Finite Group Databasebeta

← #6354848

Subgroups — #6354848

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60968C211yesmaximal, characteristic
2#60941C211yesmaximal, characteristic, complemented
3C2×#3228C211yesmaximal, characteristic, complemented
4#60236C211yesmaximal, characteristic, complemented
5#19519C214yesmaximal, complemented
6#19516C214yesmaximal, complemented
7#19430C214yesmaximal, complemented
8#19394C214yesmaximal, complemented
9#19198C212yesmaximal, complemented
10#19196C212yesmaximal, complemented
11#19193C212yesmaximal, complemented
12#19191C212yesmaximal, complemented
13#17803C211yesmaximal, characteristic, complemented
14#17747C211yesmaximal, characteristic, complemented
15#17714C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#532C2211yescharacteristic
19#3253C2214yes—
20#3246C2214yescomplemented
21#3229—21no—
22#3228—22no—
23#3228—22no—
24#3228C2212yes—
25#3228C2212yes—
26#3228C2212yescomplemented
27#3228C2212yescomplemented
28#3227—21no—
29#3227C2212yes—
30#3227C2212yes—
31#3208—21no—
32#3206C2212yes—
33#3205—21no—
34#3205C2212yes—
35#3192—22no—
36#3187—22no—
37#3184—22no—
38#3167—22no—
39C2×#500C2211yescharacteristic, complemented
40C2×#498C2211yescharacteristic, complemented
41C2×#486C2211yescharacteristic, complemented
42C2×#486C2211yescharacteristic, complemented
43#3109—22no—
44#3109C2214yes—
45#3103—22no—
46#3101C2214yescomplemented
47C2×#482—22no—
48C2×#478—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#1697C2218yescomplemented
65#1695C2218yescomplemented
66#1691C2218yescomplemented
67#1681C2218yescomplemented
68#1677C2218yescomplemented
69#1675C2218yescomplemented
70#1671C2218yescomplemented
71#1572C2212yescomplemented
72#1572C2212yescomplemented
73#1571C2212yescomplemented
74#1570C2212yescomplemented
75#1564C2212yescomplemented
76#1564C2212yescomplemented
77#1559C2214yescomplemented
78#1554C2214yescomplemented
79#1551C2214yescomplemented
80#1549C2212yescomplemented
81#1548C2212yescomplemented
82#1548C2212yescomplemented
83#1545C2212yescomplemented
84#1543C2214yes—
85C22×#41—21no—
86C22×#41—21no—
87C2×#131—22no—
88C2×#131—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#500C2312yes—
138#500C2312yes—
139#498—22no—
140#498—22no—
141#498—22no—
142#498C2312yes—
143#498C2312yes—
144#493—22no—
145#492—22no—
146#487—22no—
147#486—22no—
148#486—22no—
149#486—22no—
150#486—22no—
151#486—22no—
152#486C2312yes—
153#486C2312yes—
154#486C2312yes—
155#486C2312yes—
156#483—24no—
157#482—24no—
158#482—24no—
159#482—24no—
160#482—24no—
161#481—24no—
162#481—24no—
163#481—24no—
164#478—24no—
165#478—24no—
166#478—24no—
167#478—24no—
168#476—24no—
169#476C2318yes—
170#476C2318yes—
171#475—24no—
172#469—28no—
173#465—28no—
174#461C2318yes—
175#460—24no—
176#459—24no—
177#459C2318yes—
178#450—24no—
179#448—24no—
180#447—24no—
181#447—24no—
182#442—28no—
183C4×SD16—28no—
184#436—24no—
185#436—22no—
186#436—22no—
187#436C2314yes—
188#436C2314yes—
189#435—24no—
190#435—24no—
191C8×D4—24no—
192C8×D4—22no—
193C8×D4—22no—
194C8×D4C2314yes—
195C8×D4C2314yes—
196#432—21no—
197#432—21no—
198#432C2312yes—
199#432C2312yes—
200#431—21no—
201#431—21no—
202#431—21no—
203#431—21no—
204#431D412yes—
205#431D412yes—
206#431C2312yes—
207#431C2312yes—
208#431C2312yes—
209#431C2312yes—
210#431C2311yescharacteristic
211#431C2311yescharacteristic
212#431C2311yescharacteristic
213#431C2311yescharacteristic
214C2×#107—22no—
215C2×#106—22no—
216#423C2312yes—
217#423C2312yes—
218C2×C4⋊C8D412yes—
219C2×C4⋊C8C2312yes—
220C2×#102—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#333C2318yes—
242#333C2318yes—
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×SD16—28no—
273C2×D8—28no—
274#131—84no—
275#131—84no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—24no—
279C2×M16—24no—
280C22×C8—22no—
281C22×C8—24no—
282C22×C8—24no—
