Finite Group Databasebeta

← #4933812

Subgroups — #4933812

Order 512.

3882 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2435C211yesmaximal, characteristic
3#16558C212yesmaximal, complemented
4#16367C214yesmaximal, complemented
5#15671C212yesmaximal, complemented
6#15662C212yesmaximal, complemented
7#15297C214yesmaximal, complemented
8#15275C214yesmaximal, complemented
9#15045C212yesmaximal, complemented
10#14717C214yesmaximal, complemented
11#14473C212yesmaximal, complemented
12#13867C212yesmaximal
13#13832C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2214yescomplemented
19C23×C22⋊C4C2212yes—
20#2683—28no—
21#2435—28no—
22C2×#397C2214yescomplemented
23C2×#396C2214yescomplemented
24C2×#395C2212yescomplemented
25C2×#394C2214yes—
26C2×#393C2212yes—
27C2×#393C2214yescomplemented
28C2×#391C2212yescomplemented
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2212yescomplemented
32C2×#388C2214yescomplemented
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#379C2214yes—
38C2×#379C2214yes—
39C2×#379C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1446C2214yescomplemented
42#1445C2212yescomplemented
43#1442C2212yescomplemented
44#1442C2214yescomplemented
45#1442C2212yescomplemented
46#1441C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1440C2214yescomplemented
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1438C2214yes—
59#1438C2214yes—
60#1437C2212yes—
61#1437C2214yescomplemented
62#1436C2212yescomplemented
63#1434C2212yescomplemented
64#1434C2214yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yescomplemented
67C22⋊#95C2214yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2312yes—
70C23×Q8C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4—24no—
73C23×D4—28no—
74C23×D4—42no—
75C23×D4—42no—
76C23×D4—44no—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#124—24no—
84C2×#124—28no—
85C2×#124—24no—
86C2×#124—44no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—42no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8—42no—
92C2×C22⋊Q8—48no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4—48no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#397—48no—
121#396—216no—
122#395—28no—
123#394—216no—
124#393—48no—
125#393—44no—
126#391—44no—
127#390—24no—
128#390D418yes—
129#389—28no—
130#388—48no—
131#386—28no—
132#386—28no—
133#386—44no—
134#386—44no—
135#379—216no—
136#379—48no—
137#379—44no—
138C4×C22⋊C4—24no—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—24no—
162C25—24no—
163C25—28no—
164C25—44no—
165C25—42no—
166C25—42no—
167C25—44no—
168C25—44no—
169C25C2412yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—28no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8—44no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—42no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—416no—
198C22×D4—432no—
199C22×D4—416no—
200C22×D4—432no—
201C22×D4—416no—
202C22×D4C2×D414yes—
203C22×D4#4118yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#124—416no—
230#124—416no—
231C22⋊Q8—464no—
232C22⋊Q8—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—432no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D4116yes—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—42no—
303C2×C42C2×D414yes—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323C24—21no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—216no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—48no—
352C24—416no—
353C24—416no—
354C24—416no—
355C24—416no—
356C24—48no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—88no—
361C24—88no—
362C24—88no—
363C24—84no—
364C24—88no—
365C24—84no—
366C24—88no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24C22×D414yes—
371C24C2×#4114yes—
372C24C2×#4114yes—
373C24H(5,𝔽2)14yes—
374C24H(5,𝔽2)12yes—
375C242−1+414yes—
376C24C2511yescharacteristic
377C2×Q8—416no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×Q8—88no—
386C2×Q8—84no—
387C2×D4—28no—
388C2×D4—216no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—4128no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4#124116yes—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—28no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—416no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—88no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4—84no—
470C22×C4—88no—
471C22×C4—84no—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)18yes—
478C4⋊C4—216no—
479C4⋊C4—44no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—84no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—44no—
501C42—44no—
502C42—84no—
503C23—22no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—28no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—432no—
529C23—432no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—44no—
565C23—816no—
566C23—832no—
567C23—816no—
568C23—816no—
569C23—832no—
570C23—832no—
571C23—816no—
572C23—816no—
573C23—832no—
574C23C2×#12418yes—
575C23#56314yes—
576C23C23×D411yescharacteristic
577C23C22×#4112yes—
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)12yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)14yes—
582C23C2×2−1+412yes—
583C23#58512yes—
584Q8—88no—
585Q8—816no—
586D4—416no—
587D4—416no—
588D4—432no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—416no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—816no—
648C2×C4—816no—
649C2×C4C2×#12414yes—
650C22—28no—
651C22—24no—
652C22—216no—
653C22—28no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—48no—
667C22—48no—
668C22—48no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—44no—
673C22—48no—
674C22—864no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—84no—
679C22—82no—
680C22—84no—
681C22C22×#12411yescharacteristic
682C22#348411yescharacteristic
683C22#348412yes—
684C22#348411yescharacteristic
685C22#348912yes—
686C22#349012yes—
687C22#349012yes—
688C22#351414yes—
689C22#352014yes—
690C22C2×#56311yescharacteristic
691C22#353212yes—
692C22#353614yes—
693C22#353712yes—
694C22#354214yes—
695C22#354511yescharacteristic
696C22#356112yes—
697C4—24no—
698C4—48no—
699C4—44no—
700C4—42no—
701C4—44no—
702C4—84no—
703C4—82no—
704C4—82no—
705C4—84no—
706C4—84no—
707C2—48no—
708C2—44no—
709C2—84no—
710C2#3536711yescharacteristic
711C2#3593912yes—
712C2#4572812yes—
713C2#4613611yescharacteristic
714C2#4833014yes—
715C2#4892212yes—
716C2#4954612yes—
717C2#5200411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2683C2semidirect
3. #16558C2semidirect
4. #16367C2semidirect
5. #15671C2semidirect
6. #15662C2semidirect
7. #15297C2semidirect
8. #15275C2semidirect
9. #15045C2semidirect
10. #14717C2semidirect
11. #14473C2semidirect
13. #13832C2semidirect
18. C22×C22⋊Q8C22semidirect
22. C2×#397C22semidirect
23. C2×#396C22semidirect
24. C2×#395C22semidirect
27. C2×#393C22semidirect
28. C2×#391C22semidirect
31. C2×#389C22semidirect
32. C2×#388C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
61. #1437C22semidirect
62. #1436C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#24352
#165582
#163672
#156712
#156622
#152972
#152752
#150452
#147172
#144732
#138672
#138322