Finite Group Databasebeta

← #6564399

Subgroups — #6564399

Order 512.

2147 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3324C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3#34310C211yesmaximal, characteristic
4#29864C211yesmaximal, characteristic, complemented
5#29714C212yesmaximal, complemented
6#28716C211yesmaximal, characteristic, complemented
7#28577C212yesmaximal, complemented
8#28160C211yesmaximal, characteristic, complemented
9#27650C212yesmaximal, complemented
10#25325C212yesmaximal, complemented
11#25216C214yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#24538C214yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23628C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22504C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×D42C2211yescharacteristic
20C2×#518—21no—
21#3324—216no—
22C2×#486C2212yes—
23C2×#482C2214yes—
24C2×#474C2212yes—
25C2×#468C2214yes—
26C2×#460C2212yes—
27C2×#450C2214yes—
28C2×C4×SD16C2212yes—
29C2×#436C2212yes—
30C2×#418C2214yes—
31#2868C2211yescharacteristic, complemented
32#2857C2212yescomplemented
33#2568C2212yescomplemented
34#2547—21no—
35#2547—21no—
36#2499C2214yescomplemented
37#2496—22no—
38#2496—22no—
39#2366C2211yescharacteristic
40#2365—21no—
41#2365—21no—
42#2363—21no—
43#2361C2212yes—
44#2358—22no—
45#2354C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47D4×C22⋊C4—21no—
48D4×C22⋊C4C2212yes—
49#2324—21no—
50#2321—22no—
51C4×C4⋊Q8C2211yescharacteristic, complemented
52C4×#127C2211yescharacteristic
53C4×#121C2212yes—
54#2293C2211yescharacteristic
55C2×#390C2212yes—
56#2285C2212yescomplemented
57C42×D4C2211yescharacteristic
58#2084C2211yescharacteristic, complemented
59#2083C2212yescomplemented
60#2083C2212yescomplemented
61#2073C2214yes—
62#2073C2214yescomplemented
63#2071C2212yes—
64#2071C2211yescharacteristic
65#2071C2212yes—
66#2071C2212yescomplemented
67#2071C2212yescomplemented
68#2071C2212yescomplemented
69#2066C2212yescomplemented
70#2055C2214yes—
71#2055C2212yes—
72#2055C2214yescomplemented
73#2043C2211yescharacteristic, complemented
74#2013C2211yescharacteristic
75#2013C2212yescomplemented
76#1990C2212yescomplemented
77#1988C2212yescomplemented
78#1949C2214yescomplemented
79#1939C2212yescomplemented
80#1928C2214yescomplemented
81#1894C2214yescomplemented
82#1877C2214yescomplemented
83#1877C2214yescomplemented
84#1872C2214yescomplemented
85#1851C2214yescomplemented
86#1831C2211yescharacteristic, complemented
87#1825C2214yescomplemented
88#1804C2214yescomplemented
89#1803C2212yescomplemented
90#1802C2214yescomplemented
91#1797C2212yescomplemented
92C4×C4⋊C8C2211yescharacteristic, complemented
93C4×#103C2212yescomplemented
94#1736C2214yescomplemented
95#1728C2212yescomplemented
96C23×D4—21no—
97C23×D4C2312yes—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100C22×M16C2314yes—
101#556—28no—
102#556—28no—
103D42—24no—
104D42D418yes—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#127C2311yescharacteristic
110C2×#126—24no—
111C2×#124—22no—
112C2×C22⋊Q8C2314yes—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×#120—22no—
116#518—48no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C2×#117C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2312yes—
139#486—28no—
140#482—216no—
141#474—28no—
142#468—216no—
143#460—28no—
144#450—216no—
145C4×SD16—28no—
146#436—28no—
147C2×#107C2314yes—
148C2×#106C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152#418—216no—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C4×C8C2312yes—
164#400—24no—
165#397—24no—
166#397—24no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2314yes—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—22no—
175#390—22no—
176#390C2312yes—
177#390C2311yescharacteristic
178#390C2312yes—
179#390C2312yes—
180#390C2311yescharacteristic
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#388—21no—
189#388—22no—
190#388—21no—
191#388—22no—
192#388—22no—
193#387—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#385—22no—
201#385—22no—
202#384—21no—
203#384—21no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#382C2314yes—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380C2312yes—
215#379—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4C2312yes—
230C43C2311yescharacteristic
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2314yes—
243C25—22no—
244C22×Q8—22no—
245C22×Q8C2412yes—
246C22×D4—22no—
247C22×D4—28no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—24no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4—44no—
256C22×D4C2×D411yescharacteristic
257C22×D4C2×D418yes—
258C22×D4C2412yes—
259C22×D4C2411yescharacteristic
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C2×SD16—24no—
268C2×SD16—28no—
269C2×SD16—42no—
270C2×SD16—42no—
271C2×SD16—44no—
272C2×SD16—44no—
273C2×M16—28no—
274C2×M16—44no—
275C2×M16—44no—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—42no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284#127—22no—
285#127C2×D414yes—
286#126—416no—
287#124—48no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290#121—24no—
291#121—28no—
292#121C2×D418yes—
