Finite Group Databasebeta

← #641726

Subgroups — #641726

Order 512.

2800 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2629C211yesmaximal, characteristic, complemented
3C2×#2290C211yesmaximal, characteristic
4C2×#2285C211yesmaximal, characteristic, complemented
5#16562C214yesmaximal
6#16561C214yesmaximal, complemented
7#15221C212yesmaximal, complemented
8#15220C212yesmaximal, complemented
9#14864C214yesmaximal, complemented
10#14861C214yesmaximal, complemented
11#14340C211yesmaximal, characteristic, complemented
12#14283C211yesmaximal, characteristic, complemented
13#14276C211yesmaximal, characteristic, complemented
14#13951C212yesmaximal, complemented
15#13871C212yesmaximal
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2212yes—
20C23×C42C2211yescharacteristic
21#2851—28no—
22#2629—28no—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28#2290—28no—
29C2×#388C2212yes—
30C2×#387C2214yes—
31C2×#387C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35#2285—28no—
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#382C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49#1445C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×#125—24no—
80C2×#125—24no—
81C2×#125—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4Q818yes—
85C2×#117—28no—
86C2×#117—28no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—44no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42D418yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#398—216no—
124#396—216no—
125#389—28no—
126#389—28no—
127#389—44no—
128#388—28no—
129#387—216no—
130#387—28no—
131#386—28no—
132#385—216no—
133#385—28no—
134#384—24no—
135#384D418yescomplemented
136#382—28no—
137#382—24no—
138#380—28no—
139#380—28no—
140#380—216no—
141#380—44no—
142C4×C4⋊C4—216no—
143C4×C4⋊C4—24no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—24no—
146#376—24no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4C2×Q814yes—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C2×D414yes—
193C23×C4C2×D418yes—
194C23×C4C2×Q814yes—
195C23×C4C2×Q814yes—
196C23×C4#4114yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210#125—416no—
211C4×D4—232no—
212#117—432no—
213#117—416no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D4116yes—
252C2×C4⋊C4C2×Q814yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×Q814yes—
279C2×C22⋊C4C2×Q814yes—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—232no—
287C2×C42—22no—
288C2×C42—28no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—28no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—44no—
296C2×C42C2×D414yes—
297C2×C42C2×D418yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×Q814yes—
301C2×C42#4114yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24C22×D412yes—
343C24C22×Q812yes—
344C24C22×Q812yes—
345C24C2×#4112yes—
346C24C2511yescharacteristic
347C2×D4—28no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—44no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4C22⋊Q8116yes—
442C22×C4C22×D418yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q812yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—216no—
483C42—28no—
484C42—28no—
485C42—416no—
486C42—44no—
487C42—44no—
488C42#121116yes—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23C2×C22⋊Q814yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C23×Q811yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4111yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23#58512yes—
543C23#58512yes—
544C23#58512yes—
545C23#58512yes—
546D4—416no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—232no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—432no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—416no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4C2×#12118yes—
629C2×C4C2×#12114yes—
630C2×C4C2×C22⋊Q814yes—
631C2×C4C2×C22⋊Q814yes—
632C2×C4#53714yes—
633C2×C4#54614yes—
634C2×C4D4×Q814yes—
635C2×C4#55214yes—
636C2×C4#55414yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—82no—
658C22—82no—
659C22C22×#12111yescharacteristic
660C22C22×C22⋊Q811yescharacteristic
661C22C2×#53711yescharacteristic
662C22#344912yes—
663C22#344912yes—
664C22#345511yescharacteristic
665C22#345511yescharacteristic
666C22#345812yes—
667C22C2×#54611yescharacteristic
668C22C2×D4×Q811yescharacteristic
669C22C2×#55211yescharacteristic
670C22C2×#55411yescharacteristic
671C22#348512yes—
672C22#348712yes—
673C22#349312yes—
674C22#349312yes—
675C22#350412yes—
676C22#351211yescharacteristic
677C22#351211yescharacteristic
678C22#356212yes—
679C22#356212yes—
680C22#356512yes—
681C22#356512yes—
682C4—28no—
683C4—24no—
684C4—42no—
685C4—48no—
686C4—42no—
687C4—44no—
688C4—42no—
689C4—84no—
690C4—82no—
691C4—84no—
692C4—82no—
693C4—84no—
694C4#241718yes—
695C2—44no—
696C2—82no—
697C2#3463411yescharacteristic
698C2C2×#241711yescharacteristic
699C2#3515012yes—
700C2#3533311yescharacteristic
701C2#3755312yes—
702C2#3769112yes—
703C2#3770011yescharacteristic
704C2#3770111yescharacteristic
705C2#3787912yes—
706C2#3788811yescharacteristic
707C2#3788911yescharacteristic
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
1. C2×#2851C2semidirect
2. C2×#2629C2semidirect
4. C2×#2285C2semidirect
6. #16561C2semidirect
7. #15221C2semidirect
8. #15220C2semidirect
9. #14864C2semidirect
10. #14861C2semidirect
11. #14340C2semidirect
12. #14283C2semidirect
13. #14276C2semidirect
14. #13951C2semidirect
135. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#26292
C2×#22902
C2×#22852
#165622
#165612
#152212
#152202
#148642
#148612
#143402
#142832
#142762
#139512
#138712