Finite Group Databasebeta

← #412569

Subgroups — #412569

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic, complemented
2C2×#1808C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×C8×C22⋊C4C211yesmaximal, characteristic
5#15132C212yesmaximal
6#11315C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11199C212yesmaximal
9#10862C211yesmaximal, characteristic, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10862C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#397—22no—
21C2×#389C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1942—24no—
26#1808—24no—
27#1808—24no—
28C8×C22⋊C4—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1443—22no—
38#1442—22no—
39#1440—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8D414yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107D414yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87#397—48no—
88#389—24no—
89#389—42no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380—42no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—42no—
113#336—22no—
114#336—22no—
115C25C2×D411yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—28no—
159C22×C8—28no—
160C22×C8—28no—
161C22×C8—44no—
162C22×C8—44no—
163C22×C8—44no—
164C22×C8—44no—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8#4112yes—
169C22×C8#4112yes—
170C22×C8#4112yes—
171C22×C8#4112yes—
172C22×C8#4112yes—
173C22×C8#4112yes—
174C22×C8#4112yes—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
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—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4D814yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226#107—28no—
227#107—44no—
228#107—44no—
229#107—44no—
230C22⋊C8—28no—
231C4×C8—44no—
232C4×C8—44no—
233#95—22no—
234#95—22no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—82no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24#12112yes—
257C24C2×D812yes—
258C24#13512yes—
259C24C22×D411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4#12112yes—
347C22×C4C2×D812yes—
348C22×C4C2×D812yes—
349C22×C4#13512yes—
350C22×C4#13512yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364C22×C42−1+411yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—416no—
387C2×C8#12114yes—
388C2×C8#12314yes—
389C2×C8#12314yes—
390C2×C8#12314yes—
391C2×C8#12414yes—
392C2×C8#12614yes—
393C2×C8#12614yes—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—88no—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—88no—
414C22⋊C4—88no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23#46614yes—
458C23C2×#12111yescharacteristic
459C23#53211yescharacteristic
460C23#53211yescharacteristic
461C23D4211yescharacteristic
462C23#54711yescharacteristic
463C23#54711yescharacteristic
464C23#54711yescharacteristic
465C23C22×D811yescharacteristic
466C23C2×#13511yescharacteristic
467C23C2×#13511yescharacteristic
468C23C2×#13511yescharacteristic
469C23#57611yescharacteristic
470C23#57811yescharacteristic
471C23#57811yescharacteristic
472C23#57811yescharacteristic
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×#12311yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4C2×#12311yescharacteristic
548C2×C4#52511yescharacteristic
549C2×C4C2×#12411yescharacteristic
550C2×C4C2×#12611yescharacteristic
551C2×C4C2×#12611yescharacteristic
552C2×C4#52911yescharacteristic
553C2×C4#52911yescharacteristic
554C2×C4#53611yescharacteristic
555C2×C4#53711yescharacteristic
556C2×C4#54111yescharacteristic
557C2×C4#54111yescharacteristic
558C2×C4#54211yescharacteristic
559C2×C4#54211yescharacteristic
560C2×C4#54711yescharacteristic
561C2×C4#54811yescharacteristic
562C2×C4#54811yescharacteristic
563C2×C4#55211yescharacteristic
564C2×C4#55211yescharacteristic
565C2×C4#55211yescharacteristic
566C2×C4#55211yescharacteristic
567C2×C4#55311yescharacteristic
568C2×C4#55311yescharacteristic
569C2×C4#55511yescharacteristic
570C2×C4#55611yescharacteristic
571C2×C4#55611yescharacteristic
572C8—42no—
573C8—42no—
574C8—42no—
575C8#39718yescomplemented
576C22—24no—
577C22—24no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22#241611yescharacteristic
592C22#242211yescharacteristic
593C22#248611yescharacteristic
594C22#249111yescharacteristic
595C22#249611yescharacteristic
596C22#249611yescharacteristic
597C22#285411yescharacteristic
598C22C2×#46611yescharacteristic
599C22#305011yescharacteristic
600C22#305911yescharacteristic
601C22#306111yescharacteristic
602C22#316811yescharacteristic
603C22#316811yescharacteristic
604C22#316911yescharacteristic
605C22#316911yescharacteristic
606C22#317511yescharacteristic
607C22#317511yescharacteristic
608C22#317711yescharacteristic
609C22#317711yescharacteristic
610C22D4×D811yescharacteristic
611C22#328311yescharacteristic
612C22#328511yescharacteristic
613C22#328511yescharacteristic
614C22#333411yescharacteristic
615C22#333411yescharacteristic
616C22#333411yescharacteristic
617C22#333611yescharacteristic
618C22#333611yescharacteristic
619C22#333611yescharacteristic
620C22#333611yescharacteristic
621C22#333911yescharacteristic
622C22#333911yescharacteristic
623C22#334111yescharacteristic
624C22#334211yescharacteristic
625C22#334211yescharacteristic
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—84no—
632C4—162no—
633C4—162no—
634C4—162no—
635C4C2×#39711yescharacteristic
636C4#241511yescharacteristic
637C4#242111yescharacteristic
638C4#248511yescharacteristic
639C4#248911yescharacteristic
640C4#249411yescharacteristic
641C4#249511yescharacteristic
642C4#262511yescharacteristic
643C2—44no—
644C2#1513211yescharacteristic
645C2#2477911yescharacteristic
646C2#2478011yescharacteristic
647C2#2503311yescharacteristic
648C2#2503411yescharacteristic
649C2#2766011yescharacteristic
650C2#2766111yescharacteristic
651C2#2778911yescharacteristic
652C2#2779111yescharacteristic
653C2#2783711yescharacteristic
654C2#2783811yescharacteristic
655C2#2783911yescharacteristic
656C2#2783911yescharacteristic
657C2#3312911yescharacteristic
658C2#3313911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1942C2semidirect
6. #11315C2semidirect
7. #11292C2semidirect
9. #10862C2semidirect
10. #10862C2semidirect
11. #10862C2semidirect
575. C8#397semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×#18082
C2×C8×C22⋊C42
#151322
#113152
#112922
#111992
#108622