293#120—216no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—48no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—42no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—48no—
303C4×D4—44no—
304C4×D4—48no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—82no—
308C4×D4C2×D414yes—
309C4×D4C2×D414yes—
310C4×D4#4114yes—
311#117—28no—
312#117—28no—
313#117—44no—
314#117—44no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—28no—
380#106—24no—
381C4⋊C8—24no—
382C4⋊C8—44no—
383C4⋊C8—42no—
384#103—24no—
385#103—42no—
386#103—44no—
387#103—42no—
388#102—24no—
389#102—28no—
390#102—28no—
391#102—28no—
392#102—28no—
393C22⋊C8—28no—
394C4×C8—24no—
395#95—21no—
396#95—22no—
397#95—21no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—24no—
405#95—24no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—21no—
410C24—22no—
411C24—28no—
412C24—44no—
413C24—44no—
414C24—82no—
415C24—82no—
416C24—82no—
417C24—82no—
418C24C22×D412yes—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—28no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—416no—
441C2×D4—432no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—88no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4#12114yes—
452C2×D4C22×D412yes—
453C2×D4C22×D414yes—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D414yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4114yes—
492C22×C4C2511yescharacteristic
493SD16—48no—
494SD16—88no—
495M16—88no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—82no—
530C4⋊C4—81no—
531C4⋊C4#12114yes—
532C4⋊C4C22×D412yes—
533C22⋊C4—24no—
534C22⋊C4—28no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—81no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4#12118yes—
553C42—22no—
554C42—22no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—42no—
559C42—42no—
560C42—41no—
561C42—44no—
562C42—48no—
563C42—44no—
564C42—42no—
565C42AGL(1,ℤ/8)14yes—
566C42C22×D412yes—
567C42C2×#4112yes—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—416no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—41no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—42no—
585C23—41no—
586C23—42no—
587C23—41no—
588C23—42no—
589C23—88no—
590C23—88no—
591C23—164no—
592C23—164no—
593C23—164no—
594C23—164no—
595C23C2×#12114yes—
596C23C23×D411yescharacteristic
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—88no—
602D4—24no—
603D4—28no—
604D4—28no—
605D4—416no—
606D4—44no—
607D4—42no—
608D4—48no—
609D4—88no—
610D4—84no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—48no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4—164no—
658C2×C4—164no—
659C2×C4C2×#12114yes—
660C2×C4C2×#12112yes—
661C2×C4C2×#12112yes—
662C2×C4C2×AGL(1,ℤ/8)12yes—
663C2×C4C2×AGL(1,ℤ/8)12yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4#57512yes—
666C2×C4#57712yes—
667C2×C4C23×D411yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4#58512yes—
670C2×C4#58512yes—
671C8—42no—
672C8—44no—
673C8—42no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—82no—
688C22—81no—
689C22—82no—
690C22—81no—
691C22—82no—
692C22—81no—
693C22—82no—
694C22—81no—
695C22—1616no—
696C22C22×#12111yescharacteristic
697C22#348511yescharacteristic
698C22#350011yescharacteristic
699C22C22×AGL(1,ℤ/8)11yescharacteristic
700C22C2×#57511yescharacteristic
701C22C2×#57711yescharacteristic
702C22#358611yescharacteristic
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—24no—
708C4—42no—
709C4—42no—
710C4—41no—
711C4—42no—
712C4—82no—
713C4—81no—
714C4—82no—
715C4—81no—
716C4—82no—
717C4—84no—
718C4#303614yes—
719C4#348512yes—
720C4#352112yes—
721C2—44no—
722C2—84no—
723C2—82no—
724C2#3600911yescharacteristic
725C2C2×#303611yescharacteristic
726C2#6042911yescharacteristic
727C2#6172211yescharacteristic
728C2#6173911yescharacteristic
729C2#6253911yescharacteristic
730C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36188C2semidirect
4. #29864C2semidirect
5. #29714C2semidirect
6. #28716C2semidirect
7. #28577C2semidirect
8. #28160C2semidirect
9. #27650C2semidirect
10. #25325C2semidirect
11. #25216C2semidirect
12. #24681C2semidirect
13. #24538C2semidirect
14. #23677C2semidirect
15. #23628C2semidirect
16. #22700C2semidirect
17. #22504C2semidirect
31. #2868C22semidirect
32. #2857C22semidirect
33. #2568C22semidirect
36. #2499C22semidirect
51. C4×C4⋊Q8C22semidirect
56. #2285C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #2083C22semidirect
62. #2073C22semidirect
66. #2071C22semidirect
67. #2071C22semidirect
68. #2071C22semidirect
69. #2066C22semidirect
72. #2055C22semidirect
73. #2043C22semidirect
75. #2013C22semidirect
76. #1990C22semidirect
77. #1988C22semidirect
78. #1949C22semidirect
79. #1939C22semidirect
80. #1928C22semidirect
81. #1894C22semidirect
82. #1877C22semidirect
83. #1877C22semidirect
84. #1872C22semidirect
85. #1851C22semidirect
86. #1831C22semidirect
87. #1825C22semidirect
88. #1804C22semidirect
89. #1803C22semidirect
90. #1802C22semidirect
91. #1797C22semidirect
92. C4×C4⋊C8C22semidirect
93. C4×#103C22semidirect
94. #1736C22semidirect
95. #1728C22semidirect

Maximal subgroups

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

subgroupindex
C2×#33242
#361882
#343102
#298642
#297142
#287162
#285772
#281602
#276502
#253252
#252162
#246812
#245382
#236772
#236282
#227002
#225042