Finite Group Databasebeta

← #361975

Subgroups — #361975

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2C2×#1968C212yesmaximal
3C2×#1844C211yesmaximal, characteristic, complemented
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
7#13926C211yesmaximal, characteristic, complemented
8#13906C211yesmaximal, characteristic
9#11122C212yesmaximal, complemented
10#11121C212yesmaximal
11#10861C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×SD16—24no—
16C2×C4×SD16—24no—
17C2×C4×SD16C418yescomplemented
18C22×#106C2212yes—
19C22×#106C2211yescharacteristic
20C22×#106C2212yes—
21C22×C4⋊C8C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#102C2212yes—
25C22×#102C2211yescharacteristic
26C22×C4×C8C2212yes—
27C22×C4×C8C2211yescharacteristic
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#380—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1968—28no—
35#1844—24no—
36#1803—24no—
37#1802—24no—
38C8×C4⋊C4—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1441—22no—
43#1439—22no—
44C4⋊#95—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C22×SD16—22no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16C2×C414yes—
55C22×SD16D414yes—
56C22×SD16Q814yes—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—24no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C4×SD16—232no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—42no—
95C2×#106—42no—
96C2×#106C2×C414yes—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8D414yes—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103C2×C414yes—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—42no—
111C2×#102C2×C414yes—
112C2×C4×C8—24no—
113C2×C4×C8—24no—
114C2×C4×C8—24no—
115C2×C4×C8C2×C414yes—
116C2×C4×C8D414yes—
117C2×C4×C8D414yes—
118#391—24no—
119#391—42no—
120#390—24no—
121#390—42no—
122#384—22no—
123#384—22no—
124#384—22no—
125#384—22no—
126#380—48no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C2×#95—24no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95D412yes—
133#340—24no—
134#340—24no—
135#336—24no—
136C25—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8C22×C412yes—
149C22×Q8C2×D412yes—
150C22×Q8C2×Q812yes—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22×C412yes—
163C22×D4C2×D412yes—
164C22×D4C2×Q812yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C2×D414yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C2×SD16—28no—
180C2×SD16—28no—
181C2×SD16—28no—
182C2×SD16—28no—
183C2×SD16—28no—
184C2×SD16—28no—
185C2×SD16C4⋊C4116yescomplemented
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C22×C412yes—
197C22×C8C22×C412yes—
198C22×C8C2×D414yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204C22×C8C2×Q812yes—
205C22×C8#4112yes—
206C22×C8#4112yes—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—44no—
210C4×Q8—44no—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—44no—
214C4×D4—44no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4SD1618yescomplemented
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—84no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42C22×C412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274#106—28no—
275#106—48no—
276#106—48no—
277C4⋊C8—28no—
278#103—28no—
279#103—48no—
280#102—28no—
281#102—48no—
282C4×C8—48no—
283C4×C8C22⋊C4116yescomplemented
284#95—24no—
285#95—48no—
286#95—48no—
287#95—84no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24C22×D411yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8C2×C4⋊C414yes—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4C2×C4⋊C414yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—48no—
382C22×C4—416no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—88no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4C4×D412yes—
397C22×C4C4×D412yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12712yes—
401C22×C4#12712yes—
402C22×C4C2×SD1618yes—
403C22×C4C2×SD1614yes—
404C22×C4#13514yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×Q811yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C42−1+411yescharacteristic
