Finite Group Databasebeta

← #6345529

Subgroups — #6345529

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60951C211yesmaximal, characteristic
2#60292C211yesmaximal, characteristic, complemented
3#60237C211yesmaximal, characteristic, complemented
4#18078C214yesmaximal, complemented
5#18076C214yesmaximal, complemented
6#18028C214yesmaximal, complemented
7#18026C214yesmaximal, complemented
8#17948C214yesmaximal, complemented
9#17920C214yesmaximal, complemented
10#17810C211yesmaximal, characteristic, complemented
11#17494C211yesmaximal, characteristic, complemented
12#17463C211yesmaximal, characteristic, complemented
13#17339C211yesmaximal, characteristic, complemented
14#3444C2211yescharacteristic
15C2×#532C2211yescharacteristic
16#3254C412yes—
17#3253C412yes—
18#3238C412yes—
19#3235C412yes—
20#3217C412yes—
21#3214C412yes—
22#3194C412yes—
23#3194C412yes—
24#3171C2214yes—
25#3106C2214yes—
26#3101C2212yes—
27#3100C2212yes—
28#2991—21no—
29#2990—21no—
30#2989C2212yescomplemented
31#2988C2212yescomplemented
32#2986C2212yes—
33#2981C2214yes—
34#2980—22no—
35#2979C2212yes—
36#2976—22no—
37#2974C2214yes—
38#2973—22no—
39#2969C2214yescomplemented
40#2966—22no—
41#2935C2212yes—
42#2935C2212yes—
43#2932—21no—
44#2932—21no—
45#2925—22no—
46#2920C2211yescharacteristic
47#2920C2211yescharacteristic
48#2920C2211yescharacteristic
49C2×#432C2211yescharacteristic
50C2×#431C2211yescharacteristic
51#2880—22no—
52#1594C2214yescomplemented
53#1592C2214yescomplemented
54#1590C2214yescomplemented
55#1589C2214yescomplemented
56#1588C2214yescomplemented
57#1586C2214yescomplemented
58#1586C2214yescomplemented
59#1585C2214yescomplemented
60#1583C2214yescomplemented
61#1582C2214yescomplemented
62#1581C2214yescomplemented
63#1581C2214yescomplemented
64#1580C2214yescomplemented
65SD16⋊C8C2214yescomplemented
66C8×SD16C2214yescomplemented
67C8×D8C2214yescomplemented
68#1573C2212yescomplemented
69#1573C2212yescomplemented
70#1572C2214yes—
71#1565C2212yescomplemented
72#1565C2212yescomplemented
73#1497C2212yes—
74#1496C2212yes—
75#1488C2214yescomplemented
76#1484C2214yes—
77#1483C2212yescomplemented
78#1483C2212yescomplemented
79#1480C2212yes—
80#1480C2212yes—
81#1478C2214yes—
82#1455C2214yescomplemented
83#1454C2214yes—
84#1454C2214yescomplemented
85C22×#41—21no—
86C2×#131—22no—
87C2×#131—22no—
88#533C2×C411yescharacteristic
89#533C2×C411yescharacteristic
90#533C2311yescharacteristic
91#532—21no—
92#532—21no—
93#532—21no—
94#532—21no—
95#532C2×C411yescharacteristic
96#532C2×C412yes—
97#532C2×C412yes—
98#532C2×C411yescharacteristic
99#532C2311yescharacteristic
100#532C2312yes—
101#532C2312yes—
102C2×C4⋊Q8C2×C411yescharacteristic
103C2×#127C2×C411yescharacteristic
104#529—21no—
105#529—21no—
106#529—21no—
107#529—21no—
108#529C2×C412yes—
109#529C2312yes—
110C2×#124—21no—
111#525—21no—
112#525—21no—
113#525C2×C412yes—
114C2×#121—21no—
115C2×#121C2×C412yes—
116C4×#41—21no—
117C4×#41—21no—
118C4×#41C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C22×C42C2311yescharacteristic
122#501C2×C414yes—
123#496C2×C414yes—
124#490C2×C414yes—
125#489C2×C414yes—
126#483C2×C412yes—
127#483C2×C412yes—
128#482C2×C414yes—
129#482C2×C414yes—
130#480C2×C412yes—
131#480C2×C412yes—
132#477C2×C414yes—
133#476C2×C414yes—
134#476C2×C414yes—
