Finite Group Databasebeta

← #4387587

Subgroups — #4387587

Order 512.

3710 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2405C211yesmaximal, characteristic
2C2×#2309C211yesmaximal, characteristic
3#16391C214yesmaximal, complemented
4#15983C214yesmaximal, complemented
5#15803C214yesmaximal, complemented
6#15764C214yesmaximal, complemented
7#15285C214yesmaximal, complemented
8#15114C214yesmaximal, complemented
9#14673C211yesmaximal, characteristic, complemented
10#14341C211yesmaximal, characteristic, complemented
11#14338C211yesmaximal, characteristic, complemented
12#13869C211yesmaximal, characteristic, complemented
13#13867C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C22×#123C2212yes—
16C22×#117C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2405—28no—
20C2×#397C2214yescomplemented
21C2×#394C2214yes—
22#2309—28no—
23C2×#390C2214yes—
24C2×#390C2214yes—
25C2×#387C2214yescomplemented
26C2×#387C2214yescomplemented
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#381C2212yescomplemented
32C2×#380C2212yescomplemented
33C2×#379C2211yescharacteristic
34C2×#379C2211yescharacteristic
35C2×#379C2211yescharacteristic
36C2×#379C2212yes—
37C2×#379C2212yes—
38C2×#379C2211yescharacteristic
39C2×#379C2212yes—
40#1444C2214yescomplemented
41#1442C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1438C2214yes—
60#1434C2214yescomplemented
61#1434C2214yescomplemented
62C22⋊#95C2212yes—
63C22⋊#95C2212yescomplemented
64C22⋊#95C2212yescomplemented
65C22⋊#95C2212yescomplemented
66C22⋊#95C2212yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×D4—24no—
72C23×D4—24no—
73C23×D4—44no—
74C23×D4—44no—
75C23×D4D418yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#123—28no—
84C2×#123—44no—
85C2×#123—44no—
86C2×#117—28no—
87C2×#117—44no—
88C2×#117—44no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2314yes—
121#397—48no—
122#394—216no—
123#390—216no—
124#390—216no—
125#387—48no—
126#387—48no—
127#386—28no—
128#386—44no—
129#386—44no—
130#385—28no—
131#381—44no—
132#380—44no—
133#379—28no—
134#379—24no—
135#379—24no—
136#379—28no—
137#379—24no—
138#379—24no—
139#379—44no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25—22no—
158C25—22no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25—42no—
163C25—42no—
164C25—42no—
165C25—42no—
166C25—42no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25—42no—
171C25C2×D414yes—
172C25C2×D414yes—
173C25#4114yes—
174C25#4114yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C22×D4—232no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—432no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—432no—
194C22×D4—416no—
195C22×D4—416no—
196C22×D4C2×D418yes—
197C22×D4C2×D418yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C2×D414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230#123—432no—
231#117—432no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—416no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—28no—
361C24—44no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—44no—
366C24—44no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—44no—
377C24—44no—
378C24—48no—
379C24—82no—
380C24—82no—
381C24—82no—
382C24—82no—
383C24—84no—
384C24—84no—
385C24—88no—
386C24—88no—
387C24—88no—
388C24—88no—
389C24—88no—
390C24—88no—
391C24—88no—
392C24—88no—
393C24#123116yes—
394C24C22×D414yes—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C2×#4112yes—
399C24C2×#4112yes—
400C24C2×#4112yes—
401C24C2×#4112yes—
402C24C2×#4112yes—
403C24C2×#4112yes—
404C24C2511yescharacteristic
405C2×D4—216no—
406C2×D4—216no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—4128no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—216no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
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—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—88no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4C22×D414yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)14yes—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—88no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—84no—
523C22⋊C4—88no—
524C42—416no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—28no—
565C23—28no—
566C23—416no—
567C23—416no—
568C23—416no—
569C23—416no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—88no—
601C23—88no—
602C23—88no—
603C23—88no—
604C23—88no—
605C23—88no—
606C23—832no—
607C23—832no—
608C23—832no—
609C23—88no—
610C23—88no—
611C23—88no—
612C23—88no—
613C23—832no—
614C23C2×#12314yes—
615C23C2×#12314yes—
616C23C2×#12314yes—
617C23C23×D411yescharacteristic
618C23C22×#4111yescharacteristic
619C23C22×#4111yescharacteristic
620C23C2×H(5,𝔽2)12yes—
621C23C2×H(5,𝔽2)12yes—
622C23C2×H(5,𝔽2)12yes—
623C23#58512yes—
624C23#58512yes—
625C23#58512yes—
626D4—432no—
627D4—432no—
628C2×C4—232no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—88no—
684C2×C4—88no—
685C2×C4—88no—
686C2×C4—88no—
687C2×C4#53418yes—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—48no—
710C22—48no—
711C22—48no—
712C22—48no—
713C22—48no—
714C22—48no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—48no—
719C22—48no—
720C22—832no—
721C22—832no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—84no—
727C22—84no—
728C22C22×#12311yescharacteristic
729C22C2×#53412yes—
730C22#348512yes—
731C22#349011yescharacteristic
732C22#349011yescharacteristic
733C22#349011yescharacteristic
734C22#349012yes—
735C22#349111yescharacteristic
736C22#349111yescharacteristic
737C22#349111yescharacteristic
738C22#349112yes—
739C22#349612yes—
740C22#350512yes—
741C22#354014yes—
742C22#354214yes—
743C22#354614yes—
744C22#355114yes—
745C4—28no—
746C4—44no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—84no—
751C4—84no—
752C4—84no—
753C4—84no—
754C2—44no—
755C2—44no—
756C2—44no—
757C2—84no—
758C2#3499211yescharacteristic
759C2#3536011yescharacteristic
760C2#3544811yescharacteristic
761C2#4243214yes—
762C2#4638911yescharacteristic
763C2#4699711yescharacteristic
764C2#4699811yescharacteristic
765C2#4719511yescharacteristic
766C2#4960814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16391C2semidirect
4. #15983C2semidirect
5. #15803C2semidirect
6. #15764C2semidirect
7. #15285C2semidirect
8. #15114C2semidirect
9. #14673C2semidirect
10. #14341C2semidirect
11. #14338C2semidirect
12. #13869C2semidirect
20. C2×#397C22semidirect
25. C2×#387C22semidirect
26. C2×#387C22semidirect
31. C2×#381C22semidirect
32. C2×#380C22semidirect
40. #1444C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
60. #1434C22semidirect
61. #1434C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24052
C2×#23092
#163912
#159832
#158032
#157642
#152852
#151142
#146732
#143412
#143382
#138692
#138672