Finite Group Databasebeta

← #6499897

Subgroups — #6499897

Order 512.

4574 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3496C211yesmaximal, characteristic
2C2×#3187C212yesmaximal, complemented
3C2×#2417C211yesmaximal, characteristic, complemented
4#25531C218yesmaximal, complemented
5#24432C218yesmaximal, complemented
6#24061C212yesmaximal, complemented
7#24005C212yesmaximal, complemented
8#23931C212yesmaximal, complemented
9#23919C212yesmaximal, complemented
10#21738C211yesmaximal, characteristic, complemented
11C2×#1874C212yesmaximal, complemented
12C22×H(5,𝔽2)—21no—
13#3496—216no—
14C2×#547C2214yes—
15C2×#546—24no—
16C2×D42C2214yes—
17C2×#537—21no—
18C2×#534—21no—
19C2×#534—21no—
20C2×#532C2212yes—
21C22×C22⋊Q8C2211yescharacteristic, complemented
22C22×C22⋊Q8C2211yescharacteristic, complemented
23C22×#121C2212yes—
24C22×#121C2212yes—
25C22×C4×D4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic, complemented
27#3187—216no—
28C2×#481C2214yescomplemented
29C2×#483C2214yescomplemented
30C2×#465C2218yescomplemented
31C2×#448C2214yescomplemented
32C2×#450C2214yescomplemented
33C2×#450C2214yescomplemented
34C22×#102C2214yescomplemented
35C22×C22⋊C8C2212yescomplemented
36#2417—24no—
37#2417C2218yescomplemented
38C2×#398—21no—
39C2×#396—21no—
40C2×#391C2211yescharacteristic, complemented
41C2×#389C2211yescharacteristic, complemented
42C2×#386—21no—
43C2×#380—21no—
44C2×#380C2211yescharacteristic, complemented
45#2058C2214yescomplemented
46#2041C2214yescomplemented
47#2034C2218yescomplemented
48#2024C2214yescomplemented
49#2014C2214yescomplemented
50#2000C2218yescomplemented
51#1877C2218yescomplemented
52#1874C2214yescomplemented
53#1874C2218yescomplemented
54#1874C2214yescomplemented
55#1872C2214yescomplemented
56#1869C2218yescomplemented
57#1864C2218yescomplemented
58#1853C2214yescomplemented
59#1817C2212yescomplemented
60#1786C2214yescomplemented
61#1779C2212yescomplemented
62C2×#340C2214yescomplemented
63C2×H(5,𝔽2)—216no—
64C2×H(5,𝔽2)—48no—
65C2×H(5,𝔽2)—48no—
66C22×#41—24no—
67C22×#41—22no—
68C23×Q8C2311yescharacteristic
69C23×D4—22no—
70C23×D4—22no—
71C23×D4—24no—
72C23×D4D411yescharacteristic
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4—21no—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C22×SD16C2318yes—
79C23×C8C2314yes—
80#547—216no—
81#547—216no—
82#546—264no—
83D42—216no—
84D42—216no—
85#537—216no—
86#534—216no—
87#534—216no—
88#532—216no—
89C2×C4⋊Q8C2312yes—
90C2×#127C2312yes—
91C2×#124—22no—
92C2×#123—24no—
93C2×#123—24no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2314yes—
101C2×#121—22no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121—22no—
105C2×#121—28no—
106C2×#121—22no—
107C2×#121—216no—
108C2×#121—24no—
109C2×#121D4116yes—
110C2×#121C2314yes—
111C2×#121C2314yes—
112C2×#121C2314yes—
113C2×#120—24no—
114C2×#120—24no—
115C2×C4×D4—28no—
116C2×C4×D4—22no—
117C2×C4×D4D414yes—
118C2×C4×D4C2314yes—
119C2×C4×D4C2314yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—44no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4D411yescharacteristic
139C22×C22⋊C4D411yescharacteristic
140C22×C42C2311yescharacteristic
141#483—216no—
142#481—216no—
143#465—232no—
144#450—216no—
145#450—48no—
146#448—216no—
147C2×#106C2318yes—
148C2×#103C2318yes—
149C2×#102—48no—
150C2×#102C2318yes—
151C2×#102C2318yes—
152C2×#102C2318yes—
153C2×C22⋊C8—28no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#398—28no—
159#396—28no—
160#391—22no—
161#391C2314yes—
162#389—22no—
163#389C2314yes—
164#386—28no—
165#380—22no—
166#380—28no—
167#380C2314yes—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172#340C2318yes—
173#340C2318yes—
174#336C2314yes—