418C22×C42−1+411yescharacteristic
419SD16—232no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8C2×C22⋊C414yes—
434C2×C8C2×C22⋊C414yes—
435C2×C8C2×C4⋊C414yes—
436C2×C8#11714yes—
437C2×C8#12114yes—
438C2×C8#12114yes—
439C2×C8C22⋊Q814yes—
440C2×C8C22⋊Q814yes—
441C4⋊C4—216no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—48no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C22⋊C4—48no—
464C22⋊C4—816no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—48no—
470C42C2×C22⋊C414yes—
471C42C4×D414yes—
472C42#12114yes—
473C42#12414yes—
474C23—24no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—82no—
496C23—82no—
497C23C2×C4×D411yescharacteristic
498C23C2×#12111yescharacteristic
499C23C2×#12711yescharacteristic
500C23D4211yescharacteristic
501C23#54711yescharacteristic
502C23D4×Q811yescharacteristic
503C23#55011yescharacteristic
504C23C22×SD1612yes—
505C23C2×#13512yes—
506C23#57712yes—
507C23#57712yes—
508Q8—24no—
509Q8—24no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—42no—
516D4—24no—
517D4—24no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—816no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4#39014yes—
564C2×C4#39014yes—
565C2×C4C4×SD1618yes—
566C2×C4#46518yes—
567C2×C4#49214yes—
568C2×C4#49514yes—
569C2×C4C22×C22⋊C411yescharacteristic
570C2×C4C22×C4⋊C411yescharacteristic
571C2×C4C2×#11711yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4#52011yescharacteristic
574C2×C4#52011yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4C2×C22⋊Q811yescharacteristic
580C2×C4C2×#12411yescharacteristic
581C2×C4#53511yescharacteristic
582C2×C4#53611yescharacteristic
583C2×C4#53711yescharacteristic
584C2×C4#53711yescharacteristic
585C2×C4D4211yescharacteristic
586C2×C4#54711yescharacteristic
587C2×C4#54711yescharacteristic
588C2×C4#54711yescharacteristic
589C2×C4D4×Q811yescharacteristic
590C2×C4#55011yescharacteristic
591C2×C4#55411yescharacteristic
592C2×C4#55411yescharacteristic
593C2×C4#55511yescharacteristic
594C2×C4#55511yescharacteristic
595C2×C4#55611yescharacteristic
596C2×C4#55611yescharacteristic
597C8—24no—
598C8—44no—
599C8#38018yescomplemented
600C22—44no—
601C22—48no—
602C22—84no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22C2×#39011yescharacteristic
612C22D4×C4⋊C411yescharacteristic
613C22#235411yescharacteristic
614C22#253711yescharacteristic
615C22#253811yescharacteristic
616C22#254111yescharacteristic
617C22#254211yescharacteristic
618C22C2×C4×SD1612yes—
619C22#295212yes—
620C22C2×#46512yes—
621C22#306012yes—
622C22C2×#49211yescharacteristic
623C22C2×#49511yescharacteristic
624C22#315411yescharacteristic
625C22#315411yescharacteristic
626C22D4×SD1612yes—
627C22#328412yes—
628C22#333512yes—
629C22#334012yes—
630C22Q8×SD1612yes—
631C22#338112yes—
632C22#339212yes—
633C22#339412yes—
634C4—24no—
635C4—48no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4—162no—
640C4—162no—
641C4#196818yescomplemented
642C4C2×#38011yescharacteristic
643C4#231111yescharacteristic
644C4D4×C4⋊C411yescharacteristic
645C4#235411yescharacteristic
646C4#253711yescharacteristic
647C4#253811yescharacteristic
648C4#254711yescharacteristic
649C4#256311yescharacteristic
650C2—84no—
651C2#1390611yescharacteristic
652C2C2×#196811yescharacteristic
653C2#2122911yescharacteristic
654C2C4⋊C4×SD1612yes—
655C2#2280512yes—
656C2#2822312yes—
657C2#2826412yes—
658C2#2846011yescharacteristic
659C2#2846111yescharacteristic
660C2#2852911yescharacteristic
661C2#2853011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1844C2semidirect
4. C2×#1803C2semidirect
6. C2×C8×C4⋊C4C2semidirect
7. #13926C2semidirect
9. #11122C2semidirect
11. #10861C2semidirect
17. C2×C4×SD16C4semidirect
185. C2×SD16C4⋊C4semidirect
250. C2×C4⋊C4SD16semidirect
283. C4×C8C22⋊C4semidirect
599. C8#380semidirect
641. C4#1968semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×#19682
C2×#18442
C2×#18032
C2×#18022
C2×C8×C4⋊C42
#139262
#139062
#111222
#111212
#108612