Finite Group Databasebeta

← #154186

Subgroups — #154186

Order 512.

4790 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#382C211yesmaximal, characteristic, complemented
2C22×#380C212yesmaximal
3C22×C4×C22⋊C4C211yesmaximal, characteristic
4C2×C4⋊#95C212yesmaximal, complemented
5C2×C4⋊#95C211yesmaximal, characteristic, complemented
6#7556C214yesmaximal, complemented
7#7555C214yesmaximal
8C25×C4C2211yescharacteristic
9C23×C4⋊C4C2212yes—
10C23×C4⋊C4C2212yes—
11C23×C22⋊C4C2212yes—
12C23×C22⋊C4C2214yes—
13C23×C42C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#382—24no—
16C2×#382—28no—
17C2×#380—28no—
18C2×#380—28no—
19C2×#380—28no—
20C2×C4×C22⋊C4—24no—
21C2×C4×C22⋊C4—24no—
22C2×C4×C22⋊C4C418yescomplemented
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2218yes—
26C22×#95C2212yes—
27C22×#95C2214yes—
28C4⋊#95—24no—
29C4⋊#95—28no—
30C26C2311yescharacteristic
31C24×C4—22no—
32C24×C4—22no—
33C24×C4—24no—
34C24×C4—24no—
35C24×C4—22no—
36C24×C4—22no—
37C24×C4—24no—
38C24×C4—24no—
39C24×C4C2×C414yes—
40C24×C4D414yes—
41C24×C4Q814yes—
42C24×C4C2314yes—
43C24×C4C2314yes—
44C24×C4C2312yes—
45C24×C4C2311yescharacteristic
46C24×C4C2312yes—
47C24×C4C2311yescharacteristic
48C22×C4⋊C4—24no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4D418yes—
61C22×C4⋊C4D418yes—
62C22×C22⋊C4—28no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—28no—
66C22×C22⋊C4—28no—
67C22×C22⋊C4—28no—
68C22×C22⋊C4—28no—
69C22×C22⋊C4—216no—
70C22×C22⋊C4—48no—
71C22×C22⋊C4C2×C418yes—
72C22×C42—24no—
73C22×C42—24no—
74C22×C42—24no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—24no—
78C22×C42C2×C414yes—
79C22×C42C2×C414yes—
80C22×C42D414yes—
81C22×C42D414yes—
82#382—232no—
83#380—264no—
84C4×C22⋊C4—232no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—28no—
88C2×#95—28no—
89C2×#95—28no—
90C2×#95—216no—
91C2×#95—28no—
92C2×#95—216no—
93C2×#95—28no—
94C2×#95—216no—
95C2×#95—48no—
96C2×#95—48no—
97C2×#95C2×C418yes—
98C25—21no—
99C25—21no—
100C25—22no—
101C25—22no—
102C25—24no—
103C25—21no—
104C25—21no—
105C25—22no—
106C25—22no—
107C25—24no—
108C25—24no—
109C25C22×C412yes—
110C25C2×D412yes—
111C25C2×Q812yes—
112C25#4118yes—
113C25C2411yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—28no—
136C23×C4—28no—
137C23×C4—216no—
138C23×C4—216no—
139C23×C4—216no—
140C23×C4—28no—
141C23×C4—28no—
142C23×C4—28no—
143C23×C4—28no—
144C23×C4—216no—
145C23×C4—216no—
146C23×C4—216no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—28no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—28no—
162C23×C4—28no—
163C23×C4—28no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—28no—
167C23×C4—44no—
168C23×C4—48no—
169C23×C4—48no—
170C23×C4—48no—
171C23×C4—44no—
172C23×C4—48no—
173C23×C4—48no—
174C23×C4—44no—
175C23×C4—48no—
176C23×C4—44no—
177C23×C4C4⋊C4116yes—
178C23×C4C22×C414yes—
179C23×C4C22×C412yes—
180C23×C4C22×C414yes—
181C23×C4C22×C412yes—
182C23×C4C2×D418yes—
183C23×C4C2×D418yes—
184C23×C4C2×D414yes—
185C23×C4C2×D414yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D414yes—
191C23×C4C2×D414yes—
192C23×C4C2×Q812yes—
193C23×C4C2×Q812yes—
194C23×C4#4118yes—
195C23×C4#4112yes—
196C23×C4#4112yes—
197C23×C4#4114yes—
198C23×C4#4114yes—
199C2×C4⋊C4—216no—
200C2×C4⋊C4—216no—
201C2×C4⋊C4—216no—
202C2×C4⋊C4—216no—
203C2×C4⋊C4—216no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—48no—
213C2×C22⋊C4—232no—
214C2×C22⋊C4—232no—
215C2×C22⋊C4—216no—
216C2×C22⋊C4—216no—
217C2×C22⋊C4—216no—
