—
173#460—24no—
174#459—24no—
175#459C2318yes—
176#452—24no—
177#452—24no—
178#452—24no—
179#452—24no—
180#450—24no—
181#448—24no—
182#447—24no—
183#447—24no—
184#440—28no—
185C4×D8—28no—
186#436—24no—
187#436—22no—
188#436—22no—
189#436C2314yes—
190#436C2314yes—
191#435—24no—
192#435—24no—
193C8×D4—24no—
194C8×D4—22no—
195C8×D4—22no—
196C8×D4C2314yes—
197C8×D4C2314yes—
198#432—21no—
199#432—21no—
200#432C2312yes—
201#432C2312yes—
202#431—21no—
203#431—21no—
204#431—21no—
205#431—21no—
206#431D412yes—
207#431D412yes—
208#431C2312yes—
209#431C2312yes—
210#431C2312yes—
211#431C2312yes—
212#431C2311yescharacteristic
213#431C2311yescharacteristic
214#431C2311yescharacteristic
215#431C2311yescharacteristic
216#423C2312yes—
217#423C2312yes—
218C2×C4⋊C8D412yes—
219C2×C4⋊C8C2312yes—
220C2×#103—22no—
221C2×#102—22no—
222#408—22no—
223#408—22no—
224#408—24no—
225#408—24no—
226#407—21no—
227#407—21no—
228#407—24no—
229C2×C22⋊C8—21no—
230C2×C22⋊C8—24no—
231C2×C22⋊C8D412yes—
232C4×M16—21no—
233C4×M16—21no—
234C2×C4×C8—21no—
235C2×C4×C8C2311yescharacteristic
236C2×C4×C8C2311yescharacteristic
237#335C2314yes—
238#335C2314yes—
239#334C2314yes—
240#334C2314yes—
241#332C2318yes—
242#332C2318yes—
243#332C2318yes—
244#331C2318yes—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—41no—
252C2×#41—41no—
253C2×#41—84no—
254C2×#41—84no—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—41no—
266C22×D4C2×D412yes—
267C23×C4—22no—
268C23×C4—41no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4#4111yescharacteristic
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—84no—
275C2×SD16—28no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—84no—
279C2×D8—28no—
280#131—84no—
281#131—84no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—24no—
285C2×M16—24no—
286C22×C8—22no—
287C22×C8—24no—
288C22×C8—24no—
289C22×C8C2×D412yes—
290C22×C8C2×D412yes—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8C2×D412yes—
303C4⋊Q8C2×D412yes—
304C4⋊Q8C2412yes—
305C4⋊Q8C2412yes—
306C4⋊Q8C2412yes—
307#127—21no—
308#127—21no—
309#127—21no—
310#127—21no—
311#127—21no—
312#127C2×D412yes—
313#127C2×D412yes—
314#127C2412yes—
315#125—22no—
316#125—22no—
317#125—41no—
318#125—41no—
319#124—22no—
320#124—22no—
321#124—22no—
322#124—22no—
323#124—22no—
324#124—22no—
325#124—41no—
326#124—41no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—41no—
332C22⋊Q8—41no—
333C22⋊Q8—41no—
334C22⋊Q8—41no—
335#121—22no—
336#121—22no—
337#121—22no—
338#121—22no—
339#121—22no—
340#121—22no—
341#121—22no—
342#121—22no—
343#121—22no—
344#121—22no—
345#121—41no—
346#121—41no—
347#121—41no—
348#121—41no—
349#121C2×D414yes—
350#121C2×D414yes—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—41no—
356C4×Q8—41no—
357C4×Q8—41no—
358C4×Q8—41no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—41no—
370C4×D4—41no—
371C4×D4—41no—
372C4×D4—41no—
373C4×D4C2×D414yes—
374C4×D4C2414yes—
375#117—22no—
376#117—22no—
377#117—41no—
378#117—41no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—41no—
387C2×C4⋊C4—41no—
388C2×C4⋊C4C2×D412yes—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—41no—
392C2×C22⋊C4—41no—
393C2×C22⋊C4—41no—
394C2×C42—41no—
