Finite Group Databasebeta

← #4932469

Subgroups — #4932469

Order 512.

3870 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2709C212yesmaximal, complemented
3C2×#2406C212yesmaximal, complemented
4C2×#2431C211yesmaximal, characteristic
5#16808C214yesmaximal, complemented
6#16510C212yesmaximal, complemented
7#15856C212yesmaximal, complemented
8#15549C214yesmaximal, complemented
9#15370C214yesmaximal, complemented
10#15340C212yesmaximal, complemented
11#14969C212yesmaximal, complemented
12#14777C214yesmaximal, complemented
13#13832C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2724—28no—
22#2709—216no—
23#2431—28no—
24#2406—216no—
25C2×#399C2214yes—
26C2×#399C2212yes—
27C2×#398C2212yes—
28C2×#397C2214yes—
29C2×#395C2214yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#392C2214yes—
35C2×#391C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#389C2212yes—
39C2×#387C2214yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#385C2214yes—
46C2×#385C2214yes—
47C2×#385C2214yes—
48C2×#381C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50#1445C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2214yes—
64#1438C2212yes—
65#1437C2214yes—
66#1437C2212yes—
67#1436C2212yes—
68C4⋊#95C2214yes—
69#1434C2212yes—
70C22⋊#95C2214yes—
71C22×#41—28no—
72C22×#41—24no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41—44no—
76C23×Q8C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2314yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×#127—24no—
87C2×#127—28no—
88C2×#124—24no—
89C2×#124—44no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—44no—
95C2×#123—44no—
96C2×#121—24no—
97C2×#121—42no—
98C2×#121—42no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#399—44no—
129#399—48no—
130#398—28no—
131#397—216no—
132#395—216no—
133#394—216no—
134#394—216no—
135#394—216no—
136#394—216no—
137#392—216no—
138#391—24no—
139#390—44no—
140#390D418yes—
141#389—28no—
142#387—216no—
143#386—28no—
144#386—216no—
145#386—216no—
146#386—216no—
147#386D4116yescomplemented
148#385—216no—
149#385—216no—
150#385—216no—
151#381—28no—
152C4×C22⋊C4—24no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C25C2412yes—
173C2×#41—432no—
174C2×#41—416no—
175C2×#41—432no—
176C2×#41—416no—
177C2×#41—416no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—28no—
196C22×D4—42no—
197C22×D4—41no—
198C22×D4—41no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4C2×D418yes—
211C22×D4C2×D414yes—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4C2×D418yes—
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246#127—416no—
247#124—416no—
248#123—432no—
249#123—432no—
250#121—416no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4C2×D414yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4C2×D4116yes—
318C2×C22⋊C4C2×D4116yes—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95C2×D4116yes—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24C22×D414yes—
367C24C22×D414yes—
368C24C22×D414yes—
369C24H(5,𝔽2)12yes—
370C24C2511yescharacteristic
371#41—4128no—
372C2×Q8—28no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×D4—216no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—416no—
396C2×D4—44no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—84no—
404C2×D4—88no—
405C2×D4#127116yes—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
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—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4C22×D414yes—
482C22×C4C22×D418yes—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+414yes—
490C4⋊C4—44no—
491C4⋊C4—416no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—416no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—44no—
514C42—44no—
515C42—84no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—41no—
550C23—44no—
551C23—42no—
552C23—41no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23C2×#12718yes—
576C23C23×D411yescharacteristic
577C23C23×D412yes—
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×H(5,𝔽2)12yes—
581C23C2×2−1+412yes—
582C23#58512yes—
583C23#58514yes—
584Q8—416no—
585D4—416no—
586D4—432no—
587D4—88no—
588D4—832no—
589D4—816no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—416no—
613C2×C4—432no—
614C2×C4—432no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
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—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—816no—
653C2×C4C2×#12714yes—
654C22—28no—
655C22—216no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—48no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22C22×#12711yescharacteristic
683C22#348411yescharacteristic
684C22#348412yes—
685C22#348411yescharacteristic
686C22#348412yes—
687C22#348512yes—
688C22#348514yes—
689C22#348914yes—
690C22#348912yes—
691C22#352911yescharacteristic
692C22#353312yes—
693C22#353914yes—
694C22#354012yes—
695C22#355011yescharacteristic
696C22#355414yes—
697C22#355912yes—
698C4—24no—
699C4—42no—
700C4—44no—
701C4—44no—
702C4—48no—
703C4—44no—
704C4—44no—
705C4—48no—
706C4—84no—
707C4—84no—
708C2—48no—
709C2—82no—
710C2—162no—
711C2#3539011yescharacteristic
712C2#3606012yes—
713C2#4544112yes—
714C2#4622811yescharacteristic
715C2#4783512yes—
716C2#5017514yes—
717C2#5131712yes—
718C2#5205411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2724C2semidirect
2. C2×#2709C2semidirect
3. C2×#2406C2semidirect
5. #16808C2semidirect
6. #16510C2semidirect
7. #15856C2semidirect
8. #15549C2semidirect
9. #15370C2semidirect
10. #15340C2semidirect
11. #14969C2semidirect
12. #14777C2semidirect
13. #13832C2semidirect
147. #386D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#27092
C2×#24062
C2×#24312
#168082
#165102
#158562
#155492
#153702
#153402
#149692
#147772
#138322