Finite Group Databasebeta

← #6508654

Subgroups — #6508654

Order 512.

3482 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3537C211yesmaximal, characteristic
2C22×#481C211yesmaximal, characteristic, complemented
3C22×#480C211yesmaximal, characteristic, complemented
4#35227C211yesmaximal, characteristic, complemented
5#24092C214yesmaximal, complemented
6#24091C214yesmaximal, complemented
7#24069C214yesmaximal, complemented
8#24033C214yesmaximal, complemented
9#24032C214yesmaximal, complemented
10#24013C214yesmaximal, complemented
11#21761C211yesmaximal, characteristic, complemented
12#21761C211yesmaximal, characteristic, complemented
13#21739C211yesmaximal, characteristic, complemented
14#3537—28no—
15#3537—28no—
16C2×#560—22no—
17C2×#553C2212yes—
18C2×#553C2212yes—
19C2×D42—22no—
20C2×D42C2212yes—
21C2×D42C2212yes—
22C2×#540C2212yes—
23C2×#538—21no—
24C2×#538—21no—
25C2×#535C2212yes—
26C2×#534—21no—
27C2×#534—21no—
28C2×#525C2211yescharacteristic
29C22×#121C2211yescharacteristic
30C22×#121C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic, complemented
32C2×#481—28no—
33C2×#481C2214yescomplemented
34C2×#481C2214yescomplemented
35C2×#480—28no—
36C2×#480C2214yescomplemented
37C2×#480C2214yescomplemented
38C22×#107C2212yescomplemented
39C22×#106C2212yescomplemented
40C22×#102C2212yescomplemented
41C22×#102C2212yescomplemented
42C22×C22⋊C8C2211yescharacteristic, complemented
43C22×C22⋊C8C2211yescharacteristic, complemented
44#2849—22no—
45#2623—22no—
46#2504—21no—
47#2503—21no—
48#2502—21no—
49#2501—21no—
50#2487C2214yescomplemented
51#2486C2214yescomplemented
52#2317C2212yescomplemented
53#2311C2212yescomplemented
54#2280C2211yescharacteristic, complemented
55#2059C2218yescomplemented
56#2041C2218yescomplemented
57#2034C2218yescomplemented
58#2033C2218yescomplemented
59#1893C2214yescomplemented
60#1893C2214yescomplemented
61#1875C2214yescomplemented
62#1875C2214yescomplemented
63#1869C2214yescomplemented
64#1869C2214yescomplemented
65#1864C2218yescomplemented
66#1862C2218yescomplemented
67#1854C2218yescomplemented
68#1824C2212yescomplemented
69#1824C2212yescomplemented
70#1818C2212yescomplemented
71#1786C2212yescomplemented
72#1786C2212yescomplemented
73#1786C2218yescomplemented
74#1780C2212yescomplemented
75C22×#41C2311yescharacteristic
76C23×D4—21no—
77C23×D4—21no—
78C23×D4—22no—
79C23×D4—22no—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85C23×C8C2312yes—
86C23×C8C2312yes—
87#560—232no—
88#553—28no—
89#553—28no—
90#553—28no—
91#553—28no—
92D42—232no—
93D42—28no—
94D42—28no—
95D42—28no—
96D42—28no—
97#540—28no—
98#540—28no—
99#538—216no—
100#538—216no—
101#535—28no—
102#535—28no—
103#534—216no—
104#534—216no—
105C2×#127—22no—
106C2×#127C2312yes—
107C2×#126—22no—
108C2×#126—22no—
109C2×#125C2312yes—
110C2×#124—22no—
111#525—24no—
112#525—24no—
113C2×#123—24no—
114C2×#123—22no—
115C2×#123—22no—
116C2×#123—22no—
117C2×#123—22no—
118C2×#123D412yes—
119C2×C22⋊Q8D411yescharacteristic
120C2×C22⋊Q8D411yescharacteristic
121C2×#121—22no—
122C2×#121—22no—
123C2×#121—21no—
124C2×#121—21no—
125C2×#121—21no—
126C2×#121—21no—
127C2×#121—24no—
128C2×#121—24no—
129C2×#121—22no—
130C2×#121—22no—
131C2×#121—22no—
132C2×#121—22no—
133C2×#121D418yes—
134C2×#121D418yes—
135C2×#121D411yescharacteristic
136C2×#121D411yescharacteristic
137C2×#121C2312yes—
138C2×#121C2312yes—
139C2×#121C2312yes—
140C2×#121C2312yes—
141C2×#121C2312yes—
142C2×#121C2312yes—
143C2×#120—22no—
144C2×#120—22no—
145C2×#120—22no—
146C2×#120—22no—
147C2×#120—22no—
148C2×#120—22no—
149C2×#120D412yes—
150C2×C4×D4—22no—
151C2×C4×D4—22no—
152C2×C4×D4C2312yes—
153C2×C4×D4C2312yes—
154C2×C4×D4C2312yes—
155C2×C4×D4C2312yes—
156C2×#117C2312yes—
157C2×#117C2312yes—
158C2×#117C2311yescharacteristic
159C22×C4⋊C4—24no—
160C22×C4⋊C4—24no—
161C22×C4⋊C4—21no—
162C22×C4⋊C4—21no—
163C22×C4⋊C4—21no—
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2312yes—
166C22×C4⋊C4C2312yes—
167C22×C4⋊C4C2311yescharacteristic
168C22×C4⋊C4C2311yescharacteristic
169C22×C22⋊C4—21no—
170C22×C22⋊C4—21no—
171#481—216no—
172#481—216no—
173#480—216no—
174#480—216no—
175C2×#107—28no—
176C2×#107C2318yes—
177C2×#106—28no—
178C2×#106C2318yes—
179C2×#102—28no—
