Finite Group Databasebeta

← #361268

Subgroups — #361268

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#135C211yesmaximal, characteristic
2C2×#1876C212yesmaximal
3#20684C211yesmaximal, characteristic, complemented
4#20336C211yesmaximal, characteristic, complemented
5#11376C212yesmaximal, complemented
6#11191C212yesmaximal, complemented
7#10822C211yesmaximal, characteristic, complemented
8#10797C211yesmaximal, characteristic, complemented
9#10791C212yesmaximal, complemented
10#10790C211yesmaximal, characteristic, complemented
11#10790C211yesmaximal, characteristic, complemented
12C22×#135C2211yescharacteristic
13C22×#131C2212yes—
14C2×C4×#41C2211yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C4×#135—28no—
17C2×C4×Q16C411yescharacteristic, complemented
18C2×C4×Q16C411yescharacteristic, complemented
19C2×C4×SD16C411yescharacteristic, complemented
20C2×C4×SD16C411yescharacteristic, complemented
21C2×C4×SD16C411yescharacteristic, complemented
22C2×C4×SD16C411yescharacteristic, complemented
23C2×C4×D8C411yescharacteristic, complemented
24C2×C4×D8C411yescharacteristic, complemented
25C2×#427C2211yescharacteristic
26C22×#104C2212yes—
27C22×#104C2212yes—
28C2×#416C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#408C2212yes—
31C22×C4×C8C2211yescharacteristic
32#1876—28no—
33#1818—22no—
34#1817—21no—
35#1817—21no—
36C2×#341C2212yes—
37C2×#341C2212yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40#1729—22no—
41#1729C2212yes—
42#1729C2212yes—
43#1728—21no—
44#1728C2211yescharacteristic
45#1728C2211yescharacteristic
46C2×#336—21no—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C2×#336C2211yescharacteristic
50C4×#95C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C22×#41C2311yescharacteristic
53C2×#135—22no—
54C2×#135—24no—
55C2×#135—22no—
56C2×#135D418yes—
57C22×Q16C2×C411yescharacteristic
58C22×SD16C2×C411yescharacteristic
59C22×SD16C2×C411yescharacteristic
60C22×D8C2×C411yescharacteristic
61C2×#131—24no—
62C2×#131—24no—
63C2×#131—24no—
64C2×#131—24no—
65C2×#131—42no—
66C2×#131—42no—
67C22×M16—22no—
68C22×M16C2312yes—
69C22×M16C2312yes—
70C23×C8—22no—
71C23×C8C2311yescharacteristic
72C23×C8C2312yes—
73C23×C8C2311yescharacteristic
74C4×#41—24no—
75C4×#41—24no—
76C4×#41—42no—
77C4×#41—42no—
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×#117—21no—
91C2×#117—21no—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C42—21no—
95C22×C42—21no—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98C4×Q16—24no—
99C4×Q16—24no—
100C4×SD16—24no—
101C4×SD16—24no—
102C4×SD16—24no—
103C4×SD16—24no—
104C4×D8—24no—
105C4×D8—24no—
106#427—24no—
107C2×#107C2×C411yescharacteristic
108C2×#107C2×C411yescharacteristic
109C2×#106C2×C411yescharacteristic
110C2×#106C2×C411yescharacteristic
111C2×#104—24no—
112C2×#104—24no—
113C2×#104—24no—
114C2×#104—24no—
115C2×#104—24no—
116C2×#104—42no—
117#416—24no—
118#416—24no—
119C2×#103C2×C411yescharacteristic
120C2×#103C2×C411yescharacteristic
121C2×#103C2×C411yescharacteristic
122C2×#103C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125C2×#102C2×C411yescharacteristic
126C2×#102C2×C411yescharacteristic
127#408—24no—
128#408—42no—
129C2×C22⋊C8—22no—
130C2×C22⋊C8—22no—
131C2×C4×C8—24no—
132C2×C4×C8C2×C411yescharacteristic
133C2×C4×C8C2×C411yescharacteristic
134C2×C4×C8C2×C411yescharacteristic
135C2×C4×C8C2×C411yescharacteristic
136C2×#95—21no—
137C2×#95—21no—
138#341—24no—
139#341—24no—
140#339—42no—
141#339—42no—
142#336—22no—
143#336—22no—
144#336—24no—
145#336—22no—
146#336—22no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—42no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#336D412yes—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—21no—
158C2×#41—22no—
159C2×#41—21no—
160C2×#41—22no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—22no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41C2×D414yes—
167C2×#41C2×D414yes—
168C2×#41#4112yes—
169C2×#41#4114yes—
170C2×#41#4112yes—
171C22×Q8C22×C411yescharacteristic
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C22×C411yescharacteristic
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186C23×C4C2411yescharacteristic
187#135—216no—
188#135—216no—
189C2×Q16—22no—
190C2×Q16—22no—
191C2×Q16C22⋊C414yes—
192C2×SD16—22no—
193C2×SD16—22no—
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16C22⋊C414yes—
197C2×SD16C22⋊C414yes—
198C2×D8—22no—
199C2×D8—22no—
200C2×D8C22⋊C414yes—
201#131—48no—
202#131—48no—
203#131—48no—
204#131—48no—
205C2×M16—28no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—44no—
218C2×M16—44no—
