Finite Group Databasebeta

← #848996

Subgroups — #848996

Order 512.

2854 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2717C212yesmaximal
2C2×#2486C211yesmaximal, characteristic
3C2×C4×#124C211yesmaximal, characteristic
4#16688C212yesmaximal, complemented
5#16629C212yesmaximal, complemented
6#15717C214yesmaximal, complemented
7#15485C214yesmaximal, complemented
8#15158C212yesmaximal, complemented
9#15132C212yesmaximal, complemented
10#15132C212yesmaximal, complemented
11#14699C212yesmaximal, complemented
12#14505C212yesmaximal
13#14130C212yesmaximal, complemented
14#13907C212yesmaximal, complemented
15#13872C211yesmaximal, characteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2717—216no—
23#2486—28no—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#395C2214yes—
29C4×#124—28no—
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2214yes—
39C2×#386C2212yes—
40C2×#385C2214yes—
41C2×#382C2211yescharacteristic
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C43C2211yescharacteristic
51#1444C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×#124—24no—
85C2×#124—24no—
86C2×#124D418yes—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×C4×Q8—24no—
91C2×C4×Q8—42no—
92C2×C4×Q8—42no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4D418yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4—42no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#398—28no—
131#397—216no—
132#397—216no—
133#397—28no—
134#395—216no—
135#391—24no—
136#390—24no—
137#389—28no—
138#389—28no—
139#389—44no—
140#389—44no—
141#388—28no—
142#388—28no—
143#387—216no—
144#386—28no—
145#385—216no—
146#382—24no—
147#381—28no—
148#380—28no—
149#380—24no—
150#380—28no—
151#380—216no—
152#380—42no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C43—24no—
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8C2×D414yes—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4C2×D414yes—
194C22×D4C2×D414yes—
195C22×D4#4114yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#124—232no—
232#123—432no—
233C4×Q8—416no—
234C4×D4—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—416no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4#4118yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4C2×D418yes—
302C2×C22⋊C4C2×D418yes—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×D414yes—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—42no—
319C2×C42—41no—
320C2×C42—41no—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—42no—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24C2511yescharacteristic
370C2×Q8—28no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4#121116yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—216no—
427C22×C4—216no—
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—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—88no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4118yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+414yes—
488C22×C42−1+414yes—
489C4⋊C4—216no—
490C4⋊C4—44no—
491C4⋊C4—416no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—84no—
512C42—28no—
513C42—28no—
514C42—48no—
515C42—44no—
516C42—48no—
517C42—48no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C2×#12114yes—
563C23C2×#12114yes—
564C23C23×D411yescharacteristic
565C23C23×D411yescharacteristic
566C23C22×#4111yescharacteristic
567C23C22×#4112yes—
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×2−1+412yes—
571C23C2×2−1+412yes—
572C23#58511yescharacteristic
573C23#58511yescharacteristic
574C23#58512yes—
575Q8—416no—
576D4—416no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
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—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—816no—
657C2×C4—816no—
658C2×C4C2×#12114yes—
659C2×C4#53414yes—
660C2×C4#54218yes—
661C2×C4#54718yes—
662C2×C4#55114yes—
663C2×C4#56314yes—
664C22—28no—
665C22—28no—
666C22—24no—
667C22—24no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—44no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—44no—
679C22—44no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22C22×#12111yescharacteristic
687C22C2×#53411yescharacteristic
688C22#344911yescharacteristic
689C22C2×#54212yes—
690C22C2×#54712yes—
691C22D4×#4112yes—
692C22C2×#55111yescharacteristic
693C22#348212yes—
694C22#348912yes—
695C22#349611yescharacteristic
696C22#349611yescharacteristic
697C22#349912yes—
698C22#350511yescharacteristic
699C22#350511yescharacteristic
700C22#351512yes—
701C22#351612yes—
702C22#352112yes—
703C22C2×#56311yescharacteristic
704C22#354712yes—
705C22#355011yescharacteristic
706C22#355111yescharacteristic
707C22#356312yes—
708C22#356512yes—
709C4—24no—
710C4—42no—
711C4—44no—
712C4—44no—
713C4—42no—
714C4—42no—
715C4—44no—
716C4—48no—
717C4—82no—
718C4—84no—
719C4—84no—
720C4—84no—
721C4#271718yescomplemented
722C2—44no—
723C2—44no—
724C2#3500011yescharacteristic
725C2#3625912yes—
726C2#3737611yescharacteristic
727C2#3891512yes—
728C2#3939812yes—
729C2#4010612yes—
730C2#4614711yescharacteristic
731C2#4619011yescharacteristic
732C2C2×#271711yescharacteristic
733C2#5984311yescharacteristic
734C2#5994611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #16688C2semidirect
5. #16629C2semidirect
6. #15717C2semidirect
7. #15485C2semidirect
8. #15158C2semidirect
9. #15132C2semidirect
10. #15132C2semidirect
11. #14699C2semidirect
13. #14130C2semidirect
14. #13907C2semidirect
721. C4#2717semidirect

Maximal subgroups

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

subgroupindex
C2×#27172
C2×#24862
C2×C4×#1242
#166882
#166292
#157172
#154852
#151582
#151322
#146992
#145052
#141302
#139072
#138722