Finite Group Databasebeta

← #6470365

Subgroups — #6470365

Order 512.

4002 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3192C211yesmaximal, characteristic, complemented
2C22×#407C211yesmaximal, characteristic, complemented
3C2×#2847C212yesmaximal, complemented
4#32936C214yesmaximal, complemented
5#29742C214yesmaximal, complemented
6#24557C214yesmaximal, complemented
7#24553C214yesmaximal, complemented
8#24384C214yesmaximal, complemented
9#24378C214yesmaximal, complemented
10#21942C211yesmaximal, characteristic, complemented
11#21764C212yesmaximal, complemented
12C25×C4C2211yescharacteristic
13C23×M16C2212yescomplemented
14C22×#121C2212yes—
15C22×#121C2212yescomplemented
16C22×#121C2212yescomplemented
17C22×#120—22no—
18#3192—28no—
19C2×#481C2214yescomplemented
20C2×#469C2214yescomplemented
21C2×#469C2214yescomplemented
22C2×#466C2214yescomplemented
23C2×#447C2214yescomplemented
24C2×#407—28no—
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#407C2212yescomplemented
28C2×#407C2214yescomplemented
29C22×C22⋊C8C2214yescomplemented
30#2847—22no—
31#2847—22no—
32#2847—22no—
33#2847—22no—
34#2847C2214yes—
35#2847C2214yes—
36#2847C2214yescomplemented
37#2847C2214yescomplemented
38C2×#386—22no—
39C2×#386—22no—
40C2×#386—22no—
41C2×#380C2212yes—
42C2×#380C2212yescomplemented
43C2×#380C2212yescomplemented
44#2077C2218yescomplemented
45#2034C2218yescomplemented
46#2033C2218yescomplemented
47#1999C2218yescomplemented
48#1895C2218yescomplemented
49#1895C2218yescomplemented
50#1893C2218yescomplemented
51#1874C2214yescomplemented
52#1874C2214yescomplemented
53#1853C2214yescomplemented
54#1852C2214yescomplemented
55#1824C2214yescomplemented
56#1809C2214yescomplemented
57#1809C2214yescomplemented
58#1807C2214yescomplemented
59#1786C2214yescomplemented
60C26D411yescharacteristic
61C23×D4—22no—
62C23×D4—22no—
63C23×D4—22no—
64C23×D4C2312yes—
65C24×C4—24no—
66C24×C4—24no—
67C24×C4—44no—
68C24×C4—44no—
69C24×C4D411yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4D412yes—
72C24×C4D412yes—
73C24×C4D411yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×D8C2314yescomplemented
80C22×M16—28no—
81C22×M16—44no—
82C22×M16—44no—
83C22×M16C2314yes—
84C22×M16C2312yes—
85C22×M16C2312yes—
86C23×C8C2314yes—
87C2×#121—22no—
88C2×#121—22no—
89C2×#121—22no—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121—28no—
94C2×#121—28no—
95C2×#121—48no—
96C2×#121—48no—
97C2×#121C2314yes—
98C2×#121C2314yes—
99C2×#121C2314yes—
100C2×#121C2314yes—
101C2×#121C2314yes—
102C2×#121C2314yes—
103C2×#120—24no—
104C2×#120—24no—
105C2×#120—24no—
106C2×#120—24no—
107C2×#120—416no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117#481—216no—
118#469—216no—
119#469—48no—
120#466—48no—
121#447—216no—
122C2×#107C2314yescomplemented
123C2×#106C2314yescomplemented
124C2×#106C2314yescomplemented
125C2×#102C2318yescomplemented
126C2×#102C2318yescomplemented
127C2×#102C2318yescomplemented
128#407—24no—
129#407—28no—
130#407—216no—
131#407D418yescomplemented
132C2×C22⋊C8—216no—
133C2×C22⋊C8C2318yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136#386—24no—
137#386—24no—
138#386—24no—
139#386—24no—
140#386—24no—
141#386—24no—
142#386—24no—
143#386—24no—
144#386—24no—
145#386—24no—
146#386—24no—
147#386—24no—
148#380—22no—
149#380—22no—
150#380—22no—
151#380—22no—
152#380—22no—
