Finite Group Databasebeta

← #1420382

Subgroups — #1420382

Order 512.

4104 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×#2443C214yesmaximal, complemented
3C2×#2366C211yesmaximal, characteristic, complemented
4#15450C214yesmaximal, complemented
5#15342C212yesmaximal, complemented
6#15338C214yesmaximal, complemented
7#15261C212yesmaximal, complemented
8#15084C212yesmaximal, complemented
9#14796C214yesmaximal, complemented
10#14419C212yesmaximal, complemented
11#14419C211yesmaximal, characteristic, complemented
12#14223C212yesmaximal, complemented
13#14015C212yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×#123C2212yes—
18C22×#121C2214yes—
19C22×C4×D4C2214yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2541—28no—
23#2443—232no—
24C2×#399C2214yes—
25C2×#398C2212yes—
26C2×#397C2214yes—
27C2×#394C2214yes—
28C2×#392C2214yes—
29#2366—28no—
30C2×#390C2214yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48#1445C2212yes—
49#1444C2214yes—
50#1442C2212yes—
51#1441C2212yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2214yes—
63#1438C2212yes—
64#1436C2214yes—
65#1434C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2314yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×#127—28no—
82C2×#127—24no—
83C2×#127—24no—
84C2×#127—28no—
85C2×#127—42no—
86C2×#127—42no—
87C2×#127D418yes—
88C2×#123—216no—
89C2×#123—44no—
90C2×#121—216no—
91C2×#121—216no—
92C2×#121—48no—
93C2×C4×D4—216no—
94C2×C4×D4—24no—
95C2×C4×D4—216no—
96C2×C4×D4—42no—
97C2×C4×D4—48no—
98C2×C4×D4—42no—
99C22×C4⋊C4—216no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#399—216no—
129#398D4116yescomplemented
130#397—216no—
131#394—216no—
132#392—216no—
133#390—216no—
134#390—24no—
135#390—24no—
136#388—216no—
137#388—48no—
138#387—216no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—216no—
143#386—216no—
144#386—216no—
145#386—44no—
146#385—216no—
147#385—216no—
148#385—44no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25C2412yes—
167C25C2412yes—
168C25C2412yes—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—22no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—42no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4C2×D4116yes—
205C22×D4C2×D414yes—
206C22×D4C2×D414yes—
207C22×D4#4118yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238#127—232no—
239#127—416no—
240#127—416no—
241#123—432no—
242#121—464no—
243C4×D4—416no—
244C4×D4—464no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—432no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D4116yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42C2×D414yes—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95C2×D4116yes—
335#95C2×D418yes—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24C22×D414yes—
362C24C22×D414yes—
363C24C2×#4114yes—
364C24C2×#4114yes—
365C24C2511yescharacteristic
366C2×D4—232no—
367C2×D4—28no—
368C2×D4—232no—
369C2×D4—44no—
370C2×D4—416no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—44no—
391C2×D4—48no—
392C2×D4—88no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4#123116yescomplemented
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4C22×D414yes—
472C22×C4C22×D418yes—
473C22×C4C22×D418yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C2×#4114yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)12yes—
482C4⋊C4—232no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—416no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—88no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—88no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C42—28no—
506C42—28no—
507C42—44no—
508C42—416no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23C2×#12318yes—
574C23C23×D411yescharacteristic
575C23C23×D412yes—
576C23C22×#4112yes—
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×H(5,𝔽2)11yescharacteristic
580C23#58514yes—
581C23#58512yes—
582D4—416no—
583D4—464no—
584D4—88no—
585D4—816no—
586D4—88no—
587D4—832no—
588D4—816no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—432no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—816no—
645C2×C4—816no—
646C2×C4—816no—
647C2×C4C2×#12314yes—
648C2×C4D4218yes—
649C2×C4D4214yes—
650C22—28no—
651C22—216no—
652C22—44no—
653C22—44no—
654C22—48no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—82no—
671C22—84no—
672C22—84no—
673C22—82no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—84no—
679C22C22×#12311yescharacteristic
680C22C2×D4212yes—
681C22C2×D4211yescharacteristic
682C22D4×#4114yes—
683C22#348411yescharacteristic
684C22#348511yescharacteristic
685C22#348512yes—
686C22#348714yes—
687C22#349812yes—
688C22#350512yes—
689C22#350512yes—
690C22#351314yes—
691C22#353312yes—
692C22#353914yes—
693C22#354012yes—
694C22#354311yescharacteristic
695C4—24no—
696C4—24no—
697C4—28no—
698C4—42no—
699C4—48no—
700C4—48no—
701C4—44no—
702C4—84no—
703C4—84no—
704C4—82no—
705C2—48no—
706C2—84no—
707C2—162no—
708C2#3535111yescharacteristic
709C2D4×#12312yes—
710C2#4425014yes—
711C2#4538712yes—
712C2#4818412yes—
713C2#5205411yescharacteristic
714C2#5900112yes—
715C2#5912611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2443C2semidirect
3. C2×#2366C2semidirect
4. #15450C2semidirect
5. #15342C2semidirect
6. #15338C2semidirect
7. #15261C2semidirect
8. #15084C2semidirect
9. #14796C2semidirect
10. #14419C2semidirect
11. #14419C2semidirect
12. #14223C2semidirect
13. #14015C2semidirect
129. #398D4semidirect
404. C2×D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×#24432
C2×#23662
#154502
#153422
#153382
#152612
#150842
#147962
#144192
#142232
#140152