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42#4111yescharacteristic
402C2×C42#4111yescharacteristic
403C2×C42#4111yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—28no—
408#107—28no—
409C4⋊C8—22no—
410C4⋊C8—22no—
411C4⋊C8—22no—
412C4⋊C8C2×D412yes—
413C4⋊C8C2×D412yes—
414C4⋊C8C2×D412yes—
415C4⋊C8C2×D412yes—
416C4⋊C8C2414yes—
417C4⋊C8C2414yes—
418C4⋊C8C2414yes—
419#103—24no—
420#103—24no—
421#103—28no—
422#102—28no—
423#102—28no—
424#102—24no—
425#102—24no—
426#102—28no—
427C22⋊C8—22no—
428C22⋊C8—22no—
429C22⋊C8—22no—
430C22⋊C8—22no—
431C22⋊C8—22no—
432C22⋊C8—22no—
433C22⋊C8—24no—
434C22⋊C8—24no—
435C22⋊C8—24no—
436C22⋊C8—24no—
437C22⋊C8C2×D412yes—
438C22⋊C8C2×D412yes—
439C22⋊C8C2×D412yes—
440C22⋊C8C2×D412yes—
441#97—22no—
442#97—22no—
443C4×C8—22no—
444C4×C8—22no—
445C4×C8—22no—
446C4×C8—22no—
447C4×C8C2×D412yes—
448C4×C8C2×D412yes—
449C4×C8C2411yescharacteristic
450C4×C8C2411yescharacteristic
451C4×C8C2411yescharacteristic
452C4×C8C2411yescharacteristic
453C24—22no—
454C24—41no—
455C24#12111yescharacteristic
456#41—44no—
457#41—44no—
458#41—84no—
459#41—84no—
460#41—84no—
461#41—84no—
462#41—84no—
463#41—84no—
464#41—84no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—41no—
471C2×Q8—41no—
472C2×Q8—41no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8—82no—
476C2×Q8—82no—
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—44no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—41no—
491C2×D4—41no—
492C2×D4—41no—
493C2×D4—44no—
494C2×D4—82no—
495C2×D4—82no—
496C2×D4—82no—
497C2×D4—82no—
498C2×D4—84no—
499C2×D4—84no—
500C2×D4C22×D414yes—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—41no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—84no—
513C22×C4—84no—
514C22×C4#12111yescharacteristic
515C22×C4#12111yescharacteristic
516C22×C4#12111yescharacteristic
517C22×C4#12111yescharacteristic
518C22×C4#12111yescharacteristic
519C22×C4#12111yescharacteristic
520C22×C4#12111yescharacteristic
521C22×C4#12111yescharacteristic
522C22×C4#12111yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C2×#4111yescharacteristic
530C22×C4C2×#4111yescharacteristic
531C22×C4C2×#4111yescharacteristic
532Q16—84no—
533Q16—84no—
534SD16—84no—
535SD16—84no—
536SD16—88no—
537D8—88no—
538M16—44no—
539M16—84no—
540M16—84no—
541M16—82no—
542C2×C8—24no—
543C2×C8—24no—
544C2×C8—24no—
545C2×C8—24no—
546C2×C8—24no—
547C2×C8—84no—
548C2×C8—84no—
549C2×C8—82no—
550C2×C8C22×D412yes—
551C2×C8C22×D412yes—
552C2×C8C22×D412yes—
553C2×C8C22×D412yes—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—41no—
564C4⋊C4—41no—
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⋊C4C22×D414yes—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—24no—
577C22⋊C4—24no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—42no—
583C22⋊C4—42no—
584C42—42no—
585C42—42no—
586C42#12111yescharacteristic
587C42#12111yescharacteristic
588C42#12111yescharacteristic
589C42#12111yescharacteristic
590C42#12111yescharacteristic
591C42#12111yescharacteristic
592C42C22×D411yescharacteristic
593C42C22×D411yescharacteristic
594C42C22×D411yescharacteristic
595C42C22×D411yescharacteristic
596C42C22×D411yescharacteristic
