Finite Group Databasebeta

← #414651

Subgroups — #414651

Order 512.

2056 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C22×C4×SD16C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic, complemented
7#11323C212yesmaximal, complemented
8#11121C212yesmaximal
9#11050C212yesmaximal, complemented
10#11023C212yesmaximal
11#10834C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—22no—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#475—24no—
18C2×#475—24no—
19C2×#475—24no—
20C2×C4×SD16—24no—
21C2×C4×SD16—24no—
22C2×C4×SD16—24no—
23C22×#106C2211yescharacteristic
24C22×#106C2212yes—
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2212yes—
30C22×C4×C8C2211yescharacteristic
31#2542—24no—
32#2542—24no—
33C2×#399—22no—
34C2×#391C2212yes—
35C2×#390C2212yes—
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1990—24no—
40#1802—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1441—22no—
47C4⋊#95—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×C22⋊Q8—216no—
70C2×C22⋊Q8—48no—
71C2×C22⋊Q8—48no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8D414yes—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4D414yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—24no—
96C22×C42—21no—
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99#475—232no—
100C4×SD16—232no—
101C2×#106—24no—
102C2×#106—22no—
103C2×#106—24no—
104C2×#106—24no—
105C2×#106—22no—
106C2×#106—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8D414yes—
110C2×#103—24no—
111C2×#103—22no—
112C2×#103—24no—
113C2×#103—22no—
114C2×#103—24no—
115C2×#103—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—42no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8D414yes—
125#399—216no—
126#391—24no—
127#391—42no—
128#390—24no—
129#390—24no—
130#384—24no—
131#384—24no—
132#384—22no—
133#384—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95D412yes—
139#340—24no—
140#340—24no—
141#340—24no—
142#336—24no—
143#336—42no—
144C25—21no—
145C22×Q8—24no—
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—21no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8C2×D412yes—
162C22×Q8C2×Q812yes—
163C22×Q8#4112yes—
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×D4—21no—
175C22×D4C2×D412yes—
176C22×D4C2×Q812yes—
177C22×D4#4112yes—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C2×D414yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×SD16—28no—
193C2×SD16—28no—
194C2×SD16—28no—
195C2×SD16—28no—
196C2×SD16—28no—
197C2×SD16—28no—
198C2×SD16—28no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8C2×D412yes—
216C22×C8C2×D414yes—
217C22×C8C2×D412yes—
218C22×C8C2×Q812yes—
219C22×C8#4112yes—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222C4⋊Q8—44no—
223C4⋊Q8—44no—
224C22⋊Q8—464no—
225C4×Q8—28no—
226C4×Q8—28no—
227C4×Q8—44no—
228C4×Q8—44no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—44no—
232C4×D4—44no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4SD1618yescomplemented
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—48no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42#4112yes—
294#106—28no—
295#106—216no—
296C4⋊C8—28no—
297#103—28no—
298#103—216no—
299#102—28no—
300#102—48no—
301C4×C8—28no—
302#95—24no—
303#95—48no—
304#95—84no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24C22×D411yescharacteristic
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—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
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—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—48no—
345C2×Q8—416no—
346C2×Q8—88no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8C22⋊Q814yescomplemented
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—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
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×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4C22⋊Q814yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—416no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—88no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4#12012yes—
419C22×C4#12012yes—
420C22×C4#12112yes—
421C22×C4#12112yes—
422C22×C4#12712yes—
423C22×C4#12712yes—
424C22×C4C2×SD1618yes—
425C22×C4C2×SD1614yes—
426C22×C4AGL(1,ℤ/8)14yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×Q811yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C42−1+411yescharacteristic
439C22×C42−1+411yescharacteristic
440C22×C42−1+411yescharacteristic
441SD16—232no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—42no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8C22⋊Q814yescomplemented
463C4⋊C4—216no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—432no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C22⋊C4—432no—
486C22⋊C4—48no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—48no—
492C42#12014yes—
493C42#12014yes—
494C42#12114yes—
495C42#12114yes—
496C23—24no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23C2×#12011yescharacteristic
520C23C2×#12111yescharacteristic
521C23C2×#12711yescharacteristic
522C23#54611yescharacteristic
523C23#54711yescharacteristic
524C23#54811yescharacteristic
525C23D4×Q811yescharacteristic
526C23C22×SD1612yes—
527C23C2×AGL(1,ℤ/8)12yes—
528C23#57712yes—
529C23#57812yes—
530Q8—24no—
531Q8—24no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—816no—
539D4—24no—
540D4—24no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—84no—
580C2×C4—816no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4#39214yes—
587C2×C4#39214yes—
588C2×C4#45018yes—
589C2×C4#46018yes—
590C2×C4#49214yes—
591C2×C4#49614yes—
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12011yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4#52511yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4#53611yescharacteristic
603C2×C4#53611yescharacteristic
604C2×C4#53611yescharacteristic
605C2×C4#53711yescharacteristic
606C2×C4#53711yescharacteristic
607C2×C4#53711yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#54811yescharacteristic
613C2×C4D4×Q811yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4#55011yescharacteristic
616C2×C4#55211yescharacteristic
617C2×C4#55211yescharacteristic
618C2×C4#55411yescharacteristic
619C2×C4#55611yescharacteristic
620C8—24no—
621C8—42no—
622C8—44no—
623C22—48no—
624C22—44no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—82no—
633C22—82no—
634C22C2×#39211yescharacteristic
635C22#240711yescharacteristic
636C22#240911yescharacteristic
637C22#241511yescharacteristic
638C22#241711yescharacteristic
639C22#254011yescharacteristic
640C22#254211yescharacteristic
641C22C2×#45012yes—
642C22#301312yes—
643C22C2×#46012yes—
644C22#304612yes—
645C22C2×#49211yescharacteristic
646C22C2×#49611yescharacteristic
647C22#315411yescharacteristic
648C22#315611yescharacteristic
649C22#329812yes—
650C22#333212yes—
651C22#333512yes—
652C22#335312yes—
653C22#335912yes—
654C22#337312yes—
655C22Q8×SD1612yes—
656C22#338612yes—
657C4—24no—
658C4—48no—
659C4—44no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4—162no—
664C4#201418yes—
665C4#240711yescharacteristic
666C4#240911yescharacteristic
667C4#241511yescharacteristic
668C4#241711yescharacteristic
669C4#253611yescharacteristic
670C4#253811yescharacteristic
671C4#261611yescharacteristic
672C4#284811yescharacteristic
673C2—84no—
674C2#1517711yescharacteristic
675C2C2×#201411yescharacteristic
676C2#2342711yescharacteristic
677C2#2412012yes—
678C2#2433012yes—
679C2#2467812yes—
680C2#2481312yes—
681C2#2840011yescharacteristic
682C2#2841511yescharacteristic
683C2#2853011yescharacteristic
684C2#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
1. C22×#475C2semidirect
4. C2×#1990C2semidirect
6. #13926C2semidirect
7. #11323C2semidirect
9. #11050C2semidirect
11. #10834C2semidirect
59. C22×SD16Q8semidirect
269. C2×C4⋊C4SD16semidirect
349. C2×Q8C22⋊Q8semidirect
462. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C22×C4×SD162
C2×#25422
C2×#19902
C2×#18022
#139262
#113232
#111212
#110502
#110232
#108342