175#336C2314yes—
176C25—22no—
177C25—22no—
178C25C2×D411yescharacteristic
179C25C2×D411yescharacteristic
180H(5,𝔽2)—4128no—
181C2×#41—216no—
182C2×#41—232no—
183C2×#41—48no—
184C2×#41—416no—
185C2×#41—416no—
186C2×#41—48no—
187C22×Q8—22no—
188C22×Q8—21no—
189C22×Q8—24no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8C2412yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—216no—
196C22×D4—232no—
197C22×D4—216no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—416no—
212C22×D4—48no—
213C22×D4—48no—
214C22×D4—48no—
215C22×D4—48no—
216C22×D4—416no—
217C22×D4SD1618yescomplemented
218C22×D4C2×D414yes—
219C22×D4C2×D418yes—
220C22×D4C2414yes—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—82no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D418yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C2×SD16—216no—
248C2×SD16—48no—
249C2×SD16—48no—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2418yes—
254C4⋊Q8—28no—
255#127—28no—
256#124—216no—
257#123—232no—
258#123—232no—
259C22⋊Q8—232no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—44no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266#121—216no—
267#121—216no—
268#121—216no—
269#121—232no—
270#121—28no—
271#121—216no—
272#121—216no—
273#121—28no—
274#121—28no—
275#121—28no—
276#121C2×D4116yes—
277#121C2×D4116yes—
278#120—232no—
279#120—232no—
280C4×D4—216no—
281C4×D4—28no—
282C4×D4—216no—
283C4×D4—28no—
284C4×D4—28no—
285C4×D4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—48no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D418yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—216no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4C2×D414yes—
330C2×C22⋊C4C2×D414yes—
331C2×C42—24no—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334#106—216no—
335#103—216no—
336#102—216no—
337#102—216no—
338#102—48no—
339C22⋊C8—28no—
340C22⋊C8—28no—
341C22⋊C8—44no—
342C22⋊C8—44no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347C24—22no—
348C24—22no—
349C24—28no—
350C24—24no—
351C24—28no—
352C24—41no—
353C24—41no—
354C24—41no—
355C24—41no—
356C24—41no—
357C24—41no—
358C24—41no—
359C24—41no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—44no—
364C24—82no—
365C24—82no—
366C24—82no—
367C24—82no—
368C24—82no—
369C24—82no—
370C24—82no—
371C24—82no—
372C24#12014yes—
373C24C2×SD1614yes—
374C24C2×SD1614yes—
375C24AGL(1,ℤ/8)14yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379#41—464no—
380#41—4128no—
381C2×Q8—28no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—88no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—232no—
394C2×D4—216no—
395C2×D4—28no—
396C2×D4—216no—
397C2×D4—216no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—416no—
405C2×D4—44no—
406C2×D4—4128no—
407C2×D4—48no—
408C2×D4—464no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—464no—
415C2×D4—44no—
416C2×D4—88no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—816no—
424C2×D4C2×SD16116yes—
425C2×D4C22×D418yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—216no—
435C22×C4—28no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—416no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—48no—
476C22×C4—42no—
477C22×C4—88no—
478C22×C4—84no—
479C22×C4—816no—
480C22×C4#12018yes—
481C22×C4#12014yes—
482C22×C4C2×SD1614yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C2511yescharacteristic
492SD16—816no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—28no—
496C2×C8—88no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4C22×D418yes—
513C4⋊C4H(5,𝔽2)14yescomplemented
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—28no—
517C22⋊C4—216no—
518C22⋊C4—216no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—84no—
