Finite Group Databasebeta

← #414937

Subgroups — #414937

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#1940C211yesmaximal, characteristic, complemented
3C2×#1805C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5#13941C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic, complemented
7#11256C212yesmaximal, complemented
8#11077C212yesmaximal, complemented
9#11076C212yesmaximal
10#11023C212yesmaximal
11#10843C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#440—24no—
16C2×#440—24no—
17C2×#440—24no—
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×#97C2211yescharacteristic
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1940—24no—
32#1805—24no—
33#1802—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1441—21no—
40#1441—21no—
41#1439—22no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C22×SD16—22no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
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×Q8D414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#440—232no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—42no—
95C2×C4⋊C8—42no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—24no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—24no—
106C2×#102—42no—
107C2×#102—42no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97D414yes—
111#391—24no—
112#391—42no—
113#390—24no—
114#390—42no—
115#384—22no—
116#384—22no—
117#382—22no—
118#382—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
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×Q8C2×D412yes—
149C22×Q8C2×Q812yes—
150C22×Q8#4112yes—
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×D4C2×D412yes—
163C22×D4C2×Q812yes—
164C22×D4#4112yes—
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—82no—
174C23×C4—82no—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C2×SD16—28no—
181C2×SD16—28no—
182C2×SD16—28no—
183C2×SD16—28no—
184C2×SD16—28no—
185C2×SD16—28no—
186C2×SD16—28no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×Q812yes—
207C22×C8#4112yes—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—44no—
211C4×Q8—44no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—44no—
215C4×D4—44no—
216C2×C4⋊C4—21no—
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—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
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—22no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
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×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274#107—28no—
275C4⋊C8—48no—
276C4⋊C8—44no—
277#103—28no—
278#103—48no—
279#102—28no—
280#102—48no—
281#97—28no—
282#95—22no—
283#95—22no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—42no—
288#95—82no—
289#95—82no—
290#95—82no—
291#95—82no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24C22×D411yescharacteristic
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—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
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×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8C22⋊Q814yes—
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—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
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×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4C22⋊Q814yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
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—84no—
410C22×C4—84no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4#12112yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4#13714yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q811yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C22×C42−1+411yescharacteristic
434SD16—232no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8C22⋊Q814yes—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—48no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C22⋊C4—48no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—48no—
484C42#12014yes—
485C42#12314yes—
486C42#12314yes—
487C42#12414yes—
488C23—24no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23C2×#12111yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×#12111yescharacteristic
514C23#54611yescharacteristic
515C23#54611yescharacteristic
516C23D4×Q811yescharacteristic
517C23#55011yescharacteristic
518C23C2×#13712yes—
519C23#57512yes—
520C23#57612yes—
521C23#57712yes—
522Q8—24no—
523Q8—24no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530D4—24no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—816no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#39414yes—
593C2×C4#39414yes—
594C2×C4C2×#12011yescharacteristic
595C2×C4C2×C22⋊Q811yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×#12311yescharacteristic
599C2×C4C2×#12311yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4#52511yescharacteristic
603C2×C4C2×#12411yescharacteristic
604C2×C4#53511yescharacteristic
605C2×C4#53511yescharacteristic
606C2×C4#53511yescharacteristic
607C2×C4#53711yescharacteristic
608C2×C4#53711yescharacteristic
609C2×C4#53711yescharacteristic
610C2×C4#55211yescharacteristic
611C2×C4#55211yescharacteristic
612C2×C4#55311yescharacteristic
613C2×C4#55311yescharacteristic
614C2×C4#55311yescharacteristic
615C2×C4#55311yescharacteristic
616C2×C4#55411yescharacteristic
617C2×C4#55411yescharacteristic
618C2×C4#55411yescharacteristic
619C2×C4#55511yescharacteristic
620C2×C4#55511yescharacteristic
621C2×C4#55611yescharacteristic
622C8—24no—
623C8—42no—
624C8—82no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—84no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—82no—
633C22—84no—
634C22—82no—
635C22—84no—
636C22C2×#39411yescharacteristic
637C22#241411yescharacteristic
638C22#241411yescharacteristic
639C22#241711yescharacteristic
640C22#241711yescharacteristic
641C22#253511yescharacteristic
642C22#253811yescharacteristic
643C22#303712yes—
644C22#303911yescharacteristic
645C22#304011yescharacteristic
646C22#304111yescharacteristic
647C22#304211yescharacteristic
648C22#304512yes—
649C22#305612yes—
650C22#306212yes—
651C22#330912yes—
652C22#331112yes—
653C22#331412yes—
654C22#332012yes—
655C22#338512yes—
656C22#338812yes—
657C22#340012yes—
658C22#340312yes—
659C4—24no—
660C4—24no—
661C4—44no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—162no—
667C4—162no—
668C4#248511yescharacteristic
669C4#248511yescharacteristic
670C4#248711yescharacteristic
671C4#248711yescharacteristic
672C4#256211yescharacteristic
673C4#256311yescharacteristic
674C4#261511yescharacteristic
675C4#284811yescharacteristic
676C2—84no—
677C2#1518111yescharacteristic
678C2#2354411yescharacteristic
679C2#2354711yescharacteristic
680C2#2463912yes—
681C2#2464611yescharacteristic
682C2#2464711yescharacteristic
683C2#2482412yes—
684C2#2483311yescharacteristic
685C2#2483411yescharacteristic
686C2#2815512yes—
687C2#2827712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
2. C2×#1940C2semidirect
3. C2×#1805C2semidirect
6. #13926C2semidirect
7. #11256C2semidirect
8. #11077C2semidirect
11. #10843C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#19402
C2×#18052
C2×#18022
#139412
#139262
#112562
#110772
#110762
#110232
#108432