153#380—22no—
154#380C2314yes—
155#380C2314yes—
156#380C2314yes—
157#380C2314yes—
158#380C2314yes—
159#380C2314yes—
160C2×#95—22no—
161C2×#95—22no—
162C2×#95—22no—
163#340C2318yescomplemented
164#340C2318yescomplemented
165#340C2318yescomplemented
166C25—24no—
167C25—28no—
168C25—22no—
169C25—42no—
170C25—42no—
171C25—44no—
172C25C2×D412yes—
173C25C2×D411yescharacteristic
174C25C2×D411yescharacteristic
175C25C2×D412yes—
176C25C2×D411yescharacteristic
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—48no—
189C22×D4—48no—
190C22×D4—48no—
191C22×D4—48no—
192C22×D4—48no—
193C22×D4—48no—
194C22×D4—48no—
195C22×D4—48no—
196C22×D4—48no—
197C22×D4C2414yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—432no—
235C23×C4—432no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D414yes—
250C23×C4#4114yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×D8—28no—
257C2×D8—44no—
258C2×D8—44no—
259C2×M16—24no—
260C2×M16—28no—
261C2×M16—28no—
262C2×M16—42no—
263C2×M16—42no—
264C2×M16—44no—
265C2×M16—42no—
266C2×M16—42no—
267C2×M16—432no—
268C2×M16C2×D418yescomplemented
269C22×C8—28no—
270C22×C8—28no—
271C22×C8—44no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274#121—28no—
275#121—28no—
276#121—28no—
277#121—28no—
278#121—216no—
279#121—216no—
280#121—48no—
281#121—48no—
282#121—48no—
283#121—48no—
284#121—48no—
285#121—48no—
286#121—48no—
287#121—48no—
288#120—416no—
289#120—416no—
290#120—416no—
291#120—416no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320#107—44no—
321#106—28no—
322#106—44no—
323#102—216no—
324#102—48no—
325#102—48no—
326C22⋊C8—28no—
327C22⋊C8—216no—
328C22⋊C8C2×D4116yescomplemented
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—41no—
354C24—41no—
355C24—42no—
356C24—42no—
357C24—41no—
358C24—41no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—416no—
364C24—432no—
365C24—416no—
366C24—432no—
367C24—44no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—416no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24#12114yes—
376C24AGL(1,ℤ/8)18yes—
377C24C22×D411yescharacteristic
378C24C22×D411yescharacteristic
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—44no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—832no—
410C2×D4—832no—
411C2×D4—832no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—432no—
476C22×C4—432no—
477C22×C4—416no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4#12118yes—
491C22×C4#12114yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4114yes—
500C22×C4C2×#4112yes—
501C22×C4H(5,𝔽2)14yes—
502C22×C4H(5,𝔽2)14yes—
503C22×C4C2511yescharacteristic
504D8—88no—
505M16—48no—
506M16—416no—
507M16—48no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—416no—
514C2×C8C22×D418yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—28no—
542C23—216no—
543C23—22no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—216no—
551C23—216no—
552C23—216no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—44no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—44no—
568C23—48no—
569C23—44no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—44no—
577C23—44no—
578C23—48no—
579C23—44no—
580C23—48no—
581C23—4128no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—88no—
593C23—88no—
594C23—88no—
595C23—88no—
596C23—88no—
597C23—88no—
598C23—88no—
599C23C2×#12112yes—