283C22×C8C2×D412yes—
284C22×C8C2×D412yes—
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⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8C2×D412yes—
297C4⋊Q8C2×D412yes—
298C4⋊Q8C2412yes—
299C4⋊Q8C2412yes—
300C4⋊Q8C2412yes—
301#127—21no—
302#127—21no—
303#127—21no—
304#127—21no—
305#127—21no—
306#127C2×D412yes—
307#127C2×D412yes—
308#127C2412yes—
309#125—22no—
310#125—22no—
311#125—41no—
312#125—41no—
313#124—22no—
314#124—22no—
315#124—22no—
316#124—22no—
317#124—22no—
318#124—22no—
319#124—41no—
320#124—41no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—41no—
326C22⋊Q8—41no—
327C22⋊Q8—41no—
328C22⋊Q8—41no—
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—22no—
338#121—22no—
339#121—41no—
340#121—41no—
341#121—41no—
342#121—41no—
343#121C2×D414yes—
344#121C2×D414yes—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—41no—
350C4×Q8—41no—
351C4×Q8—41no—
352C4×Q8—41no—
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—22no—
362C4×D4—22no—
363C4×D4—41no—
364C4×D4—41no—
365C4×D4—41no—
366C4×D4—41no—
367C4×D4C2×D414yes—
368C4×D4C2414yes—
369#117—22no—
370#117—22no—
371#117—41no—
372#117—41no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4C2×D412yes—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—41no—
386C2×C22⋊C4—41no—
387C2×C22⋊C4—41no—
388C2×C42—41no—
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42#4111yescharacteristic
396C2×C42#4111yescharacteristic
397C2×C42#4111yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—24no—
402#107—24no—
403#106—24no—
404#106—24no—
405#106—28no—
406#106—28no—
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8—22no—
410C4⋊C8C2×D412yes—
411C4⋊C8C2×D412yes—
412C4⋊C8C2×D412yes—
413C4⋊C8C2×D412yes—
414C4⋊C8C2414yes—
415C4⋊C8C2414yes—
416C4⋊C8C2414yes—
417#103—28no—
418#102—28no—
419#102—28no—
420#102—24no—
421#102—24no—
422#102—24no—
423#102—24no—
424#102—28no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8—22no—
428C22⋊C8—22no—
429C22⋊C8—22no—
430C22⋊C8—22no—
431C22⋊C8—24no—
432C22⋊C8—24no—
433C22⋊C8—24no—
434C22⋊C8—24no—
435C22⋊C8C2×D412yes—
436C22⋊C8C2×D412yes—
437C22⋊C8C2×D412yes—
438C22⋊C8C2×D412yes—
439#97—22no—
440#97—22no—
441C4×C8—22no—
442C4×C8—22no—
443C4×C8—22no—
444C4×C8—22no—
445C4×C8C2×D412yes—
446C4×C8C2×D412yes—
447C4×C8C2411yescharacteristic
448C4×C8C2411yescharacteristic
449C4×C8C2411yescharacteristic
450C4×C8C2411yescharacteristic
451C24—22no—
452C24—41no—
453C24#12111yescharacteristic
454#41—44no—
455#41—44no—
456#41—84no—
457#41—84no—
458#41—84no—
459#41—84no—
460#41—84no—
461#41—84no—
462#41—84no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—41no—
469C2×Q8—41no—
470C2×Q8—41no—
471C2×Q8—82no—
472C2×Q8—82no—
473C2×Q8—82no—
474C2×Q8—82no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—44no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—41no—
489C2×D4—41no—
490C2×D4—41no—
491C2×D4—44no—
492C2×D4—82no—
493C2×D4—82no—
494C2×D4—82no—
495C2×D4—82no—
496C2×D4—84no—
497C2×D4—84no—
498C2×D4C22×D414yes—
499C22×C4—24no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—41no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4#12111yescharacteristic
513C22×C4#12111yescharacteristic
514C22×C4#12111yescharacteristic
515C22×C4#12111yescharacteristic
516C22×C4#12111yescharacteristic
517C22×C4#12111yescharacteristic
518C22×C4#12111yescharacteristic
519C22×C4#12111yescharacteristic
520C22×C4#12111yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C2×#4111yescharacteristic
528C22×C4C2×#4111yescharacteristic
529C22×C4C2×#4111yescharacteristic
530SD16—88no—
531D8—88no—
532M16—44no—
533M16—84no—
534M16—84no—
535M16—82no—
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8—24no—
541C2×C8—84no—
542C2×C8—84no—
543C2×C8—82no—
544C2×C8C22×D412yes—
545C2×C8C22×D412yes—
546C2×C8C22×D412yes—
547C2×C8C22×D412yes—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—41no—
