Finite Group Databasebeta

← #403806

Subgroups — #403806

Order 512.

1630 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
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2042C212yesmaximal
3C2×#2024C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5#14708C211yesmaximal, characteristic
6#11208C212yesmaximal, complemented
7#11050C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic
9#10932C212yesmaximal, complemented
10#10931C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8—21no—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C4×C4⋊Q8—28no—
20C2×#391—21no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389—21no—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2212yes—
31C2×C43C2211yescharacteristic
32#2042—28no—
33#2024—24no—
34#2023—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1437—21no—
44#1437—21no—
45#1437—21no—
46#1437—21no—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D414yes—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—24no—
59C2×#124—42no—
60C2×C4×Q8—28no—
61C2×C4×Q8—44no—
62C2×C4×Q8—44no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—22no—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—42no—
97#391—28no—
98#391—22no—
99#391—22no—
100#391—24no—
101#390—22no—
102#390—22no—
103#390—24no—
104#389—28no—
105#389—24no—
106#389D418yes—
107#384—24no—
108#382—42no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C43—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—22no—
126#336—22no—
127#336—24no—
128#336—24no—
129C25—21no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8#4112yes—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4#4112yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C4⋊Q8—216no—
184C4⋊Q8—216no—
185#124—48no—
186#124—48no—
187C4×Q8—432no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—28no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42C2×D414yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C2×C42#4114yes—
265#103—48no—
266#103—48no—
267#102—48no—
268#102—48no—
269#95—28no—
270#95—28no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—42no—
277C24—22no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24C22×D411yescharacteristic
293C2×Q8—216no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—48no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—42no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4#12012yes—
398C22×C4#12012yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4#12412yes—
402C22×C4#12412yes—
403C22×C4#13512yes—
404C22×C4#13512yes—
405C22×C4AGL(1,ℤ/8)12yes—
406C22×C4AGL(1,ℤ/8)12yes—
407C22×C4#13712yes—
408C22×C4#13712yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+411yescharacteristic
418C22×C42−1+411yescharacteristic
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C4⋊C4—216no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—416no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4#12118yes—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C42—28no—
456C42—28no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—24no—
463C42—44no—
464C42—44no—
465C42—416no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23C2×#12011yescharacteristic
502C23C2×#12111yescharacteristic
503C23#52511yescharacteristic
504C23#52511yescharacteristic
505C23C2×#12411yescharacteristic
506C23#53511yescharacteristic
507C23#53611yescharacteristic
508C23C2×#13511yescharacteristic
509C23C2×#13511yescharacteristic
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23C2×AGL(1,ℤ/8)11yescharacteristic
512C23C2×#13711yescharacteristic
513C23C2×#13711yescharacteristic
514C23#57511yescharacteristic
515C23#57511yescharacteristic
516Q8—432no—
517Q8—44no—
518Q8—44no—
519Q8—82no—
520Q8—82no—
521D4—44no—
522D4—44no—
523D4—82no—
524D4—82no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4#46314yes—
600C2×C4#46414yes—
601C2×C4#48914yes—
602C2×C4C2×#12112yes—
603C2×C4#52512yes—
604C2×C4#52912yes—
605C2×C4#52912yes—
606C2×C4#54212yes—
607C2×C4#54212yes—
608C2×C4#54612yes—
609C2×C4#54612yes—
610C2×C4#54811yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#54812yes—
613C2×C4#54812yes—
614C2×C4#55012yes—
615C2×C4#55211yescharacteristic
616C2×C4#55211yescharacteristic
617C2×C4#55912yes—
618C8—82no—
619C8—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—161no—
628C22—161no—
629C22—161no—
630C22—161no—
631C22—161no—
632C22—161no—
633C22—161no—
634C22—161no—
635C22#239811yescharacteristic
636C22#240411yescharacteristic
637C22#243611yescharacteristic
638C22#253611yescharacteristic
639C22#254811yescharacteristic
640C22#263211yescharacteristic
641C22#284711yescharacteristic
642C22#300211yescharacteristic
643C22#300211yescharacteristic
644C22#300311yescharacteristic
645C22#300411yescharacteristic
646C22C2×#46311yescharacteristic
647C22C2×#46411yescharacteristic
648C22#303611yescharacteristic
649C22#303711yescharacteristic
650C22#310511yescharacteristic
651C22#310611yescharacteristic
652C22#310711yescharacteristic
653C22#310811yescharacteristic
654C22#311411yescharacteristic
655C22#311511yescharacteristic
656C22#311611yescharacteristic
657C22#311711yescharacteristic
658C22#313111yescharacteristic
659C22C2×#48911yescharacteristic
660C22#314011yescharacteristic
661C22#314111yescharacteristic
662C22#320711yescharacteristic
663C22#321011yescharacteristic
664C22#321411yescharacteristic
665C22#321611yescharacteristic
666C22#323311yescharacteristic
667C22#323311yescharacteristic
668C22#323811yescharacteristic
669C22#323911yescharacteristic
670C4—24no—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—48no—
677C4—82no—
678C4—84no—
679C4—161no—
680C4—161no—
681C4#242612yes—
682C4#242612yes—
683C4#245812yes—
684C4#245812yes—
685C2—162no—
686C2#1488311yescharacteristic
687C2#2381711yescharacteristic
688C2#2382111yescharacteristic
689C2#2393211yescharacteristic
690C2#2393511yescharacteristic
691C2#2590611yescharacteristic
692C2#2590811yescharacteristic
693C2#2820311yescharacteristic
694C2#2820411yescharacteristic
695C2#2875011yescharacteristic
696C2#2875411yescharacteristic
697C2#3006111yescharacteristic
698C2#3006411yescharacteristic
699C2#3290811yescharacteristic
700C2#3290911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊Q8C2semidirect
3. C2×#2024C2semidirect
6. #11208C2semidirect
7. #11050C2semidirect
9. #10932C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20422
C2×#20242
C2×#20232
#147082
#112082
#110502
#110492
#109322
#109312
#107952