Finite Group Databasebeta

← #364723

Subgroups — #364723

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#2324C211yesmaximal, characteristic
3C2×#1875C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic, complemented
5#13795C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#480—24no—
18C2×#480—24no—
19C2×#480C418yescomplemented
20C22×#107C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2324—24no—
26#2324—24no—
27C2×#390—21no—
28C2×#390C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388—21no—
31C2×#386—21no—
32C2×#385—21no—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1875—24no—
36#1824—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C22⋊#95—21no—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121C2×C414yes—
61C2×#121D414yes—
62C2×#121D414yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—42no—
74C2×#117—42no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2×C414yescomplemented
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42—21no—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#480—232no—
103C2×#107—24no—
104C2×#107—24no—
105C2×#107C2×C418yescomplemented
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—42no—
111C2×#102C2×C418yes—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8C2×C414yescomplemented
118#390—28no—
119#390—22no—
120#390—22no—
121#389—22no—
122#389—22no—
123#388—28no—
124#386—28no—
125#385—28no—
126#380—22no—
127#380D414yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—42no—
142#336—24no—
143#336—42no—
144C25—21no—
145C25C2×D411yescharacteristic
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—24no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—82no—
157C22×D4—82no—
158C22×D4C22×C412yes—
159C22×D4C2×D412yes—
160C22×D4C2×D412yes—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4C22×C412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8C22×C414yes—
206#121—28no—
207#121—28no—
208#121—28no—
209#121C22⋊C4116yes—
210C4×D4—48no—
211C4×D4—48no—
212#117—48no—
213#117—48no—
214#117—48no—
215#117—48no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4#4114yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4—82no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—82no—
290C2×C42#4114yes—
291#107—216no—
292#102—216no—
293#102—48no—
294C22⋊C8—28no—
295C22⋊C8—44no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—82no—
300#95—82no—
301#95—82no—
302#95—82no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24C4×D412yes—
323C24#12012yes—
324C24#12112yes—
325C24C22×D411yescharacteristic
326C2×D4—216no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—88no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—88no—
346C2×D4C2×C22⋊C414yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—48no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—88no—
420C22×C4C2×C22⋊C414yes—
421C22×C4C4×D414yes—
422C22×C4C4×D412yes—
423C22×C4#12012yes—
424C22×C4#12114yes—
425C22×C4#12112yes—
426C22×C4#12312yes—
427C22×C4#12312yes—
428C22×C4#12314yes—
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4111yescharacteristic
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+412yes—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C4⋊C4C2×C22⋊C414yes—
473C22⋊C4—216no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C42—24no—
482C42—24no—
483C42—48no—
484C42—42no—
485C42—42no—
486C42—84no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23#38614yes—
533C23C2×C4×D411yescharacteristic
534C23C2×#12011yescharacteristic
535C23C2×#12111yescharacteristic
536C23C2×#12311yescharacteristic
537C23#52511yescharacteristic
538C23#53411yescharacteristic
539C23#53611yescharacteristic
540C23#57612yes—
541C23#57612yes—
542C23#57712yes—
543C23#57812yes—
544D4—28no—
545D4—44no—
546D4—44no—
547D4—44no—
548D4—816no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—816no—
597C2×C4—816no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4#38614yes—
611C2×C4C22×C22⋊C411yescharacteristic
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C4×#4112yes—
614C2×C4#52012yes—
615C2×C4C2×#12111yescharacteristic
616C2×C4C2×#12311yescharacteristic
617C2×C4#52912yes—
618C2×C4#53912yes—
619C2×C4#54612yes—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4#54712yes—
623C2×C4#54712yes—
624C2×C4#54712yes—
625C2×C4#55112yes—
626C2×C4#55212yes—
627C8—44no—
628C8—82no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—82no—
644C22—84no—
645C22—82no—
646C22—84no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22C2×#38611yescharacteristic
652C22C4×#12311yescharacteristic
653C22#232211yescharacteristic
654C22#240511yescharacteristic
655C22#240611yescharacteristic
656C22#248611yescharacteristic
657C22#253411yescharacteristic
658C22#294712yes—
659C22#294812yes—
660C22#300811yescharacteristic
661C22#300811yescharacteristic
662C22#300911yescharacteristic
663C22#301111yescharacteristic
664C22#304112yes—
665C22#304612yes—
666C22#309612yes—
667C22#309812yes—
668C22#311912yes—
669C22#312412yes—
670C22#319812yes—
671C22#320312yes—
672C22#324112yes—
673C22#324312yes—
674C4—24no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—84no—
680C4—81no—
681C4—81no—
682C4—82no—
683C4—84no—
684C4#233912yes—
685C4#235912yes—
686C4#241912yes—
687C4#242512yes—
688C2—44no—
689C2—84no—
690C2#1396811yescharacteristic
691C2#2093111yescharacteristic
692C2#2093411yescharacteristic
693C2#2158612yes—
694C2#2209312yes—
695C2#2401111yescharacteristic
696C2#2401311yescharacteristic
697C2#2406911yescharacteristic
698C2#2407311yescharacteristic
699C2#2765512yes—
700C2#2812912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1824C2semidirect
5. #13795C2semidirect
6. #11219C2semidirect
7. #11189C2semidirect
8. #10907C2semidirect
9. #10788C2semidirect
10. #10731C2semidirect
11. #10713C2semidirect
19. C2×#480C4semidirect
82. C22×C4⋊C4C2×C4semidirect
105. C2×#107C2×C4semidirect
117. C2×C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#23242
C2×#18752
C2×#18242
#137952
#112192
#111892
#109072
#107882
#107312
#107132