Finite Group Databasebeta

← #1423417

Subgroups — #1423417

Order 512.

4624 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#559C211yesmaximal, characteristic, complemented
2C2×#2709C212yesmaximal, complemented
3C2×#2541C211yesmaximal, characteristic
4#15983C212yesmaximal, complemented
5#15673C214yesmaximal, complemented
6#15314C214yesmaximal, complemented
7#15261C212yesmaximal, complemented
8#14883C212yesmaximal, complemented
9#14730C214yesmaximal, complemented
10#14419C211yesmaximal, characteristic, complemented
11#14174C212yesmaximal, complemented
12#14023C214yesmaximal, complemented
13#13968C212yesmaximal, complemented
14C2×#559—28no—
15C2×#559—216no—
16C22×#127C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C22×#124C2214yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#121C2214yes—
23C22×C4×Q8C2212yes—
24C22×C4×D4C2214yes—
25C22×C4×D4C2211yescharacteristic
26#2709—216no—
27#2541—28no—
28C2×#395C2212yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#393C2214yes—
33C2×#392C2214yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#381C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C22×#95C2212yes—
46#1446C2212yes—
47#1442C2212yes—
48#1441C2214yes—
49#1441C2212yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2212yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C23×Q8C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2314yes—
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83#559—2128no—
84C2×#127—28no—
85C2×#127—28no—
86C2×#127—24no—
87C2×#127—24no—
88C2×#127—42no—
89C2×#127—42no—
90C2×#127D418yes—
91C2×#124—216no—
92C2×#124—216no—
93C2×#124—216no—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—216no—
98C2×#123D4116yescomplemented
99C2×#121—216no—
100C2×#121—216no—
101C2×#121—48no—
102C2×C4×Q8—216no—
103C2×C4×Q8—28no—
104C2×C4×D4—216no—
105C2×C4×D4—216no—
106C2×C4×D4—24no—
107C2×C4×D4—216no—
108C2×C4×D4—42no—
109C2×C4×D4—42no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#395—28no—
136#394—216no—
137#394—216no—
138#394—216no—
139#393—48no—
140#392—216no—
141#390—24no—
142#390—24no—
143#387—216no—
144#386—28no—
145#386—216no—
146#386—216no—
147#386—44no—
148#386—44no—
149#386—48no—
150#381—44no—
151C4×C4⋊C4—24no—
152C2×#95—44no—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25C2412yes—
170C25C2412yes—
171C25C2412yes—
172C22×Q8—28no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—44no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4C2×D4116yes—
213C22×D4C2×D414yes—
214C22×D4C2×D414yes—
215C22×D4C2×D418yes—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—44no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—44no—
237C23×C4C2×D418yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248#127—232no—
249#127—416no—
250#127—416no—
251#124—2128no—
252#123—264no—
253#123—264no—
254#121—464no—
255C4×Q8—264no—
256C4×D4—2128no—
257C4×D4—416no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4C2×D4116yes—
271C2×C4⋊C4#4118yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
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×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D4116yes—
316C2×C22⋊C4C2×D418yes—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42C2×D414yes—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24C22×D414yes—
369C24C22×D414yes—
370C24C22×D414yes—
371C24C22×D414yes—
372C24C2511yescharacteristic
373C2×Q8—216no—
374C2×Q8—416no—
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×D4—232no—
379C2×D4—28no—
380C2×D4—232no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—416no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—48no—
401C2×D4—416no—
402C2×D4—44no—
403C2×D4—48no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—88no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—88no—
412C2×D4—88no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4#120116yes—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
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—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—88no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4—88no—
485C22×C4C22×D418yes—
486C22×C4C22×D418yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4H(5,𝔽2)12yes—
496C4⋊C4—216no—
497C4⋊C4—232no—
498C4⋊C4—216no—
499C4⋊C4—416no—
500C4⋊C4—44no—
501C22⋊C4—232no—
502C22⋊C4—216no—
503C22⋊C4—232no—
504C22⋊C4—216no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—88no—
517C42—232no—
518C42—28no—
519C42—28no—
520C42—44no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—28no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—42no—
583C23—44no—
584C23—44no—
585C23C2×#12018yes—
586C23C23×D411yescharacteristic
587C23C23×D412yes—
588C23C22×#4112yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23#58514yes—
593C23#58512yes—
594Q8—432no—
595D4—416no—
596D4—464no—
597D4—88no—
598D4—816no—
599D4—88no—
600D4—832no—
601D4—816no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—816no—
657C2×C4—816no—
658C2×C4—816no—
659C2×C4—816no—
660C2×C4C2×#12014yes—
661C2×C4#54618yes—
662C2×C4#55914yescomplemented
663C22—28no—
664C22—216no—
665C22—44no—
666C22—48no—
667C22—44no—
668C22—48no—
669C22—48no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—48no—
677C22—44no—
678C22—48no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—82no—
684C22—82no—
685C22—84no—
686C22—82no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22—84no—
692C22C22×#12011yescharacteristic
693C22C2×#54612yes—
694C22D4×#4114yes—
695C22C2×#55911yescharacteristic
696C22#348411yescharacteristic
697C22#348511yescharacteristic
698C22#348512yes—
699C22#348814yes—
700C22#349612yes—
701C22#349612yes—
702C22#350812yes—
703C22#352414yes—
704C22#353312yes—
705C22#354012yes—
706C22#354311yescharacteristic
707C22#356214yes—
708C4—24no—
709C4—24no—
710C4—28no—
711C4—48no—
712C4—42no—
713C4—44no—
714C4—48no—
715C4—82no—
716C4—84no—
717C4—84no—
718C2—48no—
719C2—84no—
720C2—162no—
721C2#3534711yescharacteristic
722C2#3621212yes—
723C2#4392014yes—
724C2#4514112yes—
725C2#5068412yes—
726C2#5205411yescharacteristic
727C2#5914311yescharacteristic
728C2#5962812yes—
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
1. C22×#559C2semidirect
2. C2×#2709C2semidirect
4. #15983C2semidirect
5. #15673C2semidirect
6. #15314C2semidirect
7. #15261C2semidirect
8. #14883C2semidirect
9. #14730C2semidirect
10. #14419C2semidirect
11. #14174C2semidirect
12. #14023C2semidirect
13. #13968C2semidirect
98. C2×#123D4semidirect
662. C2×C4#559semidirect

Maximal subgroups

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

subgroupindex
C22×#5592
C2×#27092
C2×#25412
#159832
#156732
#153142
#152612
#148832
#147302
#144192
#141742
#140232
#139682