135#474C2×C414yes—
136#469C2×C414yes—
137#469C2×C414yes—
138#468C2×C414yes—
139#468C2×C414yes—
140#461C2×C414yes—
141#460C2×C414yes—
142#459C2×C414yes—
143#459C2×C414yes—
144#450C2×C412yes—
145#450C2×C412yes—
146#449C2×C414yes—
147#449C2×C414yes—
148#447C2×C412yes—
149#447C2×C412yes—
150#446—22no—
151#446C2314yes—
152C8×Q8—22no—
153C8×Q8C2314yes—
154C4×SD16C2314yes—
155C4×SD16C2314yes—
156C4×D8C2314yes—
157C4×D8C2314yes—
158#436—24no—
159#436—22no—
160#436C2314yes—
161#436C2314yes—
162#436C2314yes—
163#435—24no—
164#435—24no—
165#435—24no—
166#435—24no—
167#435—24no—
168#435—24no—
169#435—24no—
170#435—24no—
171C8×D4—22no—
172C8×D4—24no—
173C8×D4C2314yes—
174C8×D4C2314yes—
175C8×D4C2314yes—
176#433—24no—
177#433—24no—
178#432—21no—
179#432—21no—
180#432—21no—
181#432—21no—
182#432—22no—
183#432C2×C412yes—
184#432C2×C412yes—
185#432D412yes—
186#432D412yes—
187#432C2312yes—
188#432C2312yes—
189#431—21no—
190#431—21no—
191#431—22no—
192#431C2312yes—
193#431C2312yes—
194#431C2312yes—
195#431C2312yes—
196#431C2312yes—
197#431C2312yes—
198#431C2312yes—
199#431C2312yes—
200#423—21no—
201#423—22no—
202#423C2×C412yes—
203#423D411yescharacteristic
204#423D411yescharacteristic
205C2×C4⋊C8—22no—
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×C4⋊C8C2311yescharacteristic
209C2×C4⋊C8C2311yescharacteristic
210#408—22no—
211#408—22no—
212#408—24no—
213#408—24no—
214#408—24no—
215#407—21no—
216#407—24no—
217#407C2×C411yescharacteristic
218#407C2×C411yescharacteristic
219#407D412yes—
220C2×C22⋊C8—21no—
221C2×C22⋊C8—21no—
222C2×C22⋊C8—24no—
223#405—24no—
224C4×M16C2312yes—
225C4×M16C2312yes—
226C4×M16C2312yes—
227C2×#97—21no—
228C2×C4×C8—21no—
229#335C2314yes—
230#335C2314yes—
231#334C2314yes—
232#334C2314yes—
233Q8⋊C8C2314yes—
234Q8⋊C8C2314yes—
235D4⋊C8C2314yes—
236D4⋊C8C2314yes—
237D4⋊C8C2314yes—
238D4⋊C8C2314yes—
239D4⋊C8C2314yes—
240D4⋊C8C2314yes—
241#322C2314yes—
242#322C2314yes—
243#322C2314yes—
244C82C2314yes—
245C2×#41—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—21no—
250C2×#41—22no—
251C2×#41—82no—
252C2×#41—82no—
253C2×#41C22×C412yes—
254C22×Q8—21no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C22×C412yes—
259C23×C4—22no—
260C23×C4C22×C411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4#4111yescharacteristic
263C2×SD16C22×C414yes—
264C2×SD16C22×C414yes—
265C2×D8C22×C414yes—
266C2×D8C22×C414yes—
267#131—84no—
268#131—84no—
269C2×M16—22no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16C22×C412yes—
274C2×M16C22×C412yes—
275C2×M16C2×D412yes—
276C2×M16C2×D412yes—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—24no—
282C22×C8—24no—
283C22×C8—24no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8C22×C411yescharacteristic
287C4⋊Q8C22×C412yes—
288C4⋊Q8C22×C412yes—
289C4⋊Q8C22×C411yescharacteristic
290C4⋊Q8C22×C411yescharacteristic
291#127—21no—
292#127—21no—
293#127C22×C412yes—
294#127C22×C412yes—
295#127C22×C411yescharacteristic
296#126—22no—
297#126—22no—
298#125—21no—
299#125—21no—
300#125C22×C412yes—
301#124—22no—
302#124—22no—
303#124—21no—
304#124—22no—
305#124—22no—
306#124—21no—
307#124C22×C412yes—
308#123—22no—
309#123—22no—
310#123—22no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C22⋊Q8C22×C412yes—
316C22⋊Q8C22×C412yes—
317#121—22no—
318#121—22no—
319#121—22no—