523C42—24no—
524C42—24no—
525C23—21no—
526C23—22no—
527C23—21no—
528C23—22no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—28no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—416no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—416no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23—82no—
561C23—88no—
562C23—88no—
563C23—88no—
564C23—88no—
565C23—164no—
566C23—164no—
567C23—164no—
568C23—164no—
569C23—164no—
570C23—164no—
571C23—164no—
572C23—164no—
573C23#450116yescomplemented
574C23C2×#12014yes—
575C23C2×#12012yes—
576C23C22×SD1614yes—
577C23C22×SD1614yes—
578C23C22×SD1612yes—
579C23C2×AGL(1,ℤ/8)12yes—
580C23C2×AGL(1,ℤ/8)14yes—
581C23C23×D411yescharacteristic
582Q8—416no—
583Q8—48no—
584Q8—84no—
585D4—232no—
586D4—216no—
587D4—416no—
588D4—432no—
589D4—432no—
590D4—432no—
591D4—48no—
592D4—88no—
593D4—88no—
594D4—88no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—432no—
616C2×C4—48no—
617C2×C4—416no—
618C2×C4—42no—
619C2×C4—416no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—84no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—816no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4C2×#12012yes—
641C2×C4C2×#12014yes—
642C2×C4C22×SD1614yes—
643C2×C4C23×D412yes—
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4#58512yes—
646C8—48no—
647C22—22no—
648C22—24no—
649C22—22no—
650C22—24no—
651C22—22no—
652C22—24no—
653C22—22no—
654C22—24no—
655C22—22no—
656C22—24no—
657C22—22no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—416no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—88no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—1616no—
676C22—1616no—
677C22C2×#45014yes—
678C22C2×#45018yes—
679C22C22×#12011yescharacteristic
680C22#348411yescharacteristic
681C22#348511yescharacteristic
682C22C23×SD1611yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22#358511yescharacteristic
685C22#358611yescharacteristic
686C4—22no—
687C4—24no—
688C4—24no—
689C4—42no—
690C4—44no—
691C4—48no—
692C4—42no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#349614yes—
697C2—24no—
698C2—28no—
699C2—44no—
700C2—88no—
701C2#3534711yescharacteristic
702C2C22×#45011yescharacteristic
703C2#6037111yescharacteristic
704C2#6156211yescharacteristic
705C2#6156911yescharacteristic
706C2#6169711yescharacteristic
707C2#6170411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3187C2semidirect
3. C2×#2417C2semidirect
4. #25531C2semidirect
5. #24432C2semidirect
6. #24061C2semidirect
7. #24005C2semidirect
8. #23931C2semidirect
9. #23919C2semidirect
10. #21738C2semidirect
11. C2×#1874C2semidirect
21. C22×C22⋊Q8C22semidirect
22. C22×C22⋊Q8C22semidirect
26. C23×C4⋊C4C22semidirect
28. C2×#481C22semidirect
29. C2×#483C22semidirect
30. C2×#465C22semidirect
31. C2×#448C22semidirect
32. C2×#450C22semidirect
33. C2×#450C22semidirect
34. C22×#102C22semidirect
35. C22×C22⋊C8C22semidirect
37. #2417C22semidirect
40. C2×#391C22semidirect
41. C2×#389C22semidirect
44. C2×#380C22semidirect
45. #2058C22semidirect
46. #2041C22semidirect
47. #2034C22semidirect
48. #2024C22semidirect
49. #2014C22semidirect
50. #2000C22semidirect
51. #1877C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
54. #1874C22semidirect
55. #1872C22semidirect
56. #1869C22semidirect
57. #1864C22semidirect
58. #1853C22semidirect
59. #1817C22semidirect
60. #1786C22semidirect
61. #1779C22semidirect
62. C2×#340C22semidirect
217. C22×D4SD16semidirect
513. C4⋊C4H(5,𝔽2)semidirect
573. C23#450semidirect

Maximal subgroups

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

subgroupindex
C2×#34962
C2×#31872
C2×#24172
#255312
#244322
#240612
#240052
#239312
#239192
#217382
C2×#18742