Finite Group Databasebeta

← #674141

Subgroups — #674141

Order 512.

8538 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D42C211yesmaximal, characteristic, complemented
2C2×#2856C211yesmaximal, characteristic, complemented
3C2×#2408C214yesmaximal, complemented
4C2×C4×#121C212yesmaximal, complemented
5#15281C214yesmaximal, complemented
6#15257C214yesmaximal, complemented
7#15126C214yesmaximal, complemented
8#15123C214yesmaximal, complemented
9#14154C211yesmaximal, characteristic, complemented
10#13906C212yesmaximal, complemented
11#13902C214yesmaximal, complemented
12C24×D4C2214yes—
13C2×D42—28no—
14C2×D42—216no—
15C22×#127C2211yescharacteristic
16C22×#121C2214yes—
17C22×#121C2212yes—
18C22×#121C2214yescomplemented
19C22×#121C2212yescomplemented
20C22×#120C2214yes—
21C22×#120C2214yescomplemented
22C22×C4×D4C2212yes—
23C22×C4×D4C2214yescomplemented
24C23×C42C2211yescharacteristic, complemented
25#2856—28no—
26#2408—232no—
27C2×#394C2218yescomplemented
28C2×#392C2218yescomplemented
29C4×#121—216no—
30C2×#390C2212yes—
31C2×#390C2214yes—
32C2×#390C2214yescomplemented
33C2×#390C2212yescomplemented
34C2×#390C2214yescomplemented
35C2×#390C2214yescomplemented
36C2×#389C2212yescomplemented
37C2×#388C2214yescomplemented
38C2×#386C2214yescomplemented
39C2×#384C2211yescharacteristic, complemented
40C2×#380C2212yescomplemented
41C2×#380C2214yescomplemented
42C2×#380C2214yescomplemented
43C2×#379C2218yescomplemented
44C2×#379C2214yescomplemented
45C2×C4×C4⋊C4C2212yescomplemented
46C2×C4×C22⋊C4C2214yescomplemented
47#1440C2214yescomplemented
48#1439C2218yescomplemented
49#1439C2218yescomplemented
50#1438C2214yescomplemented
51#1438C2218yescomplemented
52#1438C2218yescomplemented
53#1438C2214yescomplemented
54C4⋊#95C2214yescomplemented
55C26C2314yes—
56C23×D4—216no—
57C23×D4—216no—
58C23×D4—216no—
59C23×D4—216no—
60C23×D4—216no—
61C23×D4—48no—
62C23×D4—48no—
63C23×D4C2314yes—
64C23×D4C2318yes—
65C23×D4C2314yes—
66C23×D4C2314yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C24×C4C2314yes—
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72D42—2128no—
73C2×#127—28no—
74C2×#127D418yescomplemented
75C2×#121—216no—
76C2×#121—28no—
77C2×#121—28no—
78C2×#121—216no—
79C2×#121—216no—
80C2×#121—28no—
81C2×#121—216no—
82C2×#121—216no—
83C2×#121—44no—
84C2×#121—48no—
85C2×#121—44no—
86C2×#121D4116yescomplemented
87C2×#120—216no—
88C2×#120—232no—
89C2×#120—232no—
90C2×#120—48no—
91C2×C4×D4—216no—
92C2×C4×D4—216no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—48no—
96C2×C4×D4D4116yescomplemented
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2318yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2314yes—
116C22×C42—24no—
117C22×C42—28no—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122#394—232no—
123#392—232no—
124#390—216no—
125#390—28no—
126#390—216no—
127#390—28no—
128#390—216no—
129#390—48no—
130#389—28no—
131#388—216no—
132#386—216no—
133#384—24no—
134#380—28no—
135#380—216no—
136#380—48no—
137#379—232no—
138#379—48no—
139C4×C4⋊C4—28no—
140C4×C22⋊C4—216no—
141C2×#95C2318yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C25—28no—
147C25—216no—
148C25—216no—
149C25—216no—
150C25—48no—
151C25—48no—
152C25—44no—
153C25—48no—
154C25—44no—
155C25C2414yes—
156C25C2414yes—
157C25C2411yescharacteristic
158C22×D4—2128no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—216no—
162C22×D4—216no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—216no—
167C22×D4—216no—
168C22×D4—216no—
169C22×D4—28no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—216no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—464no—