320#121—22no—
321#121—21no—
322#121—21no—
323#121—21no—
324#121—22no—
325#121—22no—
326#121—21no—
327#121C22×C414yes—
328#121C22×C414yes—
329#121C22×C412yes—
330#121C22×C412yes—
331#120—22no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8C22×C412yes—
337C4×Q8C2412yes—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4—22no—
354C4×D4C22×C414yes—
355C4×D4C22×C412yes—
356C4×D4C2414yes—
357C4×D4C2412yes—
358#117—21no—
359#117—21no—
360#117—22no—
361#117—22no—
362#117—22no—
363#117C22×C412yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4C22×C412yes—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C42—22no—
369C2×C42C22×C411yescharacteristic
370C2×C42C22×C411yescharacteristic
371C2×C42C22×C411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42#4111yescharacteristic
376C2×C42#4111yescharacteristic
377C2×C42#4111yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107C22×C414yes—
382#107C22×C414yes—
383#106C22×C414yes—
384#106C22×C414yes—
385C4⋊C8—24no—
386C4⋊C8—24no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393C4⋊C8C22×C414yes—
394C4⋊C8C22×C414yes—
395C4⋊C8C2×D412yes—
396C4⋊C8C2×D412yes—
397C4⋊C8C2×D412yes—
398C4⋊C8C2×D412yes—
399C4⋊C8C2412yes—
400C4⋊C8C2412yes—
401C4⋊C8C2412yes—
402C4⋊C8C2412yes—
403#103C22×C414yes—
404#103C22×C414yes—
405#102C22×C414yes—
406#102C22×C414yes—
407#102C22×C414yes—
408#102C22×C414yes—
409#102C22×C414yes—
410#102C22×C414yes—
411C22⋊C8—22no—
412C22⋊C8—22no—
413C22⋊C8—22no—
414C22⋊C8—22no—
415C22⋊C8—22no—
416C22⋊C8—22no—
417C22⋊C8—24no—
418C22⋊C8—24no—
419C22⋊C8—24no—
420C22⋊C8—24no—
421C22⋊C8—24no—
422C22⋊C8—24no—
423C22⋊C8C22×C412yes—
424C22⋊C8C22×C412yes—
425C22⋊C8C22×C412yes—
426C22⋊C8C22×C412yes—
427C22⋊C8C2×D414yes—
428C22⋊C8C2×D414yes—
429#97—24no—
430#97—22no—
431#97—22no—
432#97—22no—
433#97C22×C414yes—
434#97C2×D414yes—
435C4×C8—24no—
436C4×C8—22no—
437C4×C8—22no—
438C4×C8C2414yes—
439C4×C8C2414yes—
440C4×C8C2414yes—
441C24—22no—
442C24C4×D411yescharacteristic
443#41—44no—
444#41—44no—
445#41—44no—
446#41—81no—
447#41—81no—
448#41—82no—
449#41—82no—
450#41—84no—
451#41—82no—
452#41—84no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—82no—
458C2×Q8C23×C412yes—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—44no—
472C2×D4—82no—
473C2×D4—82no—
474C2×D4—84no—
475C2×D4C23×C412yes—
476C2×D4C23×C414yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—81no—
489C22×C4C4×D411yescharacteristic
490C22×C4C4×D411yescharacteristic
491C22×C4C4×D411yescharacteristic
492C22×C4C4×D411yescharacteristic
493C22×C4C4×D411yescharacteristic
494C22×C4C4×D411yescharacteristic
495C22×C4C4×D411yescharacteristic
496C22×C4C4×D411yescharacteristic
497C22×C4C4×D411yescharacteristic
498C22×C4C23×C411yescharacteristic
499C22×C4C23×C411yescharacteristic
500C22×C4C23×C411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4111yescharacteristic
505C22×C4C2×#4111yescharacteristic
506C22×C4C2×#4111yescharacteristic
507SD16—44no—
508SD16—44no—
509D8—44no—
510D8—44no—
511M16—42no—
512M16—42no—
513M16—42no—
514M16—42no—
515M16—84no—
516M16—84no—
517M16—84no—
518M16—82no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—24no—
524C2×C8—42no—
525C2×C8—42no—
526C2×C8—84no—
527C2×C8—84no—
528C2×C8—84no—
529C2×C8—81no—
530C2×C8—81no—
531C2×C8C23×C414yes—
