Finite Group Databasebeta

← #6472776

Subgroups — #6472776

Order 512.

3368 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic, complemented
3#32981C214yesmaximal, complemented
4#29891C214yesmaximal, complemented
5#27602C214yesmaximal, complemented
6#24555C218yesmaximal, complemented
7#24009C214yesmaximal, complemented
8#21940C211yesmaximal, characteristic, complemented
9#21938C211yesmaximal, characteristic, complemented
10#21758C211yesmaximal, characteristic, complemented
11#21758C211yesmaximal, characteristic, complemented
12C2×#1779C211yesmaximal, characteristic, complemented
13C25×C4C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#121C2212yes—
16C22×#120—21no—
17C22×#117C2212yescomplemented
18C2×#482C2218yescomplemented
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2214yescomplemented
22#2847—21no—
23#2847—21no—
24#2847—22no—
25#2847C2212yes—
26#2847C2212yes—
27#2847C2214yes—
28C2×#386—21no—
29C2×#386—22no—
30#2280—22no—
31#2280—22no—
32#2280C2214yes—
33#2280C2214yescomplemented
34C2×#381—21no—
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yescomplemented
40C2×#379—21no—
41C2×C4×C22⋊C4—21no—
42C2×C4×C22⋊C4—21no—
43#2077C2218yescomplemented
44#2076C2218yescomplemented
45#2034C2218yescomplemented
46#2033C2218yescomplemented
47#1893C2214yescomplemented
48#1893C2214yescomplemented
49#1875C2218yescomplemented
50#1874C2214yescomplemented
51#1874C2214yescomplemented
52#1854C2218yescomplemented
53#1853C2214yescomplemented
54#1853C2214yescomplemented
55#1852C2214yescomplemented
56#1852C2214yescomplemented
57#1825C2218yescomplemented
58#1823C2212yescomplemented
59#1823C2212yescomplemented
60#1808C2214yescomplemented
61#1807C2214yescomplemented
62#1786C2212yescomplemented
63#1786C2212yescomplemented
64#1779C2214yes—
65#1779C2212yescomplemented
66#1779C2212yescomplemented
67C26D411yescharacteristic
68C23×D4—22no—
69C23×D4—21no—
70C23×D4C2311yescharacteristic
71C24×C4—24no—
72C24×C4—24no—
73C24×C4—48no—
74C24×C4D411yescharacteristic
75C24×C4D411yescharacteristic
76C24×C4D412yes—
77C24×C4D411yescharacteristic
78C24×C4D412yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C23×C8C2312yes—
85C23×C8C2312yes—
86C2×#121—22no—
87C2×#121—22no—
88C2×#121—21no—
89C2×#121—21no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—48no—
93C2×#121C2314yes—
94C2×#121C2312yes—
95C2×#121C2312yes—
96C2×#121C2314yes—
97C2×#120—22no—
98C2×#120—22no—
99C2×#120—24no—
100C2×#120—48no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—48no—
104C2×#117C2314yes—
105C2×#117C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C22⋊C4—21no—
118C22×C42—22no—
119#482—416no—
120C2×#107C2318yescomplemented
121C2×#106C2318yescomplemented
122C2×#102C2318yescomplemented
123C2×#102C2318yescomplemented
124C2×C22⋊C8—24no—
125C2×C22⋊C8—24no—
126C2×C22⋊C8—48no—
127C2×C22⋊C8C2318yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2318yes—
133#386—24no—
134#386—22no—
135#386—24no—
136#386—24no—
137#386—24no—
138#386—22no—
139#386—24no—
140#381—24no—
141#381—24no—
142#380—22no—
143#380—22no—
144#380—21no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—22no—
151#380—22no—
152#380C2312yes—
153#380C2312yes—
154#380C2312yes—
155#380C2314yes—
156#380C2314yes—
157#380C2314yes—
158#380C2312yes—
159#380C2312yes—
160#380C2314yes—
161#380C2312yes—
162#379—24no—
163#379—24no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—24no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—22no—
171C2×#95—22no—
172C2×#95—21no—
173#340C2318yescomplemented
174#340C2318yescomplemented
175#340C2318yescomplemented
176#340C2318yescomplemented
177C25—24no—
178C25—24no—
179C25—24no—
180C25—21no—
181C25—44no—
182C25—44no—
183C25C2×D411yescharacteristic
184C25C2×D411yescharacteristic
185C25C2×D412yes—
186C25C2×D412yes—
187C25C2×D411yescharacteristic
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—44no—
196C22×D4—48no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4C2412yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—22no—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—432no—
235C23×C4—432no—
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4#4114yes—
250C23×C4#4114yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262#121—24no—
263#121—24no—
264#121—24no—
265#121—24no—
266#121—216no—
267#121—48no—
268#121—48no—
269#121—48no—
270#121—44no—
271#121—44no—