218C2×C22⋊C4—232no—
219C2×C22⋊C4—432no—
220C2×C22⋊C4—432no—
221C2×C22⋊C4—416no—
222C2×C22⋊C4—416no—
223C2×C42—216no—
224C2×C42—28no—
225C2×C42—216no—
226C2×C42—28no—
227C2×C42—28no—
228C2×C42—216no—
229C2×C42—48no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42C22⋊C4116yescomplemented
233#95—232no—
234#95—232no—
235#95—216no—
236#95—432no—
237#95—416no—
238C24—22no—
239C24—24no—
240C24—22no—
241C24—24no—
242C24—24no—
243C24—28no—
244C24—24no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—28no—
253C24—28no—
254C24—28no—
255C24—28no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—28no—
261C24—28no—
262C24—28no—
263C24—28no—
264C24—28no—
265C24—28no—
266C24—28no—
267C24—28no—
268C24—28no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—41no—
275C24—41no—
276C24—42no—
277C24—42no—
278C24—41no—
279C24—42no—
280C24—42no—
281C24—44no—
282C24—42no—
283C24—42no—
284C24—44no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24C2×C4⋊C414yes—
294C24#11718yes—
295C24#12418yes—
296C24#12518yes—
297C24C23×C411yescharacteristic
298C24C22×D411yescharacteristic
299C24C22×D411yescharacteristic
300C24C22×D411yescharacteristic
301C24C22×D412yes—
302C24C22×D412yes—
303C24C22×Q811yescharacteristic
304C24C2×#4111yescharacteristic
305C24C2×#4111yescharacteristic
306C24C2×#4114yes—
307C24C2×#4114yes—
308C24C2×#4112yes—
309C24C2×#4112yes—
310C24H(5,𝔽2)14yes—
311C242−1+414yes—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—232no—
347C22×C4—264no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—41no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—416no—
387C22×C4—432no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—432no—
392C22×C4—432no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4C2×C22⋊C418yes—
432C22×C4C2×C22⋊C414yes—
433C22×C4C2×C4⋊C414yes—
434C22×C4C2×C4⋊C414yes—
435C22×C4#11718yes—
436C22×C4#11714yes—
437C22×C4C4×D418yes—
438C22×C4C4×D418yes—
439C22×C4#12018yes—
440C22×C4#12014yes—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4#12118yes—
444C22×C4#12118yes—
445C22×C4#12118yes—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q814yes—
448C22×C4C22⋊Q818yes—
449C22×C4C22⋊Q818yes—
450C22×C4#12318yes—
451C22×C4#12314yes—
452C22×C4#12318yes—
453C22×C4#12418yes—
454C22×C4#12414yes—
455C22×C4#12618yes—
456C22×C4#12714yes—
457C4⋊C4—432no—
458C4⋊C4—432no—
459C22⋊C4—264no—
460C22⋊C4—464no—
461C42—232no—
462C42—416no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—216no—
469C23—216no—
470C23—28no—
471C23—28no—
472C23—216no—
473C23—216no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—28no—
481C23—48no—
482C23—44no—
483C23—44no—
484C23—48no—
485C23—48no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—44no—
505C23—44no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—44no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23#38218yescomplemented
522C23C22×C22⋊C411yescharacteristic
523C23C22×C4⋊C411yescharacteristic
524C23C2×#11711yescharacteristic
525C23C2×#11712yes—
526C23C2×#11712yes—
527C23C2×C4×D412yes—
528C23C2×C4×D412yes—
529C23#52014yes—
530C23C2×#12011yescharacteristic
531C23C2×#12012yes—
532C23C2×#12111yescharacteristic
533C23C2×#12111yescharacteristic
534C23C2×#12112yes—
535C23C2×#12112yes—
536C23C2×#12112yes—
537C23C2×C22⋊Q811yescharacteristic
538C23C2×C22⋊Q811yescharacteristic
539C23C2×C22⋊Q812yes—
540C23C2×C22⋊Q812yes—
541C23C2×#12312yes—
542C23C2×#12312yes—
543C23C2×#12311yescharacteristic