180C2×#102—28no—
181C2×#102C2318yes—
182C2×#102C2318yes—
183C2×#102C2318yes—
184C2×#102C2318yes—
185C2×C22⋊C8—24no—
186C2×C22⋊C8—24no—
187C2×C22⋊C8C2312yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2318yes—
192#400—24no—
193#399—24no—
194#398—24no—
195#397—24no—
196#391—22no—
197#390—22no—
198#389C2318yes—
199#386—24no—
200#386—24no—
201#385—24no—
202#385—24no—
203#384—22no—
204#382—22no—
205#381—21no—
206#381—24no—
207#380C2318yes—
208#380C2312yes—
209#380C2312yes—
210#379—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213#340C2318yes—
214#340C2318yes—
215#336C2318yes—
216C25—21no—
217C25—21no—
218C25—22no—
219C25C2×D411yescharacteristic
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—44no—
223C22×Q8C2×D411yescharacteristic
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—216no—
227C22×D4—216no—
228C22×D4—28no—
229C22×D4—28no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—48no—
246C22×D4—88no—
247C22×D4—88no—
248C22×D4—84no—
249C22×D4—84no—
250C22×D4C2×D412yes—
251C22×D4C2×D412yes—
252C22×D4C2×D411yescharacteristic
253C22×D4C2×D412yes—
254C22×D4C2×D414yes—
255C22×D4C2×D414yes—
256C22×D4C2412yes—
257C22×D4C2412yes—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—44no—
266C23×C4—44no—
267C23×C4—84no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D414yes—
272C23×C4C2×D414yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C22×C8—24no—
278C22×C8—24no—
279C22×C8—44no—
280C22×C8—44no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283#127—216no—
284#127—24no—
285#127—24no—
286#126—216no—
287#126—216no—
288#125—24no—
289#125—24no—
290#124—216no—
291#123—216no—
292#123—216no—
293#123—216no—
294#123—216no—
295#123—28no—
296#123—88no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299#121—216no—
300#121—216no—
301#121—216no—
302#121—216no—
303#121—216no—
304#121—216no—
305#121—24no—
306#121—24no—
307#121—28no—
308#121—28no—
309#121—28no—
310#121—28no—
311#121—24no—
312#121—24no—
313#121—24no—
314#121—24no—
315#121—24no—
316#121—24no—
317#121—24no—
318#121—24no—
319#121C2×D418yes—
320#121C2×D418yes—
321#121C2×D418yes—
322#121C2×D418yes—
323#120—216no—
324#120—216no—
325#120—216no—
326#120—216no—
327#120—216no—
328#120—216no—
329#120—28no—
330C4×D4—216no—
331C4×D4—216no—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—24no—
335C4×D4—24no—
336C4×D4—24no—
337C4×D4—24no—
338C4×D4—24no—
339C4×D4—24no—
340#117—22no—
341#117—22no—
342#117—24no—
343#117—24no—
344#117—44no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—28no—
361C2×C4⋊C4—44no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—44no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D412yes—
368C2×C4⋊C4C2×D414yes—
369C2×C4⋊C4C2×D414yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—28no—
389C2×C22⋊C4C2×D412yes—
390C2×C22⋊C4C2×D412yes—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D412yes—
393C2×C22⋊C4C2×D412yes—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42C2412yes—
397#107—216no—
398#106—216no—
399#102—216no—
400#102—216no—
401#102—48no—
402#102—48no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—48no—
408#95—22no—
409#95—22no—
410#95—28no—
411#95—28no—
412C24—24no—
413C24—24no—
414C24—28no—
415C24—22no—
416C24—22no—
417C24—24no—
418C24—24no—
419C24—42no—
420C24—44no—
421C24—44no—
422C24—82no—
423C24—82no—
424C24—84no—
425C24—82no—
426C24—82no—
427C24—84no—
428C24—84no—
429C24#12014yes—
430C24C22×D412yes—
431C24C22×D411yescharacteristic
432C24C22×D411yescharacteristic
433C24C22×D411yescharacteristic
434#41—84no—
435#41—84no—
436C2×Q8—22no—
437C2×Q8—42no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—28no—
447C2×D4—28no—
448C2×D4—28no—
449C2×D4—24no—
450C2×D4—28no—
451C2×D4—28no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—28no—
458C2×D4—28no—
459C2×D4—48no—
460C2×D4—48no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—48no—
464C2×D4—48no—
465C2×D4—84no—
466C2×D4—84no—
467C2×D4—82no—
468C2×D4—82no—
469C2×D4—84no—
470C2×D4—84no—
471C2×D4—84no—
472C2×D4—88no—
473C2×D4—88no—
474C2×D4—84no—
475C2×D4—84no—
476C2×D4—84no—
477C2×D4—84no—
478C2×D4—816no—
479C2×D4—816no—
480C2×D4—832no—
481C2×D4—832no—
482C2×D4C22×D414yes—
