Finite Group Databasebeta

← #414653

Subgroups — #414653

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C22×C4×SD16C211yesmaximal, characteristic
3C2×#2568C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1803C211yesmaximal, characteristic, complemented
6#13906C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#11051C212yesmaximal
10#11023C212yesmaximal
11#10834C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#474—24no—
17C2×#474—24no—
18C2×#474—24no—
19C2×C4×SD16—24no—
20C2×C4×SD16—24no—
21C2×C4×SD16—24no—
22C22×#107C2212yes—
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2212yes—
27C22×#102C2212yes—
28C22×#102C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2568—24no—
31#2568—24no—
32C2×#400—22no—
33C2×#391—22no—
34C2×#391C2212yes—
35C2×#390C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380—22no—
39C2×C4×C4⋊C4C2211yescharacteristic
40#1990—24no—
41#1803—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1439—22no—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16—22no—
57C22×SD16—22no—
58C22×SD16—22no—
59C22×SD16Q814yescomplemented
60C23×C8C2311yescharacteristic
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8D414yes—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8D414yes—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4D414yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—24no—
92C22×C42—21no—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#474—232no—
96C4×SD16—232no—
97C2×#107—24no—
98C2×#107—24no—
99C2×#107—24no—
100C2×#106—22no—
101C2×#106—22no—
102C2×#106—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8D414yes—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—42no—
112C2×#102—22no—
113C2×#102—24no—
114C2×#102—22no—
115C2×#102—24no—
116C2×#102—22no—
117C2×#102—24no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120C2×C4×C8D414yes—
121#400—216no—
122#391—216no—
123#391—24no—
124#391—24no—
125#390—24no—
126#390—42no—
127#384—22no—
128#384—22no—
129#382—24no—
130#382—24no—
131#380—48no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95D412yes—
138#340—24no—
139#340—24no—
140#340—24no—
141#336—24no—
142#336—42no—
143C25—21no—
144C22×Q8—24no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8C2×D412yes—
161C22×Q8C2×Q812yes—
162C22×Q8#4112yes—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C2×D412yes—
175C22×D4C2×Q812yes—
176C22×D4#4112yes—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2×D414yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16—28no—
194C2×SD16—28no—
195C2×SD16—28no—
196C2×SD16—28no—
197C2×SD16—28no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8C2×D412yes—
215C22×C8C2×D414yes—
216C22×C8C2×D412yes—
217C22×C8C2×Q812yes—
218C22×C8#4112yes—
219C4⋊Q8—28no—
220C4⋊Q8—28no—
221C4⋊Q8—44no—
222C4⋊Q8—44no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—44no—
226C4×Q8—44no—
227C4×D4—28no—
228C4×D4—28no—
229C4×D4—44no—
230C4×D4—44no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—84no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42#4112yes—
289#107—216no—
290#106—28no—
291C4⋊C8—28no—
292#103—28no—
293#103—48no—
294#102—28no—
295#102—216no—
296C4×C8—28no—
297#95—216no—
298#95—216no—
299#95—24no—
300#95—48no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24C22×D411yescharacteristic
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—48no—
341C2×Q8—416no—
342C2×Q8—88no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8C22⋊Q814yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4C22⋊Q814yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—416no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—88no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4#12012yes—
416C22×C4#12012yes—
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4#12712yes—
420C22×C4#12712yes—
421C22×C4#13514yes—
422C22×C4AGL(1,ℤ/8)14yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×Q811yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C42−1+411yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437SD16—232no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—42no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8C22⋊Q814yes—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—41no—
466C4⋊C4—42no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—42no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C22⋊C4—48no—
481C22⋊C4—816no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—48no—
487C42#12014yes—
488C42#12014yes—
489C42#12114yes—
490C42#12114yes—
491C23—24no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—82no—
513C23—82no—
514C23C2×#12011yescharacteristic
515C23C2×#12111yescharacteristic
516C23C2×#12711yescharacteristic
517C23#54611yescharacteristic
518C23#54711yescharacteristic
519C23#54811yescharacteristic
520C23D4×Q811yescharacteristic
521C23C2×#13512yes—
522C23C2×AGL(1,ℤ/8)12yes—
523C23#57712yes—
524C23#57812yes—
525Q8—24no—
526Q8—24no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—42no—
533Q8—816no—
534D4—24no—
535D4—24no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—84no—
574C2×C4—82no—
575C2×C4—816no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4#39214yes—
582C2×C4#39214yes—
583C2×C4#44918yes—
584C2×C4#46318yes—
585C2×C4#49514yes—
586C2×C4#49614yes—
587C2×C4C2×#12011yescharacteristic
588C2×C4C2×#12011yescharacteristic
589C2×C4C2×#12111yescharacteristic
590C2×C4C2×#12111yescharacteristic
591C2×C4C2×C22⋊Q811yescharacteristic
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#53611yescharacteristic
598C2×C4#53611yescharacteristic
599C2×C4#53611yescharacteristic
600C2×C4#53711yescharacteristic
601C2×C4#53711yescharacteristic
602C2×C4#53711yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#54711yescharacteristic
605C2×C4#54811yescharacteristic
606C2×C4#54811yescharacteristic
607C2×C4#54811yescharacteristic
608C2×C4D4×Q811yescharacteristic
609C2×C4D4×Q811yescharacteristic
610C2×C4#55011yescharacteristic
611C2×C4#55211yescharacteristic
612C2×C4#55211yescharacteristic
613C2×C4#55411yescharacteristic
614C2×C4#55611yescharacteristic
615C8—24no—
616C8—42no—
617C8—44no—
618C22—44no—
619C22—48no—
620C22—82no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22C2×#39211yescharacteristic
630C22#240711yescharacteristic
631C22#240911yescharacteristic
632C22#241511yescharacteristic
633C22#241711yescharacteristic
634C22#254011yescharacteristic
635C22#254211yescharacteristic
636C22C2×#44912yes—
637C22#301312yes—
638C22C2×#46312yes—
639C22#304612yes—
640C22C2×#49511yescharacteristic
641C22C2×#49611yescharacteristic
642C22#315411yescharacteristic
643C22#315611yescharacteristic
644C22#330212yes—
645C22#332412yes—
646C22#334012yes—
647C22#335312yes—
648C22#336412yes—
649C22#336812yes—
650C22#338112yes—
651C22#338612yes—
652C4—24no—
653C4—44no—
654C4—84no—
655C4—82no—
656C4—84no—
657C4—82no—
658C4—162no—
659C4#201218yes—
660C4#240711yescharacteristic
661C4#240911yescharacteristic
662C4#241511yescharacteristic
663C4#241711yescharacteristic
664C4#253611yescharacteristic
665C4#253811yescharacteristic
666C4#261611yescharacteristic
667C4#284811yescharacteristic
668C2—84no—
669C2#1517711yescharacteristic
670C2C2×#201211yescharacteristic
671C2#2342711yescharacteristic
672C2#2411912yes—
673C2#2432812yes—
674C2#2468112yes—
675C2#2481612yes—
676C2#2840311yescharacteristic
677C2#2842011yescharacteristic
678C2#2852911yescharacteristic
679C2#2854011yescharacteristic
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×#2568C2semidirect
4. C2×#1990C2semidirect
5. C2×#1803C2semidirect
7. #11325C2semidirect
8. #11124C2semidirect
11. #10834C2semidirect
59. C22×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C22×C4×SD162
C2×#25682
C2×#19902
C2×#18032
#139062
#113252
#111242
#110512
#110232
#108342