219C2×M16#4114yes—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—28no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—22no—
236C22×C8—44no—
237C22×C8—44no—
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C22×C411yescharacteristic
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D414yes—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—41no—
250C4×Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—44no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—44no—
265#117—22no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—44no—
270#117—44no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305#107—22no—
306#107—22no—
307#106—22no—
308#106—22no—
309#104—48no—
310#104—48no—
311#103—22no—
312#103—22no—
313#103—22no—
314#103—22no—
315#102—22no—
316#102—22no—
317#102—22no—
318#102—22no—
319C22⋊C8—44no—
320C22⋊C8—44no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325#95—24no—
326#95—42no—
327C24—21no—
328C24—21no—
329C24C2×#4111yescharacteristic
330#41—24no—
331#41—24no—
332#41—24no—
333#41—24no—
334#41—24no—
335#41—24no—
336#41—24no—
337#41—24no—
338#41—24no—
339#41—24no—
340#41—42no—
341#41—42no—
342#41—42no—
343#41—42no—
344#41#12318yes—
345#41#12318yes—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8C2×C22⋊C412yes—
355C2×Q8C2×C22⋊C412yes—
356C2×Q8#11712yes—
357C2×Q8#11712yes—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4C2×C22⋊C412yes—
375C2×D4C2×C22⋊C412yes—
376C2×D4#11712yes—
377C2×D4#11712yes—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
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—42no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4112yes—
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C42−1+411yescharacteristic
431Q16—28no—
432SD16—28no—
433SD16—28no—
434D8—28no—
435M16—24no—
436M16—24no—
437M16—416no—
438M16—42no—
439M16—42no—
440M16—42no—
441M16—42no—
442M16—42no—
443M16—42no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—416no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8C2×C22⋊C412yes—
476C2×C8C2×C22⋊C412yes—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—22no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×C4×D411yescharacteristic
530C23#52912yes—
531C23#54612yes—
532C23#55211yescharacteristic
533C23#55311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8#38114yes—
542D4—28no—
543D4—28no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4—44no—
550D4—44no—
551D4—41no—
552D4—41no—
553D4#38114yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4#44312yes—
600C2×C4#44412yes—
601C2×C4C22×C22⋊C411yescharacteristic
602C2×C4C2×#11711yescharacteristic
603C2×C4C2×#11711yescharacteristic
604C2×C4C4×#4111yescharacteristic
605C2×C4#51811yescharacteristic
606C2×C4#52011yescharacteristic
607C2×C4C2×#12312yes—
608C2×C4C2×#12312yes—
609C2×C4#52512yes—
610C2×C4#53812yes—
611C2×C4#53912yes—
612C2×C4D4212yes—
613C2×C4#54612yes—
614C2×C4#54611yescharacteristic
615C2×C4#54712yes—
616C2×C4#54811yescharacteristic
617C2×C4#55112yes—
618C2×C4#55312yes—
619C8—22no—
620C8—22no—
621C8—41no—
622C8—41no—
623C8—42no—
624C8—42no—
625C8—44no—
626C8—82no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—24no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22#229011yescharacteristic
650C22D4×C22⋊C411yescharacteristic
651C22#235211yescharacteristic
652C22#241912yes—
653C22#244412yes—
654C22C2×#44311yescharacteristic
655C22C2×#44411yescharacteristic
656C22#295711yescharacteristic
657C22#295711yescharacteristic
658C4—24no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—84no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4C2×#38111yescharacteristic
670C4#231811yescharacteristic
671C4#233911yescharacteristic
672C4#235311yescharacteristic
673C4#241812yes—
674C4#244312yes—
675C2—22no—
676C2—22no—
677C2—44no—
678C2—82no—
679C2#1389411yescharacteristic
680C2#2113711yescharacteristic
681C2#2114011yescharacteristic
682C2#2245211yescharacteristic
683C2#2245411yescharacteristic
684C2#2278211yescharacteristic
685C2#2278311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20684C2semidirect
4. #20336C2semidirect
5. #11376C2semidirect
6. #11191C2semidirect
7. #10822C2semidirect
8. #10797C2semidirect
9. #10791C2semidirect
10. #10790C2semidirect
11. #10790C2semidirect
17. C2×C4×Q16C4semidirect
18. C2×C4×Q16C4semidirect
19. C2×C4×SD16C4semidirect
20. C2×C4×SD16C4semidirect
21. C2×C4×SD16C4semidirect
22. C2×C4×SD16C4semidirect
23. C2×C4×D8C4semidirect
24. C2×C4×D8C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1352
C2×#18762
#206842
#203362
#113762
#111912
#108222
#107972
#107912
#107902