Finite Group Databasebeta

← #415603

Subgroups — #415603

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#127C211yesmaximal, characteristic
2C2×#1862C212yesmaximal, complemented
3#14699C211yesmaximal, characteristic, complemented
4#11324C211yesmaximal, characteristic, complemented
5#11267C211yesmaximal, characteristic, complemented
6#11123C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#11008C211yesmaximal, characteristic
9#11008C211yesmaximal, characteristic
10#10795C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4—21no—
15C22×C4×D4C2212yes—
16C22×#107C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C4×#127—28no—
22C2×#390—21no—
23C2×#390—21no—
24C2×#390C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387—21no—
28C2×#384C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1862—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1436—21no—
39#1436—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45C23×D4—21no—
46C23×D4—21no—
47C23×D4C2312yes—
48C24×C4—22no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#127—24no—
52C2×#127—24no—
53C2×#127—22no—
54C2×#127Q814yes—
55C2×C4×D4—28no—
56C2×C4×D4—28no—
57C2×C4×D4—24no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4—44no—
63C2×C4×D4—44no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C2×C4×D4D418yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—42no—
88C2×#107—42no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102—42no—
99C2×#102—42no—
100C2×#102—42no—
101#390—28no—
102#390—28no—
103#390—24no—
104#390—24no—
105#389—22no—
106#389—22no—
107#389—24no—
108#387—44no—
109#384—24no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C43—24no—
121#340—24no—
122#340—42no—
123#340—42no—
124#336—24no—
125#336—24no—
126#336—22no—
127#336—22no—
128C25—22no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—24no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4C2×D414yes—
155C22×D4C2×Q814yes—
156C22×D4#4114yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—28no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8#4114yes—
190#127—216no—
191#127—216no—
192C4×D4—48no—
193C4×D4—48no—
194C4×D4—432no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—432no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4#4112yes—
231C2×C4⋊C4#4112yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42D814yes—
262C2×C42C2×D414yes—
263C2×C42C2×Q812yes—
264C2×C42#4112yes—
265#107—48no—
266#102—216no—
267#102—48no—
268#102—48no—
269#102—48no—
270#95—24no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24C22×D411yescharacteristic
296C2×D4—216no—
297C2×D4—216no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4C22⋊Q818yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—416no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4#12012yes—
411C22×C4#12012yes—
412C22×C4C22⋊Q812yes—
413C22×C4C22⋊Q812yes—
414C22×C4C2×D812yes—
415C22×C4C2×D812yes—
416C22×C4C2×D812yes—
417C22×C4#13512yes—
418C22×C4AGL(1,ℤ/8)12yes—
419C22×C4AGL(1,ℤ/8)12yes—
420C22×C4AGL(1,ℤ/8)12yes—
421C22×C4#13712yes—
422C22×C4C22×D412yes—
423C22×C4C22×Q811yescharacteristic
424C22×C4C2×#4112yes—
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)11yescharacteristic
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—84no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C42—28no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—28no—
471C42—44no—
472C42—44no—
473C42—42no—
474C42—42no—
475C23—28no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23C2×#12011yescharacteristic
511C23C2×C22⋊Q811yescharacteristic
512C23#52511yescharacteristic
513C23#52511yescharacteristic
514C23#53211yescharacteristic
515C23#53511yescharacteristic
516C23#53511yescharacteristic
517C23C22×D811yescharacteristic
518C23C2×#13511yescharacteristic
519C23C2×AGL(1,ℤ/8)11yescharacteristic
520C23C2×AGL(1,ℤ/8)11yescharacteristic
521C23C2×AGL(1,ℤ/8)11yescharacteristic
522C23C2×#13711yescharacteristic
523C23#57511yescharacteristic
524C23#57511yescharacteristic
525D4—28no—
526D4—28no—
527D4—432no—
528D4—432no—
529D4—44no—
530D4—44no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—44no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—832no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4#44714yes—
611C2×C4#47414yes—
612C2×C4#47814yes—
613C2×C4C2×C22⋊Q812yes—
614C2×C4#52512yes—
615C2×C4#52512yes—
616C2×C4#53212yes—
617C2×C4#53412yes—
618C2×C4#53412yes—
619C2×C4#55112yes—
620C2×C4#55112yes—
621C2×C4#55312yes—
622C2×C4#55312yes—
623C2×C4#55312yes—
624C2×C4#55411yescharacteristic
625C2×C4#55412yes—
626C2×C4#55411yescharacteristic
627C2×C4#55911yescharacteristic
628C2×C4#55911yescharacteristic
629C8—44no—
630C8—82no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22#238811yescharacteristic
647C22#256611yescharacteristic
648C22#258011yescharacteristic
649C22#259811yescharacteristic
650C22#260311yescharacteristic
651C22#261311yescharacteristic
652C22#284811yescharacteristic
653C22C2×#44711yescharacteristic
654C22#300211yescharacteristic
655C22#300311yescharacteristic
656C22#300711yescharacteristic
657C22C2×#47411yescharacteristic
658C22C2×#47811yescharacteristic
659C22#307711yescharacteristic
660C22#307711yescharacteristic
661C22#310011yescharacteristic
662C22#310011yescharacteristic
663C22#310511yescharacteristic
664C22#310611yescharacteristic
665C22#311011yescharacteristic
666C22#311211yescharacteristic
667C22#311411yescharacteristic
668C22#311511yescharacteristic
669C22#316911yescharacteristic
670C22#317111yescharacteristic
671C22#318011yescharacteristic
672C22#318211yescharacteristic
673C22#320511yescharacteristic
674C22#320511yescharacteristic
675C22#321011yescharacteristic
676C22#321211yescharacteristic
677C22#321611yescharacteristic
678C22#321711yescharacteristic
679C22#321711yescharacteristic
680C22#321911yescharacteristic
681C4—24no—
682C4—24no—
683C4—24no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—84no—
690C4—161no—
691C4—161no—
692C4#256412yes—
693C4#257912yes—
694C4#260012yes—
695C4#260212yes—
696C2—88no—
697C2#1520311yescharacteristic
698C2#2356711yescharacteristic
699C2#2357611yescharacteristic
700C2#2919211yescharacteristic
701C2#2919511yescharacteristic
702C2#2925411yescharacteristic
703C2#2925611yescharacteristic
704C2#2936411yescharacteristic
705C2#2936711yescharacteristic
706C2#2949311yescharacteristic
707C2#2949811yescharacteristic
708C2#2962211yescharacteristic
709C2#2962911yescharacteristic
710C2#3295311yescharacteristic
711C2#3295911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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
2. C2×#1862C2semidirect
3. #14699C2semidirect
4. #11324C2semidirect
5. #11267C2semidirect
6. #11123C2semidirect
7. #11066C2semidirect
10. #10795C2semidirect
11. #10788C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1272
C2×#18622
#146992
#113242
#112672
#111232
#110662
#110082
#107952
#107882