600C23C2×#12114yes—
601C23#53412yes—
602C23#53412yes—
603C23C2×AGL(1,ℤ/8)14yes—
604C23C2×AGL(1,ℤ/8)14yes—
605C23C2×AGL(1,ℤ/8)14yes—
606C23#57514yes—
607C23C23×D411yescharacteristic
608D4—48no—
609D4—48no—
610D4—88no—
611D4—816no—
612D4—816no—
613D4—816no—
614D4—816no—
615D4—816no—
616D4—816no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—48no—
640C2×C4—44no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—48no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4—88no—
663C2×C4C2×#12112yes—
664C2×C4C2×#12114yes—
665C2×C4#53412yes—
666C2×C4#53412yes—
667C2×C4C23×D411yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×H(5,𝔽2)12yes—
670C2×C4C2×H(5,𝔽2)12yes—
671C8—28no—
672C8—44no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—22no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—22no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—22no—
689C22—22no—
690C22—416no—
691C22—432no—
692C22—432no—
693C22—416no—
694C22—416no—
695C22—42no—
696C22—42no—
697C22—41no—
698C22—41no—
699C22—42no—
700C22—41no—
701C22—41no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—84no—
706C22—84no—
707C22—84no—
708C22—84no—
709C22—84no—
710C22—84no—
711C22—84no—
712C22—1616no—
713C22#305518yes—
714C22#319214yescomplemented
715C22C22×#12111yescharacteristic
716C22C2×#53411yescharacteristic
717C22C2×#53411yescharacteristic
718C22C22×AGL(1,ℤ/8)12yes—
719C22C2×#57512yes—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—24no—
724C4—42no—
725C4—84no—
726C4—84no—
727C4—84no—
728C4#348412yes—
729C4#349012yes—
730C2—28no—
731C2—24no—
732C2—42no—
733C2—44no—
734C2—42no—
735C2—164no—
736C2#3498111yescharacteristic
737C2C2×#305512yes—
738C2C2×#319211yescharacteristic
739C2#6074111yescharacteristic
740C2#6075012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3192C2semidirect
2. C22×#407C2semidirect
3. C2×#2847C2semidirect
4. #32936C2semidirect
5. #29742C2semidirect
6. #24557C2semidirect
7. #24553C2semidirect
8. #24384C2semidirect
9. #24378C2semidirect
10. #21942C2semidirect
11. #21764C2semidirect
13. C23×M16C22semidirect
15. C22×#121C22semidirect
16. C22×#121C22semidirect
19. C2×#481C22semidirect
20. C2×#469C22semidirect
21. C2×#469C22semidirect
22. C2×#466C22semidirect
23. C2×#447C22semidirect
27. C2×#407C22semidirect
28. C2×#407C22semidirect
29. C22×C22⋊C8C22semidirect
36. #2847C22semidirect
37. #2847C22semidirect
42. C2×#380C22semidirect
43. C2×#380C22semidirect
44. #2077C22semidirect
45. #2034C22semidirect
46. #2033C22semidirect
47. #1999C22semidirect
48. #1895C22semidirect
49. #1895C22semidirect
50. #1893C22semidirect
51. #1874C22semidirect
52. #1874C22semidirect
53. #1853C22semidirect
54. #1852C22semidirect
55. #1824C22semidirect
56. #1809C22semidirect
57. #1809C22semidirect
58. #1807C22semidirect
59. #1786C22semidirect
79. C22×D8C23semidirect
122. C2×#107C23semidirect
123. C2×#106C23semidirect
124. C2×#106C23semidirect
125. C2×#102C23semidirect
126. C2×#102C23semidirect
127. C2×#102C23semidirect
131. #407D4semidirect
163. #340C23semidirect
164. #340C23semidirect
165. #340C23semidirect
268. C2×M16C2×D4semidirect
328. C22⋊C8C2×D4semidirect
714. C22#3192semidirect

Maximal subgroups

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

subgroupindex
C2×#31922
C22×#4072
C2×#28472
#329362
#297422
#245572
#245532
#243842
#243782
#219422
#217642