272#120—416no—
273#120—48no—
274#120—48no—
275#117—216no—
276#117—48no—
277#117—48no—
278#117—48no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—48no—
319#107—48no—
320#106—48no—
321#102—48no—
322#102—48no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326C22⋊C8—24no—
327C22⋊C8—216no—
328C22⋊C8—48no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—416no—
367C24—432no—
368C24—416no—
369C24—432no—
370C24—44no—
371C24—44no—
372C24—416no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24#12314yes—
377C24#13514yes—
378C24AGL(1,ℤ/8)14yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—28no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—84no—
402C2×D4—84no—
403C2×D4—832no—
404C2×D4—816no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—432no—
460C22×C4—432no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4#12314yes—
478C22×C4#12314yes—
479C22×C4#12314yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)18yes—
492C22×C4C2511yescharacteristic
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—28no—
496C2×C8—28no—
497C2×C8—84no—
498C2×C8—84no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C42—48no—
525C42—48no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—216no—
534C23—216no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—44no—
549C23—44no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—44no—
565C23—48no—
566C23—48no—
567C23—44no—
568C23—4128no—
569C23—48no—
570C23—44no—
571C23—44no—
572C23—48no—
573C23—42no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—88no—
579C23—84no—
580C23—88no—
581C23—84no—
582C23—84no—
583C23#482116yescomplemented
584C23C2×#12312yes—
585C23C2×#12312yes—
586C23C2×#12312yes—
587C23#53414yes—
588C23C2×#13512yes—
589C23C2×#13512yes—
590C23C2×#13512yes—
591C23C2×AGL(1,ℤ/8)12yes—
592C23C2×AGL(1,ℤ/8)12yes—
593C23C2×AGL(1,ℤ/8)12yes—
594C23#57514yes—
595C23C23×D411yescharacteristic
596D4—44no—
597D4—44no—
598D4—84no—
599D4—816no—
600D4—816no—
601D4—88no—
602D4—88no—
603C2×C4—216no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4C2×#12312yes—
653C2×C4#53414yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)14yes—
657C8—44no—
658C8—44no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—416no—
678C22—432no—
679C22—416no—
680C22—432no—
681C22—416no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—84no—
691C22—84no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—168no—
696C22C2×#48214yes—
697C22C2×#48214yes—
698C22C2×#48214yes—
699C22C22×#12311yescharacteristic
700C22C2×#53412yes—
701C22C22×#13511yescharacteristic
702C22C22×AGL(1,ℤ/8)11yescharacteristic
703C22C2×#57512yes—
704C4—22no—
705C4—24no—
706C4—22no—
707C4—22no—
708C4—42no—
709C4—84no—
710C4—84no—
711C4—84no—
712C4—82no—
713C4#349012yes—
714C4#349012yes—
715C2—24no—
716C2—24no—
717C2—24no—
718C2—44no—
719C2—44no—
720C2—162no—
721C2#3499211yescharacteristic
722C2C22×#48211yescharacteristic
723C2#6051111yescharacteristic
724C2#6073412yes—
725C2#6075012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2847C2
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
2. C2×#2280C2semidirect
3. #32981C2semidirect
4. #29891C2semidirect
5. #27602C2semidirect
6. #24555C2semidirect
7. #24009C2semidirect
8. #21940C2semidirect
9. #21938C2semidirect
10. #21758C2semidirect
11. #21758C2semidirect
12. C2×#1779C2semidirect
17. C22×#117C22semidirect
18. C2×#482C22semidirect
21. C22×C22⋊C8C22semidirect
33. #2280C22semidirect
39. C2×#380C22semidirect
43. #2077C22semidirect
44. #2076C22semidirect
45. #2034C22semidirect
46. #2033C22semidirect
47. #1893C22semidirect
48. #1893C22semidirect
49. #1875C22semidirect
50. #1874C22semidirect
51. #1874C22semidirect
52. #1854C22semidirect
53. #1853C22semidirect
54. #1853C22semidirect
55. #1852C22semidirect
56. #1852C22semidirect
57. #1825C22semidirect
58. #1823C22semidirect
59. #1823C22semidirect
60. #1808C22semidirect
61. #1807C22semidirect
62. #1786C22semidirect
63. #1786C22semidirect
65. #1779C22semidirect
66. #1779C22semidirect
120. C2×#107C23semidirect
121. C2×#106C23semidirect
122. C2×#102C23semidirect
123. C2×#102C23semidirect
173. #340C23semidirect
174. #340C23semidirect
175. #340C23semidirect
176. #340C23semidirect
583. C23#482semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C2×#22802
#329812
#298912
#276022
#245552
#240092
#219402
#219382
#217582
C2×#17792