Finite Group Databasebeta

← #4457505

Subgroups — #4457505

Order 512.

6333 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
1C22×#534C211yesmaximal, characteristic, complemented
2C2×#2410C212yesmaximal, complemented
3C2×#2622C211yesmaximal, characteristic, complemented
4#16657C214yesmaximal, complemented
5#16654C214yesmaximal, complemented
6#16255C214yesmaximal, complemented
7#16249C214yesmaximal, complemented
8#14757C214yesmaximal, complemented
9#14747C214yesmaximal, complemented
10#13867C212yesmaximal, complemented
11C2×#1440C211yesmaximal, characteristic, complemented
12C24×D4C2212yes—
13C2×#534—216no—
14C2×#534—28no—
15C22×#123C2214yes—
16C22×#121C2214yes—
17C22×#121C2214yescomplemented
18C22×#120C2211yescharacteristic
19C22×#120C2214yes—
20C22×#120C2212yescomplemented
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23#2622—28no—
24#2410—216no—
25C2×#397C2214yescomplemented
26C2×#394C2214yes—
27C2×#394C2214yescomplemented
28C2×#394C2214yescomplemented
29C2×#392C2214yes—
30C2×#392C2214yescomplemented
31C2×#386C2212yescomplemented
32C2×#386C2214yescomplemented
33C2×#386C2212yescomplemented
34C2×#386C2214yescomplemented
35C2×#386C2214yescomplemented
36C2×#385C2212yescomplemented
37C2×#385C2212yescomplemented
38C2×#379C2211yescharacteristic
39C2×#379C2212yescomplemented
40C2×#379C2214yescomplemented
41C22×#95C2214yescomplemented
42#1444C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2214yescomplemented
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yescomplemented
57#1438C2214yescomplemented
58#1438C2214yescomplemented
59#1438C2214yescomplemented
60C22⋊#95C2214yescomplemented
61C26C2311yescharacteristic
62C26C2312yes—
63C23×D4—28no—
64C23×D4—28no—
65C23×D4—48no—
66C23×D4—48no—
67C23×D4D4116yescomplemented
68C23×D4C2311yescharacteristic
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2314yes—
73C23×D4C2314yes—
74C24×C4C2314yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77#534—2128no—
78C2×#123—216no—
79C2×#123—216no—
80C2×#123—216no—
81C2×#121—216no—
82C2×#121—216no—
83C2×#121—216no—
84C2×#121—216no—
85C2×#121—48no—
86C2×#121—48no—
87C2×#120—216no—
88C2×#120—216no—
89C2×#120—24no—
90C2×#120—28no—
91C2×#120—28no—
92C2×#120—216no—
93C2×#120—48no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—48no—
100C22×C22⋊C4D418yescomplemented
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120#397—216no—
121#394—216no—
122#394—216no—
123#394—48no—
124#392—216no—
125#392—48no—
126#386—216no—
127#386—28no—
128#386—28no—
129#386—48no—
130#386—48no—
131#385—28no—
132#385—28no—
133#379—24no—
134#379—28no—
135#379—48no—
136C2×#95—48no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25—28no—
150C25—24no—
151C25—24no—
152C25—44no—
153C25—44no—
154C25—44no—
155C25—44no—
156C25—44no—
157C25—44no—
158C25—44no—
159C25—44no—
160C25C2×D418yes—
161C25C2×D414yes—
162C25C2×D418yes—
163C25C2411yescharacteristic
164C25C2414yes—
165C25C2412yes—
166C25C2411yescharacteristic
167C22×D4—264no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—464no—
203C22×D4—464no—
204C22×D4—432no—
205C22×D4—432no—
206C22×D4C2×D4116yes—
207C22×D4C2×D4116yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D418yes—
224C23×C4C2×D418yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232#123—2128no—
233#121—2128no—
234#121—464no—
235#120—2128no—
236#120—432no—
237#120—416no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C22⋊C4—232no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—432no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×D4116yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—232no—
323C24—216no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—216no—
339C24—21no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—48no—
345C24—416no—
346C24—48no—
347C24—416no—
348C24—416no—
349C24—48no—
350C24—48no—