597C42C22×D411yescharacteristic
598C42C2×#4111yescharacteristic
599C42C2×#4111yescharacteristic
600C42C2×#4111yescharacteristic
601C42C2511yescharacteristic
602C23—22no—
603C23—24no—
604C23—41no—
605C23—41no—
606C23—44no—
607C23—42no—
608C23—82no—
609C23—82no—
610C23—82no—
611C23—82no—
612C23—82no—
613C23C2×#12111yescharacteristic
614C23C2×#12111yescharacteristic
615C23C2×#12111yescharacteristic
616Q8—44no—
617Q8—44no—
618Q8—44no—
619Q8—44no—
620Q8—82no—
621Q8—82no—
622Q8—82no—
623Q8—82no—
624D4—28no—
625D4—44no—
626D4—48no—
627D4—44no—
628D4—44no—
629D4—44no—
630D4—44no—
631D4—82no—
632D4—82no—
633D4—82no—
634D4—82no—
635D4—84no—
636D4—84no—
637D4—84no—
638D4—82no—
639D4—82no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4C2×#12111yescharacteristic
671C2×C4C2×#12111yescharacteristic
672C2×C4C2×#12111yescharacteristic
673C2×C4C2×#12111yescharacteristic
674C2×C4C2×#12111yescharacteristic
675C2×C4C2×#12111yescharacteristic
676C2×C4C2×#12111yescharacteristic
677C2×C4C2×#12111yescharacteristic
678C2×C4C2×#12111yescharacteristic
679C2×C4#57614yes—
680C2×C4C23×D411yescharacteristic
681C2×C4C23×D411yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C8—22no—
684C8—22no—
685C8—22no—
686C8—22no—
687C8—44no—
688C8—44no—
689C8—44no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—164no—
706C22—164no—
707C22#304112yes—
708C22C22×#12111yescharacteristic
709C4—21no—
710C4—21no—
711C4—21no—
712C4—21no—
713C4—21no—
714C4—21no—
715C4—21no—
716C4—21no—
717C4—44no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4#304112yes—
722C4#306912yes—
723C4C2×#57612yes—
724C4#358412yes—
725C2—22no—
726C2—41no—
727C2—41no—
728C2—84no—
729C2—162no—
730C2C2×#304111yescharacteristic
731C2#6045011yescharacteristic
732C2C2×#306911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60940C2
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. #60969C2semidirect
3. C2×#3210C2semidirect
4. #60236C2semidirect
5. #19511C2semidirect
6. #19508C2semidirect
7. #19424C2semidirect
8. #19388C2semidirect
9. #19198C2semidirect
10. #19197C2semidirect
11. #19188C2semidirect
12. #19186C2semidirect
13. #17803C2semidirect
14. #17743C2semidirect
15. #17715C2semidirect
19. #3254C22semidirect
29. #3210C22semidirect
30. #3210C22semidirect
39. C2×#494C22semidirect
40. C2×#492C22semidirect
41. C2×#487C22semidirect
42. C2×#486C22semidirect
44. #3110C22semidirect
49. #2981C22semidirect
50. #2981C22semidirect
52. #2975C22semidirect
54. #2972C22semidirect
63. #1701C22semidirect
64. #1699C22semidirect
65. #1689C22semidirect
66. #1685C22semidirect
67. #1681C22semidirect
68. #1679C22semidirect
69. #1669C22semidirect
70. #1665C22semidirect
71. #1572C22semidirect
72. #1572C22semidirect
73. #1571C22semidirect
74. #1570C22semidirect
75. #1564C22semidirect
76. #1564C22semidirect
77. #1557C22semidirect
79. #1552C22semidirect
80. #1549C22semidirect
81. #1546C22semidirect
82. #1546C22semidirect
83. #1545C22semidirect
84. #1544C22semidirect

Maximal subgroups

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

subgroupindex
#609692
#609402
C2×#32102
#602362
#195112
#195082
#194242
#193882
#191982
#191972
#191882
#191862
#178032
#177432
#177152