Finite Group Databasebeta

← #414601

Subgroups — #414601

Order 512.

1924 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic
3C2×C4×#124C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6#14714C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#11121C212yesmaximal
10#11009C212yesmaximal
11#10832C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#475—24no—
17C2×#475—24no—
18C2×#475—24no—
19C2×#474—24no—
20C2×#474—24no—
21C2×#474—24no—
22C22×#107C2212yes—
23C22×#106C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2212yes—
28C22×#102C2212yes—
29C22×#102C2212yes—
30C4×#124—24no—
31C4×#124—24no—
32C2×#391—22no—
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#388—22no—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×C4×C22⋊C4—22no—
39C2×C43C2211yescharacteristic
40#1803—24no—
41#1802—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C4⋊#95—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yes—
61C2×#124—22no—
62C2×#124—22no—
63C2×#124—22no—
64C2×#124—22no—
65C2×#124—22no—
66C2×#124Q814yescomplemented
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8D414yes—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—22no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C42—21no—
91C22×C42D412yes—
92C22×C42D412yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#475—232no—
97#474—232no—
98C2×#107—24no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#106—24no—
102C2×#106—24no—
103C2×#106—24no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8D414yes—
109C2×C4⋊C8D414yes—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—42no—
116C2×#102—24no—
117C2×#102—24no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—42no—
122#391—22no—
123#391—22no—
124#391—48no—
125#390—22no—
126#390—22no—
127#388—216no—
128#384—22no—
129#384—22no—
130#382—22no—
131#382—22no—
132C4×C22⋊C4—216no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—24no—
136C2×#95—24no—
137C43—22no—
138C43—22no—
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—21no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C2×D414yes—
205C22×C8C2×D414yes—
206C4⋊Q8—28no—
207C4⋊Q8—28no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210#124—28no—
211#124—28no—
212#124—28no—
213#124—28no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—44no—
217C4×Q8—44no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—44no—
221C4×D4—44no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
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—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D414yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×Q812yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292#107—216no—
293#106—216no—
294C4⋊C8—28no—
295C4⋊C8—28no—
296#103—216no—
297#103—48no—
298#102—216no—
299#102—48no—
300#95—28no—
301#95—28no—
302#95—48no—
303#95—48no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C22×D411yescharacteristic
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—22no—
329C2×Q8—22no—
330C2×Q8—22no—
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—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—48no—
344C2×Q8—416no—
345C2×Q8—88no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8C22⋊Q814yes—
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—22no—
362C2×D4—22no—
363C2×D4—22no—
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×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4C22⋊Q814yes—
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—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—88no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4#12012yes—
424C22×C4#12012yes—
425C22×C4#12112yes—
426C22×C4#12112yes—
427C22×C4C22⋊Q814yes—
428C22×C4#12712yes—
429C22×C4#12712yes—
430C22×C4#13514yes—
431C22×C4AGL(1,ℤ/8)14yes—
432C22×C4#13714yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×Q811yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C42−1+411yescharacteristic
445C22×C42−1+411yescharacteristic
446C22×C42−1+411yescharacteristic
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C22⋊C4—232no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—48no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—48no—
494C42—48no—
495C42#12014yes—
496C42#12014yes—
497C42#12114yes—
498C42#12114yes—
499C23—24no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23C2×#12011yescharacteristic
523C23C2×#12111yescharacteristic
524C23C2×C22⋊Q811yescharacteristic
525C23#52511yescharacteristic
526C23C2×#12711yescharacteristic
527C23#53611yescharacteristic
528C23#53711yescharacteristic
529C23C2×#13512yes—
530C23C2×AGL(1,ℤ/8)12yes—
531C23C2×#13712yes—
532C23#57512yes—
533Q8—24no—
534Q8—24no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—816no—
542D4—24no—
543D4—24no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—84no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—816no—
591C2×C4—84no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—84no—
595C2×C4—82no—
596C2×C4#39214yes—
597C2×C4#39214yes—
598C2×C4#44918yes—
599C2×C4#46318yes—
600C2×C4#49514yes—
601C2×C4#49614yes—
602C2×C4C2×#12011yescharacteristic
603C2×C4C2×#12011yescharacteristic
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×C22⋊Q811yescharacteristic
607C2×C4C2×C22⋊Q811yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4#53611yescharacteristic
611C2×C4#53611yescharacteristic
612C2×C4#53711yescharacteristic
613C2×C4#53711yescharacteristic
614C2×C4#54611yescharacteristic
615C2×C4#54611yescharacteristic
616C2×C4#54711yescharacteristic
617C2×C4#54711yescharacteristic
618C2×C4#54811yescharacteristic
619C2×C4#54811yescharacteristic
620C2×C4#54811yescharacteristic
621C2×C4#54811yescharacteristic
622C2×C4D4×Q811yescharacteristic
623C2×C4D4×Q811yescharacteristic
624C2×C4D4×Q811yescharacteristic
625C2×C4#55011yescharacteristic
626C2×C4#55211yescharacteristic
627C2×C4#55211yescharacteristic
628C2×C4#55411yescharacteristic
629C2×C4#55611yescharacteristic
630C8—44no—
631C8—44no—
632C22—44no—
633C22—48no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22C2×#39211yescharacteristic
644C22#253611yescharacteristic
645C22#253811yescharacteristic
646C22#254011yescharacteristic
647C22#254211yescharacteristic
648C22#261611yescharacteristic
649C22#284811yescharacteristic
650C22C2×#44912yes—
651C22#300712yes—
652C22C2×#46312yes—
653C22#303712yes—
654C22#307612yes—
655C22#307912yes—
656C22#310712yes—
657C22#311512yes—
658C22C2×#49511yescharacteristic
659C22C2×#49611yescharacteristic
660C22#315211yescharacteristic
661C22#315311yescharacteristic
662C22#323312yes—
663C22#323912yes—
664C22#324812yes—
665C22#325112yes—
666C4—24no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—82no—
671C4—84no—
672C4—82no—
673C4—162no—
674C4#201218yes—
675C4#240711yescharacteristic
676C4#240711yescharacteristic
677C4#240911yescharacteristic
678C4#240911yescharacteristic
679C4#241511yescharacteristic
680C4#241511yescharacteristic
681C4#241711yescharacteristic
682C4#241711yescharacteristic
683C2—84no—
684C2#1517711yescharacteristic
685C2C2×#201211yescharacteristic
686C2#2342911yescharacteristic
687C2#2820312yes—
688C2#2828112yes—
689C2#2837211yescharacteristic
690C2#2838311yescharacteristic
691C2#2851011yescharacteristic
692C2#2852011yescharacteristic
693C2#2968912yes—
694C2#3295512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#1803C2semidirect
6. #14714C2semidirect
7. #11325C2semidirect
8. #11124C2semidirect
11. #10832C2semidirect
66. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C22×#4742
C2×C4×#1242
C2×#18032
C2×#18022
#147142
#113252
#111242
#111212
#110092
#108322