351C24—416no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—88no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—84no—
360C24—84no—
361C24—88no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24—88no—
366C24—88no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24—88no—
371C24—88no—
372C24#120116yes—
373C24C22×D418yes—
374C24C22×D414yes—
375C24C22×D414yes—
376C24C22×D414yes—
377C24C22×D414yes—
378C24C22×D412yes—
379C24H(5,𝔽2)18yes—
380C24C2511yescharacteristic
381C2×D4—232no—
382C2×D4—232no—
383C2×D4—232no—
384C2×D4—232no—
385C2×D4—232no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—4256no—
390C2×D4—416no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—416no—
404C2×D4—48no—
405C2×D4—88no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—88no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—88no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—216no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4C22×D418yes—
476C22×C4C22×D418yes—
477C22×C4H(5,𝔽2)18yes—
478C4⋊C4—232no—
479C4⋊C4—416no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—232no—
483C22⋊C4—232no—
484C22⋊C4—232no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—232no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—432no—
529C23—432no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—41no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—832no—
578C23—832no—
579C23—832no—
580C23—816no—
581C23—816no—
582C23—832no—
583C23—832no—
584C23—832no—
585C23—832no—
586C23—832no—
587C23—832no—
588C23C2×#12018yes—
589C23C2×#12014yes—
590C23#534116yes—
591C23#53518yes—
592C23C23×D412yes—
593C23C23×D411yescharacteristic
594C23C2×H(5,𝔽2)14yes—
595C23C2×H(5,𝔽2)14yes—
596C23#58514yes—
597D4—464no—
598D4—464no—
599D4—832no—
600D4—832no—
601D4—816no—
602D4—88no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—432no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
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—832no—
649C22—28no—
650C22—216no—
651C22—216no—
652C22—216no—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—48no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22—48no—
661C22—48no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—48no—
669C22—832no—
670C22—864no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—82no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—81no—
684C22#261318yes—
685C22C22×#12011yescharacteristic
686C22C2×#53414yes—
687C22C2×#53512yes—
688C22#348412yes—
689C22#348512yes—
690C22#349614yes—
691C22#349614yes—
692C22#353414yes—
693C22#353714yes—
694C22#355814yes—
695C22#356014yes—
696C4—48no—
697C4—48no—
698C4—48no—
699C4—82no—
700C4—84no—
701C4—84no—
702C4—81no—
703C2—28no—
704C2—48no—
705C2—88no—
706C2—161no—
707C2C2×#261311yescharacteristic
708C2#3534712yes—
709C2#3673914yes—
710C2#4048614yes—
711C2#4051014yes—
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×#534C2semidirect
2. C2×#2410C2semidirect
3. C2×#2622C2semidirect
4. #16657C2semidirect
5. #16654C2semidirect
6. #16255C2semidirect
7. #16249C2semidirect
8. #14757C2semidirect
9. #14747C2semidirect
10. #13867C2semidirect
11. C2×#1440C2semidirect
17. C22×#121C22semidirect
20. C22×#120C22semidirect
25. C2×#397C22semidirect
27. C2×#394C22semidirect
28. C2×#394C22semidirect
30. C2×#392C22semidirect
31. C2×#386C22semidirect
32. C2×#386C22semidirect
33. C2×#386C22semidirect
34. C2×#386C22semidirect
35. C2×#386C22semidirect
36. C2×#385C22semidirect
37. C2×#385C22semidirect
39. C2×#379C22semidirect
40. C2×#379C22semidirect
41. C22×#95C22semidirect
42. #1444C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
56. #1438C22semidirect
57. #1438C22semidirect
58. #1438C22semidirect
59. #1438C22semidirect
60. C22⋊#95C22semidirect
67. C23×D4D4semidirect
100. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5342
C2×#24102
C2×#26222
#166572
#166542
#162552
#162492
#147572
#147472
#138672
C2×#14402