Finite Group Databasebeta

← #2807592

Subgroups — #2807592

Order 512.

3942 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2676C214yesmaximal, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2537C211yesmaximal, characteristic, complemented
4#16514C212yesmaximal, complemented
5#16508C214yesmaximal, complemented
6#15761C212yesmaximal, complemented
7#15630C212yesmaximal, complemented
8#15335C214yesmaximal, complemented
9#15323C214yesmaximal, complemented
10#14811C212yesmaximal, complemented
11#14722C212yesmaximal, complemented
12#14424C211yesmaximal, characteristic, complemented
13#14176C212yesmaximal, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×C22⋊Q8C2214yes—
17C22×#121C2214yes—
18C22×#121C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2676—232no—
22#2542—28no—
23#2537—28no—
24C2×#399C2214yes—
25C2×#396C2214yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#393C2214yes—
32C2×#392C2214yes—
33C2×#392C2214yes—
34C2×#392C2214yes—
35C2×#392C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2214yes—
38C2×#390C2211yescharacteristic
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#380C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49#1446C2212yes—
50#1445C2214yes—
51#1444C2212yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2214yes—
62#1438C2212yes—
63#1438C2214yes—
64#1437C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2314yes—
76C23×D4C2312yes—
77C24×C4C2312yes—
78C24×C4C2314yes—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×#124—216no—
83C2×#124—44no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—48no—
87C2×#121—216no—
88C2×#121—216no—
89C2×#121—216no—
90C2×#121—44no—
91C2×#121—48no—
92C2×C4×D4—24no—
93C2×C4×D4—28no—
94C2×C4×D4—48no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#399—216no—
126#396—216no—
127#394—28no—
128#394—28no—
129#394—216no—
130#394—44no—
131#394D4116yescomplemented
132#393—216no—
133#392—216no—
134#392—216no—
135#392—48no—
136#392—44no—
137#390—24no—
138#390—216no—
139#390—24no—
140#388—48no—
141#387—216no—
142#386—216no—
143#386—216no—
144#386—48no—
145#385—216no—
146#384—24no—
147#384D418yescomplemented
148#380—216no—
149C4×C4⋊C4—24no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25C2414yes—
167C25C2412yes—
168C22×Q8—24no—
169C22×Q8—28no—
170C22×Q8—24no—
171C22×Q8—44no—
172C22×Q8—42no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—28no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4C2×D418yes—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C4⋊Q8—416no—
231#124—432no—
232C22⋊Q8—464no—
233#121—432no—
234#121—464no—
235C4×D4—416no—
236C4×D4—432no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D4116yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
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—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
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—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D4116yes—
306C2×C22⋊C4C2×D4116yes—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—28no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—42no—
315C2×C42C2×D414yes—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—42no—
331#95C2×D418yes—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24C22×D418yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—416no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—88no—
363C2×Q8—84no—
364C2×D4—28no—
365C2×D4—216no—
366C2×D4—416no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—44no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—88no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4C22×D418yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)14yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C4H(5,𝔽2)12yes—
470C4⋊C4—232no—
471C4⋊C4—216no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—88no—
480C22⋊C4—232no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—28no—
495C42—28no—
496C42—48no—
497C42—44no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23C23×D411yescharacteristic
557C23C23×D412yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×2−1+412yes—
563C23#58514yes—
564Q8—816no—
565D4—416no—
566D4—432no—
567D4—832no—
568D4—816no—
569D4—816no—
570D4—832no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—816no—
629C2×C4—832no—
630C2×C4#53514yes—
631C2×C4D4214yes—
632C2×C4D4218yes—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—82no—
659C22C2×#53511yescharacteristic
660C22C2×D4211yescharacteristic
661C22C2×D4212yes—
662C22#348411yescharacteristic
663C22#348611yescharacteristic
664C22#348712yes—
665C22#348914yes—
666C22#349614yes—
667C22#349812yes—
668C22#353612yes—
669C22#353612yes—
670C22#354114yes—
671C22#354214yes—
672C22#354311yescharacteristic
673C22#354512yes—
674C22#355112yes—
675C4—24no—
676C4—28no—
677C4—24no—
678C4—48no—
679C4—42no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—84no—
684C4—84no—
685C4—82no—
686C2—84no—
687C2—88no—
688C2#3670711yescharacteristic
689C2#3933912yes—
690C2#4510312yes—
691C2#4526712yes—
692C2#4787414yes—
693C2#5208311yescharacteristic
694C2#5485312yes—
695C2#5912611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2676C2semidirect
2. C2×#2542C2semidirect
3. C2×#2537C2semidirect
4. #16514C2semidirect
5. #16508C2semidirect
6. #15761C2semidirect
7. #15630C2semidirect
8. #15335C2semidirect
9. #15323C2semidirect
10. #14811C2semidirect
11. #14722C2semidirect
12. #14424C2semidirect
13. #14176C2semidirect
131. #394D4semidirect
147. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26762
C2×#25422
C2×#25372
#165142
#165082
#157612
#156302
#153352
#153232
#148112
#147222
#144242
#141762