Finite Group Databasebeta

← #627791

Subgroups — #627791

Order 512.

2752 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×#2314C211yesmaximal, characteristic, complemented
4C2×#2290C211yesmaximal, characteristic
5#15901C214yesmaximal, complemented
6#15774C214yesmaximal, complemented
7#15441C214yesmaximal, complemented
8#15308C214yesmaximal
9#15181C212yesmaximal
10#15132C212yesmaximal, complemented
11#14686C211yesmaximal, characteristic, complemented
12#14283C211yesmaximal, characteristic, complemented
13#14277C211yesmaximal, characteristic, complemented
14#14046C212yesmaximal, complemented
15#13973C212yesmaximal, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2851—28no—
21#2629—28no—
22C2×#398C2214yes—
23C2×#397C2214yes—
24C2×#396C2214yes—
25#2314—28no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30#2290—28no—
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#387C2212yes—
35C2×#387C2214yes—
36C2×#386C2212yes—
37C2×#385C2214yes—
38C2×#385C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×#376C2211yescharacteristic
52#1444C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
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×D4—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117D418yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42—24no—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120#398—216no—
121#397—48no—
122#396—216no—
123#389—28no—
124#389—28no—
125#389—28no—
126#389—48no—
127#388—28no—
128#388—216no—
129#387—28no—
130#387—216no—
131#387—216no—
132#386—28no—
133#385—28no—
134#385—216no—
135#384—24no—
136#384—24no—
137#382D418yescomplemented
138#380—28no—
139#380—28no—
140#380—28no—
141#380—28no—
142#380—48no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147#376—24no—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—42no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C2×D414yes—
197C23×C4#4114yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211#125—416no—
212C4×D4—232no—
213#117—232no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
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—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
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⋊C4—44no—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D418yes—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—416no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42#4114yes—
301C2×C42#4114yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×D4116yes—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24C22×D412yes—
344C24C2×#4112yes—
345C242−1+414yes—
346C24C2511yescharacteristic
347C2×D4—28no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—44no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—216no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4C22×D412yes—
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—216no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—28no—
483C42—28no—
484C42—48no—
485C42—44no—
486C42—44no—
487C42—88no—
488C42#121116yes—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#54114yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4111yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+412yes—
544C23#58512yes—
545C23#58512yes—
546D4—416no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—432no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
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—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4C2×#12118yes—
630C2×C4C2×#12114yes—
631C2×C4#52514yes—
632C2×C4#53614yes—
633C2×C4#54714yes—
634C2×C4#54814yes—
635C2×C4#55214yes—
636C2×C4#55614yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—44no—
642C22—42no—
643C22—44no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—42no—
649C22—44no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—81no—
657C22—82no—
658C22—81no—
659C22C22×#12111yescharacteristic
660C22C2×#52511yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#54111yescharacteristic
663C22#345511yescharacteristic
664C22#346012yes—
665C22C2×#54711yescharacteristic
666C22C2×#54811yescharacteristic
667C22C2×#55211yescharacteristic
668C22C2×#55611yescharacteristic
669C22#348812yes—
670C22#348812yes—
671C22#348912yes—
672C22#348912yes—
673C22#349212yes—
674C22#349312yes—
675C22#349512yes—
676C22#350811yescharacteristic
677C22#351211yescharacteristic
678C22#354712yes—
679C22#355212yes—
680C22#356312yes—
681C22#356512yes—
682C4—28no—
683C4—48no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—84no—
691C4—82no—
692C4—84no—
693C4—84no—
694C4#241518yes—
695C2—44no—
696C2—82no—
697C2C2×#241511yescharacteristic
698C2#3525911yescharacteristic
699C2#3531011yescharacteristic
700C2#3551812yes—
701C2#3727912yes—
702C2#3770511yescharacteristic
703C2#3771011yescharacteristic
704C2#3776712yes—
705C2#3788211yescharacteristic
706C2#3788911yescharacteristic
707C2#3796112yes—
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
3. C2×#2314C2semidirect
5. #15901C2semidirect
6. #15774C2semidirect
7. #15441C2semidirect
10. #15132C2semidirect
11. #14686C2semidirect
12. #14283C2semidirect
13. #14277C2semidirect
14. #14046C2semidirect
15. #13973C2semidirect
137. #382D4semidirect

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×#23142
C2×#22902
#159012
#157742
#154412
#153082
#151812
#151322
#146862
#142832
#142772
#140462
#139732