532C2×C8C23×C414yes—
533C2×C8C22×D414yes—
534C2×C8C22×D414yes—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4C23×C412yes—
549C4⋊C4C23×C412yes—
550C4⋊C4C23×C414yes—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—24no—
563C22⋊C4—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42C4×D411yescharacteristic
568C42C4×D411yescharacteristic
569C42C4×D411yescharacteristic
570C42C4×D411yescharacteristic
571C42C4×D411yescharacteristic
572C42C4×D411yescharacteristic
573C42C23×C411yescharacteristic
574C42C23×C411yescharacteristic
575C42C23×C411yescharacteristic
576C42C22×D411yescharacteristic
577C42C22×D411yescharacteristic
578C42C22×D411yescharacteristic
579C42C2×#4111yescharacteristic
580C42C2×#4111yescharacteristic
581C42C2×#4111yescharacteristic
582C42C2511yescharacteristic
583C23—22no—
584C23—24no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—44no—
589C23—82no—
590C23—82no—
591C23C2×C4×D411yescharacteristic
592C23C2×C4×D411yescharacteristic
593C23C2×C4×D411yescharacteristic
594Q8—24no—
595Q8—44no—
596Q8—42no—
597Q8—82no—
598Q8—81no—
599D4—24no—
600D4—24no—
601D4—24no—
602D4—44no—
603D4—42no—
604D4—48no—
605D4—44no—
606D4—82no—
607D4—82no—
608D4—82no—
609D4—82no—
610D4—81no—
611D4—84no—
612D4—82no—
613D4—82no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—84no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4C2×C4×D411yescharacteristic
648C2×C4C2×C4×D411yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C24×C411yescharacteristic
657C2×C4C23×D411yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C8—24no—
660C8—24no—
661C8—24no—
662C8—24no—
663C8—44no—
664C8—42no—
665C8—42no—
666C8—42no—
667C8—42no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—42no—
682C22—44no—
683C22—82no—
684C22—164no—
685C22C22×C4×D411yescharacteristic
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4#357312yes—
701C4#358511yescharacteristic
702C4#358511yescharacteristic
703C2—41no—
704C2—41no—
705C2—41no—
706C2—82no—
707C2—84no—
708C2#6014211yescharacteristic
709C2#6016811yescharacteristic
710C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60951C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60292C2semidirect
3. #60237C2semidirect
4. #18078C2semidirect
5. #18076C2semidirect
6. #18028C2semidirect
7. #18026C2semidirect
8. #17948C2semidirect
9. #17920C2semidirect
10. #17810C2semidirect
11. #17494C2semidirect
12. #17463C2semidirect
13. #17339C2semidirect
30. #2989C22semidirect
31. #2988C22semidirect
39. #2969C22semidirect
52. #1594C22semidirect
53. #1592C22semidirect
54. #1590C22semidirect
55. #1589C22semidirect
56. #1588C22semidirect
57. #1586C22semidirect
58. #1586C22semidirect
59. #1585C22semidirect
60. #1583C22semidirect
61. #1582C22semidirect
62. #1581C22semidirect
63. #1581C22semidirect
64. #1580C22semidirect
65. SD16⋊C8C22semidirect
66. C8×SD16C22semidirect
67. C8×D8C22semidirect
68. #1573C22semidirect
69. #1573C22semidirect
71. #1565C22semidirect
72. #1565C22semidirect
75. #1488C22semidirect
77. #1483C22semidirect
78. #1483C22semidirect
82. #1455C22semidirect
84. #1454C22semidirect

Maximal subgroups

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

subgroupindex
#609512
#602922
#602372
#180782
#180762
#180282
#180262
#179482
#179202
#178102
#174942
#174632
#173392