Finite Group Databasebeta

← #381422

Subgroups — #381422

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#436C211yesmaximal, characteristic, complemented
2C2×#1923C211yesmaximal, characteristic, complemented
3C2×#1870C212yesmaximal, complemented
4C4×#392C211yesmaximal, characteristic
5#11255C211yesmaximal, characteristic, complemented
6#11042C212yesmaximal, complemented
7#11042C211yesmaximal, characteristic, complemented
8#10964C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic, complemented
10#10797C211yesmaximal, characteristic, complemented
11#10715C212yesmaximal, complemented
12C23×M16C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2212yes—
15C2×#436—24no—
16C2×#436—28no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#392C411yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29#1923—24no—
30#1870—28no—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1444C411yescharacteristic
40#1439C412yes—
41#1439C411yescharacteristic
42#1438C412yes—
43#1438C411yescharacteristic
44C4×#95C2211yescharacteristic
45C23×D4C2×C411yescharacteristic
46C23×D4C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2312yes—
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—42no—
52C22×M16—42no—
53C22×M16—42no—
54C22×M16—42no—
55C22×M16D418yescomplemented
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—42no—
69C2×C4×D4—42no—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#436—232no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—42no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8—42no—
105C2×C4×C8—24no—
106C2×C4×C8D414yescomplemented
107#392—24no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—42no—
111C4×C22⋊C4—42no—
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C412yes—
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119#336—24no—
120#336—24no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—42no—
125#336—42no—
126#336—42no—
127C25C22×C412yes—
128C25C22×C411yescharacteristic
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—22no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D414yes—
172C23×C4#4114yes—
173C23×C4C2411yescharacteristic
174C2×M16—216no—
175C2×M16—48no—
176C2×M16—48no—
177C2×M16—48no—
178C2×M16—48no—
179C2×M16—48no—
180C2×M16—48no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—21no—
211C22×C8C2×D414yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D414yes—
215C4×D4—28no—
216C4×D4—216no—
217C4×D4—28no—
218C4×D4—48no—
219C4×D4—48no—
220C4×D4—48no—
221C4×D4—48no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4M1614yescomplemented
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262C4⋊C8—28no—
263C4⋊C8—44no—
264C22⋊C8—216no—
265C22⋊C8—48no—
266C22⋊C8—44no—
267C22⋊C8—48no—
268C22⋊C8—48no—
269C22⋊C8—44no—
270C4×C8—28no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—24no—
275#95—22no—
276#95—42no—
277#95—42no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—21no—
289C24—22no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—22no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24C4×D414yes—
303C24C2×M1612yes—
304C24C23×C411yescharacteristic
305C2×D4—28no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—44no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4C4×D412yes—
403C22×C4C4×D414yes—
404C22×C4C4×D412yes—
405C22×C4C2×M1612yes—
406C22×C4C2×M1612yes—
407C22×C4#13114yes—
408C22×C4#13112yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C42−1+411yescharacteristic
419M16—432no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8#12014yes—
454C4⋊C4—28no—
455C4⋊C4—48no—
456C22⋊C4—216no—
457C22⋊C4—28no—
458C22⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C42—22no—
464C42—22no—
465C42—24no—
466C42—42no—
467C42—41no—
468C42—41no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23C2×C4×D412yes—
532C23C2×C4×D411yescharacteristic
533C23#51811yescharacteristic
534C23#51812yes—
535C23#52011yescharacteristic
536C23C22×M1611yescharacteristic
537C23C2×#13111yescharacteristic
538C23C2×#13112yes—
539C23#56811yescharacteristic
540C23#56812yes—
541C23#56811yescharacteristic
542D4—416no—
543D4—432no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4#43514yes—
604C2×C4C2×#12011yescharacteristic
605C2×C4#52512yes—
606C2×C4#52511yescharacteristic
607C2×C4#53412yes—
608C2×C4#53412yes—
609C2×C4#53511yescharacteristic
610C2×C4#53511yescharacteristic
611C2×C4#53511yescharacteristic
612C2×C4#53512yes—
613C2×C4#53712yes—
614C2×C4#53711yescharacteristic
615C2×C4#53812yes—
616C2×C4#53812yes—
617C2×C4#53812yes—
618C2×C4#53912yes—
619C2×C4#54012yes—
620C2×C4#54011yescharacteristic
621C2×C4#54211yescharacteristic
622C8—24no—
623C8—44no—
624C8—42no—
625C8—81no—
626C8—81no—
627C8—82no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—81no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—81no—
650C22—81no—
651C22—82no—
652C22—81no—
653C22—82no—
654C22C4×#12011yescharacteristic
655C22#232112yes—
656C22#232411yescharacteristic
657C22#232512yes—
658C22#232811yescharacteristic
659C22C2×#43511yescharacteristic
660C22#293011yescharacteristic
661C22#293011yescharacteristic
662C22#293111yescharacteristic
663C22#293312yes—
664C22#293312yes—
665C22D4×M1612yes—
666C22#293512yes—
667C22#297412yes—
668C22#297511yescharacteristic
669C22#297612yes—
670C22#297812yes—
671C22#297812yes—
672C22#297912yes—
673C22#298111yescharacteristic
674C22#298911yescharacteristic
675C22#299312yes—
676C22#299311yescharacteristic
677C4—24no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—82no—
683C4—81no—
684C4—81no—
685C4#261311yescharacteristic
686C4#261511yescharacteristic
687C4#263012yes—
688C4#263111yescharacteristic
689C4#263512yes—
690C4#263611yescharacteristic
691C2—82no—
692C2—84no—
693C2#1429911yescharacteristic
694C2#2143911yescharacteristic
695C2#2144511yescharacteristic
696C2#2203812yes—
697C2#2203912yes—
698C2#2213811yescharacteristic
699C2#2213911yescharacteristic
700C2#2217412yes—
701C2#2217912yes—
702C2#2226211yescharacteristic
703C2#2226511yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#436C2semidirect
2. C2×#1923C2semidirect
3. C2×#1870C2semidirect
5. #11255C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
8. #10964C2semidirect
9. #10964C2semidirect
10. #10797C2semidirect
11. #10715C2semidirect
55. C22×M16D4semidirect
106. C2×C4×C8D4semidirect
247. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#4362
C2×#19232
C2×#18702
C4×#3922
#112552
#110422
#109642
#107972
#107152