558C4⋊C4—41no—
559C4⋊C4—41no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4—41no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4C22×D414yes—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—42no—
577C22⋊C4—42no—
578C42—42no—
579C42—42no—
580C42#12111yescharacteristic
581C42#12111yescharacteristic
582C42#12111yescharacteristic
583C42#12111yescharacteristic
584C42#12111yescharacteristic
585C42#12111yescharacteristic
586C42C22×D411yescharacteristic
587C42C22×D411yescharacteristic
588C42C22×D411yescharacteristic
589C42C22×D411yescharacteristic
590C42C22×D411yescharacteristic
591C42C22×D411yescharacteristic
592C42C2×#4111yescharacteristic
593C42C2×#4111yescharacteristic
594C42C2×#4111yescharacteristic
595C42C2511yescharacteristic
596C23—22no—
597C23—24no—
598C23—41no—
599C23—41no—
600C23—44no—
601C23—42no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23—82no—
607C23C2×#12111yescharacteristic
608C23C2×#12111yescharacteristic
609C23C2×#12111yescharacteristic
610Q8—44no—
611Q8—44no—
612Q8—82no—
613Q8—82no—
614Q8—82no—
615Q8—82no—
616Q8—82no—
617Q8—82no—
618D4—28no—
619D4—44no—
620D4—48no—
621D4—44no—
622D4—44no—
623D4—44no—
624D4—44no—
625D4—82no—
626D4—82no—
627D4—82no—
628D4—82no—
629D4—84no—
630D4—84no—
631D4—84no—
632D4—82no—
633D4—82no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—82no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4C2×#12111yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4C2×#12111yescharacteristic
667C2×C4C2×#12111yescharacteristic
668C2×C4C2×#12111yescharacteristic
669C2×C4C2×#12111yescharacteristic
670C2×C4C2×#12111yescharacteristic
671C2×C4C2×#12111yescharacteristic
672C2×C4C2×#12111yescharacteristic
673C2×C4#57714yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C23×D411yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C8—22no—
678C8—22no—
679C8—22no—
680C8—22no—
681C8—44no—
682C8—44no—
683C8—44no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—164no—
700C22—164no—
701C22#304212yes—
702C22C22×#12111yescharacteristic
703C4—21no—
704C4—21no—
705C4—21no—
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—21no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#304212yes—
716C4#306812yes—
717C4C2×#57712yes—
718C4#358412yes—
719C2—22no—
720C2—41no—
721C2—41no—
722C2—84no—
723C2—162no—
724C2C2×#304211yescharacteristic
725C2#6045011yescharacteristic
726C2C2×#306811yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60968C2
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
2. #60941C2semidirect
3. C2×#3228C2semidirect
4. #60236C2semidirect
5. #19519C2semidirect
6. #19516C2semidirect
7. #19430C2semidirect
8. #19394C2semidirect
9. #19198C2semidirect
10. #19196C2semidirect
11. #19193C2semidirect
12. #19191C2semidirect
13. #17803C2semidirect
14. #17747C2semidirect
15. #17714C2semidirect
20. #3246C22semidirect
26. #3228C22semidirect
27. #3228C22semidirect
39. C2×#500C22semidirect
40. C2×#498C22semidirect
41. C2×#486C22semidirect
42. C2×#486C22semidirect
46. #3101C22semidirect
49. #2981C22semidirect
50. #2981C22semidirect
52. #2975C22semidirect
54. #2972C22semidirect
63. #1701C22semidirect
64. #1697C22semidirect
65. #1695C22semidirect
66. #1691C22semidirect
67. #1681C22semidirect
68. #1677C22semidirect
69. #1675C22semidirect
70. #1671C22semidirect
71. #1572C22semidirect
72. #1572C22semidirect
73. #1571C22semidirect
74. #1570C22semidirect
75. #1564C22semidirect
76. #1564C22semidirect
77. #1559C22semidirect
78. #1554C22semidirect
79. #1551C22semidirect
80. #1549C22semidirect
81. #1548C22semidirect
82. #1548C22semidirect
83. #1545C22semidirect

Maximal subgroups

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

subgroupindex
#609682
#609412
C2×#32282
#602362
#195192
#195162
#194302
#193942
#191982
#191962
#191932
#191912
#178032
#177472
#177142