Finite Group Databasebeta

← #766046

Subgroups — #766046

Order 512.

3698 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2490C212yesmaximal, complemented
2C2×#2428C212yesmaximal, complemented
3#15651C214yesmaximal, complemented
4#15550C214yesmaximal, complemented
5#15320C214yesmaximal, complemented
6#15312C214yesmaximal, complemented
7#15218C211yesmaximal, characteristic, complemented
8#15127C211yesmaximal, characteristic, complemented
9#15123C212yesmaximal, complemented
10#13994C212yesmaximal, complemented
11#13970C212yesmaximal, complemented
12#13847C211yesmaximal, characteristic, complemented
13#13785C212yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×#121C2214yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2490—216no—
22#2428—216no—
23C2×#397C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#392C2214yes—
28C2×#392C2214yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2214yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×#376C2211yescharacteristic
45#1445C2212yes—
46#1444C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2214yes—
58#1438C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C23×D4C2314yes—
73C23×D4C2314yes—
74C24×C4C2312yes—
75C24×C4C2314yes—
76C2×#127—28no—
77C2×#127—42no—
78C2×#125—28no—
79C2×#125—42no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—216no—
85C2×#121—48no—
86C2×#121—48no—
87C2×#121—44no—
88C2×#121—44no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—44no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2314yes—
123#397—216no—
124#394—216no—
125#394—216no—
126#394—48no—
127#392—216no—
128#392—48no—
129#390—28no—
130#390—216no—
131#390—28no—
132#390D4116yescomplemented
133#389—28no—
134#388—216no—
135#388—48no—
136#386—216no—
137#386—48no—
138#380—28no—
139#380—28no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—42no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—28no—
144#376—42no—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C25C2414yes—
164C25C2412yes—
165C22×D4—24no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—22no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4C2×D418yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223#127—416no—
224#125—416no—
225#121—464no—
226#121—432no—
227#121—432no—
228C4×D4—432no—
229C4×D4—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4#4118yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
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⋊C4C2×D4116yes—
298C2×C22⋊C4C2×D4116yes—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—28no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C2×D418yes—
311C2×C42#4114yes—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—41no—
323#95—42no—
324#95—41no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24C22×D418yes—
359C24H(5,𝔽2)14yes—
360C24C2511yescharacteristic
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
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—88no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D418yes—
470C22×C4C22×D414yes—
471C22×C4C2×#4118yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C22⋊C4—216no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C42—28no—
503C42—216no—
504C42—44no—
505C42—44no—
506C42—88no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—44no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23C23×D412yes—
578C23C22×#4111yescharacteristic
579C23C22×#4112yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)12yes—
583C23C2×H(5,𝔽2)12yes—
584C23#58512yes—
585C23#58512yes—
586D4—416no—
587D4—432no—
588D4—832no—
589D4—832no—
590D4—832no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—832no—
658C2×C4—816no—
659C2×C4#53218yes—
660C2×C4#53814yes—
661C2×C4D4214yes—
662C2×C4#54618yes—
663C2×C4#55514yes—
664C22—48no—
665C22—44no—
666C22—48no—
667C22—44no—
668C22—48no—
669C22—48no—
670C22—44no—
671C22—48no—
672C22—44no—
673C22—44no—
674C22—48no—
675C22—44no—
676C22—48no—
677C22—44no—
678C22—82no—
679C22—84no—
680C22—84no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—84no—
685C22—82no—
686C22—84no—
687C22—82no—
688C22—84no—
689C22—84no—
690C22—82no—
691C22—84no—
692C22—82no—
693C22—84no—
694C22C2×#53212yes—
695C22C2×#53811yescharacteristic
696C22C2×D4211yescharacteristic
697C22C2×#54612yes—
698C22C2×#55511yescharacteristic
699C22#348412yes—
700C22#348512yes—
701C22#348812yes—
702C22#348812yes—
703C22#349412yes—
704C22#350811yescharacteristic
705C22#351211yescharacteristic
706C22#351312yes—
707C22#351712yes—
708C22#351912yes—
709C22#352412yes—
710C22#353711yescharacteristic
711C22#353911yescharacteristic
712C22#354212yes—
713C22#354212yes—
714C22#354612yes—
715C4—44no—
716C4—48no—
717C4—44no—
718C4—48no—
719C4—44no—
720C4—42no—
721C4—42no—
722C4—82no—
723C4—82no—
724C4—84no—
725C4#255518yes—
726C2—88no—
727C2—84no—
728C2C2×#255511yescharacteristic
729C2#3557012yes—
730C2#3557112yes—
731C2#3728311yescharacteristic
732C2#3731711yescharacteristic
733C2#4493212yes—
734C2#4502012yes—
735C2#4535212yes—
736C2#4538612yes—
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×#2490C2semidirect
2. C2×#2428C2semidirect
3. #15651C2semidirect
4. #15550C2semidirect
5. #15320C2semidirect
6. #15312C2semidirect
7. #15218C2semidirect
8. #15127C2semidirect
9. #15123C2semidirect
10. #13994C2semidirect
11. #13970C2semidirect
12. #13847C2semidirect
13. #13785C2semidirect
132. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24902
C2×#24282
#156512
#155502
#153202
#153122
#152182
#151272
#151232
#139942
#139702
#138472
#137852