544C23#52514yes—
545C23#52514yes—
546C23C2×#12411yescharacteristic
547C23C2×#12412yes—
548C23C2×#12412yes—
549C23C2×#12512yes—
550C23C2×#12612yes—
551C23#52914yes—
552C23#52914yes—
553C23C2×#12711yescharacteristic
554C23#53212yes—
555C23#53312yes—
556C23#53512yes—
557C23#53512yes—
558C23#53612yes—
559C23#53612yes—
560C23#53714yes—
561C23#54012yes—
562C23#54112yes—
563C23#54214yes—
564C23#54414yes—
565C23#54614yes—
566C23#54614yes—
567C23#54614yes—
568C23#54714yes—
569C23#54814yes—
570C23#54814yes—
571C23#54814yes—
572C23#55014yes—
573C23#55214yes—
574C23#55214yes—
575C23#55214yes—
576C23#55314yes—
577C23#55314yes—
578C23#55314yes—
579C23#55514yes—
580C23#55614yes—
581C2×C4—28no—
582C2×C4—216no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—464no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—416no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—416no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—88no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4#38018yes—
634C2×C4#385116yes—
635C2×C4#38618yes—
636C2×C4#388116yes—
637C2×C4#39018yes—
638C2×C4#39218yes—
639C2×C4#39318yes—
640C2×C4#394116yes—
641C2×C4#396116yes—
642C2×C4#39718yes—
643C2×C4#39918yes—
644C22—216no—
645C22—28no—
646C22—44no—
647C22—44no—
648C22—48no—
649C22—416no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—416no—
656C22—48no—
657C22—48no—
658C22—48no—
659C22C2×#38011yescharacteristic
660C22#228012yes—
661C22C2×#38211yescharacteristic
662C22#228312yes—
663C22#228511yescharacteristic
664C22C2×#38512yes—
665C22C2×#38611yescharacteristic
666C22C2×#38812yes—
667C22C2×#39011yescharacteristic
668C22#231112yes—
669C22#231412yes—
670C22#231712yes—
671C22#235214yes—
672C22#235214yes—
673C22#235214yes—
674C22#235414yes—
675C22C2×#39211yescharacteristic
676C22C2×#39311yescharacteristic
677C22C2×#39412yes—
678C22C2×#39612yes—
679C22C2×#39711yescharacteristic
680C22C2×#39911yescharacteristic
681C22#240714yes—
682C22#241414yes—
683C22#241514yes—
684C22#242014yes—
685C22#242114yes—
686C22#242614yes—
687C22#242714yes—
688C22#243112yes—
689C22#243212yes—
690C22#243512yes—
691C22#243612yes—
692C22#243914yes—
693C22#244114yes—
694C22#244214yes—
695C22#244214yes—
696C22#248514yes—
697C22#248914yes—
698C22#249014yes—
699C22#249414yes—
700C22#249514yes—
701C22#249912yes—
702C22#250012yes—
703C22#250312yes—
704C22#250412yes—
705C22#253514yes—
706C22#253614yes—
707C22#254014yes—
708C22#256214yes—
709C22#261911yescharacteristic
710C22#262012yes—
711C22#262111yescharacteristic
712C22#262812yes—
713C22#262912yes—
714C22#285012yes—
715C22#285012yes—
716C22#285112yes—
717C22#285112yes—
718C4—44no—
719C4—48no—
720C4—48no—
721C4—88no—
722C4—88no—
723C4#1439116yes—
724C2—48no—
725C2C2×#143911yescharacteristic
726C2#1387112yes—
727C2#1387514yes—
728C2#1387912yes—
729C2#1388112yes—
730C2#1390512yes—
731C2#1434011yescharacteristic
732C2#1468211yescharacteristic
733C2#1521512yes—
734C2#1521612yes—
735C2#1521912yes—
736C2#1522014yes—
737C2#1522312yes—
738C2#1523312yes—
739C2#1523612yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC24C23×C4
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#382C2semidirect
4. C2×C4⋊#95C2semidirect
5. C2×C4⋊#95C2semidirect
6. #7556C2semidirect
22. C2×C4×C22⋊C4C4semidirect
232. C2×C42C22⋊C4semidirect
521. C23#382semidirect

Maximal subgroups

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

subgroupindex
C22×#3822
C22×#3802
C22×C4×C22⋊C42
C2×C4⋊#952
#75562
#75552