Finite Group Databasebeta

← #647027

Subgroups — #647027

Order 512.

3062 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic, complemented
3#16988C212yesmaximal, complemented
4#16886C212yesmaximal, complemented
5#16685C214yesmaximal, complemented
6#15105C212yesmaximal, complemented
7#15015C211yesmaximal, characteristic, complemented
8#14936C211yesmaximal, characteristic, complemented
9#14927C212yesmaximal, complemented
10#14926C211yesmaximal, characteristic
11#14913C211yesmaximal, characteristic
12#14812C212yesmaximal, complemented
13#14729C212yesmaximal, complemented
14#14682C211yesmaximal, characteristic, complemented
15#14269C214yesmaximal, complemented
16#14015C212yesmaximal, complemented
17#13968C212yesmaximal, complemented
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#396C2214yes—
26C4×#121—28no—
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#388C2212yes—
30C2×#387—28no—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#382C2212yes—
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1443C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2212yes—
73#1438C2212yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77#1434C2212yes—
78#1434C2212yes—
79C22⋊#95C2211yescharacteristic
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C2×#123—28no—
96C2×#123—44no—
97C2×#123—44no—
98C2×#121—28no—
99C2×#121—24no—
100C2×C4×D4—28no—
101C2×C4×D4—24no—
102C2×C4×D4—44no—
103C2×C4×D4—44no—
104C2×C4×D4—44no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C42—44no—
132C22×C42D418yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#396—48no—
139#390—24no—
140#389—24no—
141#388—28no—
142#387—28no—
143#387—28no—
144#387—28no—
145#387—28no—
146#386—28no—
147#386—28no—
148#386—44no—
149#386—44no—
150#385—216no—
151#385—44no—
152#382—44no—
153#381—44no—
154#380—24no—
155#380—44no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—28no—
158C2×#95—24no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95—24no—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—42no—
202C22×D4#4118yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—44no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254#123—432no—
255#121—232no—
256C4×D4—416no—
257C4×D4—432no—
258C2×C4⋊C4—232no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4#4118yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—416no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—42no—
353#95—44no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24C22×D412yes—
374C24C2×#4114yes—
375C24C2×#4114yes—
376C24H(5,𝔽2)14yes—
377C24H(5,𝔽2)12yes—
378C242−1+412yes—
379C24C2511yescharacteristic
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—416no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—44no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
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—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—88no—
489C22×C4—88no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4114yes—
500C22×C4C2×#4114yes—
501C22×C4C2×#4114yes—
502C22×C4H(5,𝔽2)14yes—
503C22×C42−1+414yes—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—88no—
514C4⋊C4—88no—
515C22⋊C4—216no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C42—28no—
527C42—28no—
528C42—44no—
529C42—44no—
530C42—48no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—28no—
547C23—28no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23#52514yes—
567C23#53514yes—
568C23#53714yes—
569C23#53818yes—
570C23#53918yes—
571C23C23×D411yescharacteristic
572C23C22×#4112yes—
573C23C22×#4112yes—
574C23C2×H(5,𝔽2)12yes—
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×2−1+412yes—
578C23C2×2−1+411yescharacteristic
579C23#58512yes—
580D4—416no—
581D4—432no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—416no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—416no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
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—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—816no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4#52514yes—
665C2×C4#53414yes—
666C2×C4#53614yes—
667C22—24no—
668C22—24no—
669C22—216no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—216no—
675C22—44no—
676C22—44no—
677C22—48no—
678C22—44no—
679C22—48no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—84no—
686C22#263618yes—
687C22C2×#52511yescharacteristic
688C22C2×#52511yescharacteristic
689C22C2×#53411yescharacteristic
690C22C2×#53511yescharacteristic
691C22C2×#53611yescharacteristic
692C22C2×#53711yescharacteristic
693C22#344911yescharacteristic
694C22C2×#53812yes—
695C22C2×#53912yes—
696C22#347514yes—
697C22#351914yes—
698C22#352014yes—
699C22#352212yes—
700C22#352312yes—
701C22#353412yes—
702C22#353512yes—
703C22#355914yes—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—42no—
708C4—44no—
709C4—42no—
710C4—44no—
711C4—44no—
712C4—42no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—84no—
718C4—84no—
719C2—28no—
720C2—84no—
721C2C2×#263611yescharacteristic
722C2#3487511yescharacteristic
723C2#3507611yescharacteristic
724C2#3508411yescharacteristic
725C2#3524711yescharacteristic
726C2#3745211yescharacteristic
727C2#3753911yescharacteristic
728C2#3814114yes—
729C2#3817514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#387C2semidirect
3. #16988C2semidirect
4. #16886C2semidirect
5. #16685C2semidirect
6. #15105C2semidirect
7. #15015C2semidirect
8. #14936C2semidirect
9. #14927C2semidirect
12. #14812C2semidirect
13. #14729C2semidirect
14. #14682C2semidirect
15. #14269C2semidirect
16. #14015C2semidirect
17. #13968C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C22×#3872
#169882
#168862
#166852
#151052
#150152
#149362
#149272
#149262
#149132
#148122
#147292
#146822
#142692
#140152
#139682