Finite Group Databasebeta

← #412052

Subgroups — #412052

Order 512.

1636 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
1C2×#2547C211yesmaximal, characteristic
2C2×#2042C212yesmaximal
3C2×#1981C211yesmaximal, characteristic, complemented
4C2×#1960C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6C2×#1762C211yesmaximal, characteristic
7#14150C211yesmaximal, characteristic, complemented
8#11190C212yesmaximal, complemented
9#11067C212yesmaximal, complemented
10#11009C212yesmaximal
11#10835C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#97C2211yescharacteristic
24#2547—24no—
25#2547—24no—
26C2×#397—22no—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#382C2211yescharacteristic
30C2×#380—21no—
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2042—28no—
35#1981—24no—
36#1960—24no—
37#1804—24no—
38#1762—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1442—21no—
46#1436—22no—
47C4⋊#95C2212yes—
48C23×Q8C2312yes—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D414yes—
60C2×#124—24no—
61C2×#124—24no—
62C2×#124—24no—
63C2×#124—24no—
64C2×#124—42no—
65C2×#124—42no—
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—22no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—42no—
110C2×#102D414yes—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114#397—216no—
115#391—24no—
116#391—24no—
117#389—24no—
118#389—42no—
119#382—22no—
120#382D414yes—
121#380—28no—
122#380—28no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95D412yes—
136#340—24no—
137#340—22no—
138#340—22no—
139#336—24no—
140#336—42no—
141C25—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8C2×D414yes—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C2×D412yes—
168C22×D4C2×D412yes—
169C22×D4#4112yes—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—24no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
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×C8#4112yes—
206C4⋊Q8—28no—
207C4⋊Q8—28no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210#124—48no—
211#124—48no—
212#124—48no—
213#124—48no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—44no—
217C4×D4—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4#4112yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—42no—
271C2×C42—82no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277C2×C42#4112yes—
278C4⋊C8—44no—
279#103—48no—
280#103—48no—
281#103—48no—
282#103—48no—
283#102—28no—
284#102—28no—
285#102—48no—
286#97—44no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—82no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24C22×D411yescharacteristic
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
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—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
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×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4#12114yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
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—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
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—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4#12012yes—
417C22×C4#12012yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4#12412yes—
421C22×C4#12412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—88no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—84no—
486C42#12114yes—
487C42#12114yes—
488C42#12314yes—
489C42#12314yes—
490C23—24no—
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—44no—
504C23—44no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23C2×#12011yescharacteristic
514C23C2×#12111yescharacteristic
515C23C2×#12411yescharacteristic
516C23D4211yescharacteristic
517C23#54611yescharacteristic
518C23#54811yescharacteristic
519C23#55011yescharacteristic
520C23#57512yes—
521C23#57512yes—
522C23#57712yes—
523C23#57812yes—
524Q8—44no—
525Q8—44no—
526Q8—44no—
527Q8—44no—
528Q8—82no—
529Q8—82no—
530Q8—82no—
531Q8—82no—
532D4—24no—
533D4—24no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—816no—
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—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#39414yes—
593C2×C4#39414yes—
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×#12311yescharacteristic
598C2×C4C2×#12311yescharacteristic
599C2×C4#52511yescharacteristic
600C2×C4#52911yescharacteristic
601C2×C4#52911yescharacteristic
602C2×C4#53211yescharacteristic
603C2×C4#53211yescharacteristic
604C2×C4#53511yescharacteristic
605C2×C4#53711yescharacteristic
606C2×C4#54011yescharacteristic
607C2×C4#54011yescharacteristic
608C2×C4#54211yescharacteristic
609C2×C4#54211yescharacteristic
610C2×C4D4211yescharacteristic
611C2×C4D4211yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54611yescharacteristic
614C2×C4#54611yescharacteristic
615C2×C4#54711yescharacteristic
616C2×C4#54711yescharacteristic
617C2×C4#54811yescharacteristic
618C2×C4#55011yescharacteristic
619C2×C4#55311yescharacteristic
620C2×C4#55311yescharacteristic
621C2×C4#55511yescharacteristic
622C8—42no—
623C8—42no—
624C8—82no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—84no—
629C22—84no—
630C22—82no—
631C22—84no—
632C22—82no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22C2×#39411yescharacteristic
637C22#240711yescharacteristic
638C22#240811yescharacteristic
639C22#242611yescharacteristic
640C22#242811yescharacteristic
641C22#249011yescharacteristic
642C22#249111yescharacteristic
643C22#300511yescharacteristic
644C22#300511yescharacteristic
645C22#301011yescharacteristic
646C22#301111yescharacteristic
647C22#304012yes—
648C22#304612yes—
649C22#313712yes—
650C22#314012yes—
651C22#326712yes—
652C22#326912yes—
653C22#330712yes—
654C22#331512yes—
655C22#337612yes—
656C22#337712yes—
657C22#340112yes—
658C22#340212yes—
659C4—24no—
660C4—24no—
661C4—84no—
662C4—84no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#241411yescharacteristic
669C4#241611yescharacteristic
670C4#242011yescharacteristic
671C4#242211yescharacteristic
672C4#242711yescharacteristic
673C4#242811yescharacteristic
674C4#262411yescharacteristic
675C4#285411yescharacteristic
676C2—84no—
677C2#1512611yescharacteristic
678C2#2350611yescharacteristic
679C2#2351011yescharacteristic
680C2#2419711yescharacteristic
681C2#2420111yescharacteristic
682C2#2424911yescharacteristic
683C2#2425311yescharacteristic
684C2#2528912yes—
685C2#2538112yes—
686C2#2776412yes—
687C2#2778412yes—
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
3. C2×#1981C2semidirect
4. C2×#1960C2semidirect
7. #14150C2semidirect
8. #11190C2semidirect
9. #11067C2semidirect
11. #10835C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#20422
C2×#19812
C2×#19602
C2×#18042
C2×#17622
#141502
#111902
#110672
#110092
#108352