183C22×D4—464no—
184C22×D4—44no—
185C22×D4—48no—
186C22×D4—48no—
187C22×D4—44no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—48no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—464no—
194C22×D4—464no—
195C22×D4—464no—
196C22×D4—464no—
197C22×D4C2×D4116yes—
198C22×D4C2×D418yes—
199C22×D4C2×D418yes—
200C22×D4C2×D418yes—
201C22×D4C2×D4116yescomplemented
202C22×D4#4118yes—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—42no—
220C23×C4C2×D4116yes—
221C23×C4C2×D418yescomplemented
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230#127—232no—
231#121—2128no—
232#121—264no—
233#121—464no—
234#121—432no—
235#120—2128no—
236#120—464no—
237C4×D4—264no—
238C4×D4—464no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×D418yescomplemented
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4C2×D4116yes—
292C2×C22⋊C4C2×D4116yescomplemented
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—416no—
302C2×C42C2×D414yes—
303C2×C42C2×D418yes—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—216no—
308#95—48no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—21no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—22no—
322C24—21no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—232no—
329C24—28no—
330C24—28no—
331C24—28no—
332C24—24no—
333C24—28no—
334C24—24no—
335C24—28no—
336C24—28no—
337C24—432no—
338C24—432no—
339C24—432no—
340C24—432no—
341C24—416no—
342C24—432no—
343C24—416no—
344C24—432no—
345C24—432no—
346C24—816no—
347C24—816no—
348C24—816no—
349C24—816no—
350C24—816no—
351C24—816no—
352C24—88no—
353C24—88no—
354C24—88no—
355C24—816no—
356C24—88no—
357C24C22×D418yes—
358C24C22×D418yes—
359C24C22×D418yes—
360C24C22×D414yes—
361C24C2×#4118yes—
362C24H(5,𝔽2)12yes—
363C24C2511yescharacteristic
364C2×D4—264no—
365C2×D4—232no—
366C2×D4—232no—
367C2×D4—216no—
368C2×D4—232no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—232no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—232no—
377C2×D4—416no—
378C2×D4—432no—
379C2×D4—432no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—4512no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—432no—
388C2×D4—432no—
389C2×D4—432no—
390C2×D4—432no—
391C2×D4—432no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—816no—
401C2×D4—88no—
402C2×D4#121132yescomplemented
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—232no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—232no—
433C22×C4—216no—
434C22×C4—232no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—416no—
459C22×C4C22×D418yes—
460C22×C4C22×D418yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C2×#4114yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C4H(5,𝔽2)14yes—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C22⋊C4—232no—
479C22⋊C4—232no—
480C22⋊C4—232no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—432no—
486C22⋊C4—432no—
487C22⋊C4—416no—
488C22⋊C4—816no—
489C22⋊C4—816no—
490C22⋊C4—88no—
491C42—44no—
492C42—48no—
493C42—416no—
494C42#120116yescomplemented
495C23—28no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—22no—
512C23—28no—
513C23—24no—
514C23—28no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—22no—
523C23—28no—
524C23—216no—
525C23—216no—
526C23—28no—
527C23—28no—
528C23—4128no—
529C23—41no—
530C23—44no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—41no—
538C23—44no—
539C23—41no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—41no—
554C23—44no—
555C23—48no—
556C23—48no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—44no—