483C2×D4C22×D414yes—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—28no—
490C22×C4—28no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—28no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—48no—
507C22×C4—84no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—84no—
511C22×C4—82no—
512C22×C4—82no—
513C22×C4—816no—
514C22×C4#12014yes—
515C22×C4#12014yes—
516C22×C4#12014yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C2511yescharacteristic
530C2×C8—24no—
531C2×C8—24no—
532C2×C8—28no—
533C2×C8—28no—
534C2×C8—48no—
535C2×C8—48no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—28no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—88no—
553C4⋊C4C22×D414yes—
554C4⋊C4C22×D414yes—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—28no—
558C22⋊C4—28no—
559C22⋊C4—28no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—28no—
564C22⋊C4—28no—
565C22⋊C4—28no—
566C22⋊C4—28no—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—88no—
573C42—28no—
574C42—22no—
575C42—22no—
576C42—44no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—28no—
583C23—28no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—24no—
588C23—24no—
589C23—24no—
590C23—24no—
591C23—24no—
592C23—44no—
593C23—416no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—48no—
601C23—48no—
602C23—84no—
603C23—84no—
604C23—84no—
605C23—84no—
606C23—88no—
607C23—88no—
608C23—88no—
609C23—168no—
610C23—168no—
611C23—1616no—
612C23C2×#12012yes—
613C23C2×#12012yes—
614C23C2×#12012yes—
615C23#57614yes—
616C23#57714yes—
617C23C23×D411yescharacteristic
618Q8—82no—
619D4—28no—
620D4—28no—
621D4—48no—
622D4—416no—
623D4—48no—
624D4—416no—
625D4—44no—
626D4—44no—
627D4—84no—
628D4—84no—
629D4—82no—
630D4—84no—
631D4—88no—
632D4—88no—
633D4—88no—
634D4—88no—
635D4—88no—
636D4—88no—
637D4—88no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—48no—
666C2×C4—44no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—88no—
670C2×C4—88no—
671C2×C4—88no—
672C2×C4—88no—
673C2×C4—88no—
674C2×C4C2×#12012yes—
675C2×C4C2×#12012yes—
676C2×C4C2×#12012yes—
677C2×C4C23×D411yescharacteristic
678C2×C4C23×D411yescharacteristic
679C2×C4#58514yes—
680C8—44no—
681C8—44no—
682C22—28no—
683C22—28no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—416no—
692C22—44no—
693C22—48no—
694C22—42no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—44no—
703C22—88no—
704C22—88no—
705C22—168no—
706C22—168no—
707C22—1616no—
708C22#300814yes—
709C22#300914yes—
710C22C22×#12011yescharacteristic
711C22#348512yes—
712C22C2×#57612yes—
713C22C2×#57712yes—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—42no—
721C4—44no—
722C4—84no—
723C4—84no—
724C4#348512yes—
725C4#348512yes—
726C2—24no—
727C2—24no—
728C2—42no—
729C2—82no—
730C2—84no—
731C2—84no—
732C2#3544611yescharacteristic
733C2C2×#300811yescharacteristic
734C2C2×#300911yescharacteristic
735C2#6176614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#481C2semidirect
3. C22×#480C2semidirect
4. #35227C2semidirect
5. #24092C2semidirect
6. #24091C2semidirect
7. #24069C2semidirect
8. #24033C2semidirect
9. #24032C2semidirect
10. #24013C2semidirect
11. #21761C2semidirect
12. #21761C2semidirect
13. #21739C2semidirect
31. C23×C4⋊C4C22semidirect
33. C2×#481C22semidirect
34. C2×#481C22semidirect
36. C2×#480C22semidirect
37. C2×#480C22semidirect
38. C22×#107C22semidirect
39. C22×#106C22semidirect
40. C22×#102C22semidirect
41. C22×#102C22semidirect
42. C22×C22⋊C8C22semidirect
43. C22×C22⋊C8C22semidirect
50. #2487C22semidirect
51. #2486C22semidirect
52. #2317C22semidirect
53. #2311C22semidirect
54. #2280C22semidirect
55. #2059C22semidirect
56. #2041C22semidirect
57. #2034C22semidirect
58. #2033C22semidirect
59. #1893C22semidirect
60. #1893C22semidirect
61. #1875C22semidirect
62. #1875C22semidirect
63. #1869C22semidirect
64. #1869C22semidirect
65. #1864C22semidirect
66. #1862C22semidirect
67. #1854C22semidirect
68. #1824C22semidirect
69. #1824C22semidirect
70. #1818C22semidirect
71. #1786C22semidirect
72. #1786C22semidirect
73. #1786C22semidirect
74. #1780C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35372
C22×#4812
C22×#4802
#352272
#240922
#240912
#240692
#240332
#240322
#240132
#217612
#217392