562C23—48no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—44no—
569C23—48no—
570C23—44no—
571C23—864no—
572C23—864no—
573C23—832no—
574C23—864no—
575C23—832no—
576C23—832no—
577C23—864no—
578C23—864no—
579C23C2×#121116yes—
580C23D4214yescomplemented
581C23C23×D412yes—
582C23C23×D411yescharacteristic
583C23C23×D412yes—
584C23C22×#4112yes—
585C23C2×H(5,𝔽2)12yes—
586C23C2×H(5,𝔽2)12yes—
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)11yescharacteristic
589C23#58512yes—
590D4—264no—
591D4—464no—
592D4—4128no—
593D4—464no—
594D4—816no—
595D4—832no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—432no—
628C2×C4—464no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4C2×#12014yes—
642C2×C4C2×#12018yes—
643C2×C4C2×#12118yes—
644C2×C4#53518yes—
645C2×C4D4218yes—
646C2×C4D4218yes—
647C2×C4#54618yes—
648C2×C4#54618yes—
649C22—216no—
650C22—216no—
651C22—24no—
652C22—28no—
653C22—216no—
654C22—44no—
655C22—48no—
656C22—44no—
657C22—48no—
658C22—48no—
659C22—42no—
660C22—48no—
661C22—44no—
662C22—48no—
663C22—44no—
664C22—48no—
665C22—48no—
666C22—42no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—44no—
671C22—48no—
672C22—48no—
673C22—416no—
674C22—416no—
675C22—48no—
676C22—48no—
677C22—48no—
678C22—8128no—
679C22—88no—
680C22—88no—
681C22—88no—
682C22—88no—
683C22C22×#12011yescharacteristic
684C22C22×#12112yes—
685C22C2×#53512yes—
686C22C2×D4212yes—
687C22C2×D4212yes—
688C22C2×D4211yescharacteristic
689C22C2×#54612yes—
690C22C2×#54612yes—
691C22D4×#4112yes—
692C22#348111yescharacteristic
693C22#348411yescharacteristic
694C22#348511yescharacteristic
695C22#348812yes—
696C22#348812yes—
697C22#349612yes—
698C22#349812yes—
699C22#351412yes—
700C22#351912yes—
701C22#354311yescharacteristic
702C22#354612yes—
703C22#354911yescharacteristic
704C4—24no—
705C4—44no—
706C4—48no—
707C4—42no—
708C4—48no—
709C4—84no—
710C4—84no—
711C4#2408116yescomplemented
712C2—44no—
713C2—416no—
714C2—88no—
715C2C2×#240812yes—
716C2#3551211yescharacteristic
717C2D4×#12112yes—
718C2#3625812yes—
719C2#3891012yes—
720C2#3934212yes—
721C2#5903512yes—
722C2#5904111yescharacteristic
723C2#5912611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×D42C2semidirect
2. C2×#2856C2semidirect
3. C2×#2408C2semidirect
4. C2×C4×#121C2semidirect
5. #15281C2semidirect
6. #15257C2semidirect
7. #15126C2semidirect
8. #15123C2semidirect
9. #14154C2semidirect
10. #13906C2semidirect
11. #13902C2semidirect
18. C22×#121C22semidirect
19. C22×#121C22semidirect
21. C22×#120C22semidirect
23. C22×C4×D4C22semidirect
24. C23×C42C22semidirect
27. C2×#394C22semidirect
28. C2×#392C22semidirect
32. C2×#390C22semidirect
33. C2×#390C22semidirect
34. C2×#390C22semidirect
35. C2×#390C22semidirect
36. C2×#389C22semidirect
37. C2×#388C22semidirect
38. C2×#386C22semidirect
39. C2×#384C22semidirect
40. C2×#380C22semidirect
41. C2×#380C22semidirect
42. C2×#380C22semidirect
43. C2×#379C22semidirect
44. C2×#379C22semidirect
45. C2×C4×C4⋊C4C22semidirect
46. C2×C4×C22⋊C4C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1438C22semidirect
51. #1438C22semidirect
52. #1438C22semidirect
53. #1438C22semidirect
54. C4⋊#95C22semidirect
74. C2×#127D4semidirect
86. C2×#121D4semidirect
96. C2×C4×D4D4semidirect
201. C22×D4C2×D4semidirect
221. C23×C4C2×D4semidirect
256. C2×C4⋊C4C2×D4semidirect
292. C2×C22⋊C4C2×D4semidirect
402. C2×D4#121semidirect
494. C42#120semidirect
580. C23D42semidirect
711. C4#2408semidirect

Maximal subgroups

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

subgroupindex
C22×D422
C2×#28562
C2×#24082
C2×C4×#1212
#152812
#152572
#151262
#151232
#141542
#139062
#139022