Finite Group Databasebeta

← #4122655

Subgroups — #4122655

Order 512.

5410 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C214yesmaximal, complemented
2#17137C211yesmaximal, characteristic, complemented
3#15952C212yesmaximal, complemented
4#15947C211yesmaximal, characteristic, complemented
5#15506C212yesmaximal, complemented
6#15468C214yesmaximal, complemented
7#15369C214yesmaximal, complemented
8#15257C212yesmaximal, complemented
9#14175C212yesmaximal, complemented
10#14152C211yesmaximal, characteristic, complemented
11#14134C212yesmaximal, complemented
12#13967C214yesmaximal, complemented
13#13956C212yesmaximal, complemented
14C23×#41C2212yes—
15C22×#121C2211yescharacteristic
16C22×#120C2214yes—
17C22×#120C2212yes—
18C22×#120C2212yescomplemented
19C22×C4×D4C2212yes—
20C23×C22⋊C4C2214yescomplemented
21#2404—232no—
22C2×#393C2214yescomplemented
23C2×#393C2214yescomplemented
24C2×#392C2214yes—
25C2×#392C2212yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yescomplemented
31C2×#386C2214yescomplemented
32C2×#386C2214yescomplemented
33C2×#386C2214yescomplemented
34C2×#381C2214yescomplemented
35C2×#381C2212yescomplemented
36C2×#379C2212yescomplemented
37C2×C4×C22⋊C4C2211yescharacteristic, complemented
38C22×#95C2211yescharacteristic, complemented
39#1446C2212yescomplemented
40#1446C2212yescomplemented
41#1445C2211yescharacteristic, complemented
42#1445C2211yescharacteristic, complemented
43#1443C2214yescomplemented
44#1443C2214yescomplemented
45#1442C2212yescomplemented
46#1440C2214yescomplemented
47#1440C2212yescomplemented
48#1439C2212yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1438C2214yes—
53#1438C2211yescharacteristic
54#1438C2212yes—
55#1438C2211yescharacteristic
56#1438C2212yes—
57#1438C2212yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2212yescomplemented
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1434C2212yescomplemented
64C22⋊#95C2214yescomplemented
65C22⋊#95C2214yescomplemented
66C22⋊#95C2214yescomplemented
67C22⋊#95C2214yescomplemented
68C26C2312yes—
69C22×#41—216no—
70C22×#41—28no—
71C22×#41—44no—
72C22×#41—48no—
73C22×#41—44no—
74C23×Q8C2312yes—
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2314yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C2×#121—24no—
86C2×#121—42no—
87C2×#121—42no—
88C2×#120—216no—
89C2×#120—216no—
90C2×#120—216no—
91C2×#120—28no—
92C2×#120—48no—
93C2×#120—48no—
94C2×#120—44no—
95C2×C4×D4—44no—
96C2×C4×D4—48no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—216no—
100C22×C22⋊C4—48no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2312yes—
120#393—216no—
121#393—216no—
122#392—216no—
123#392D4116yescomplemented
124#386—216no—
125#386—216no—
126#386—216no—
127#386—216no—
128#386—216no—
129#386—216no—
130#386—48no—
131#386—48no—
132#381—216no—
133#381—44no—
134#379—44no—
135C4×C22⋊C4—42no—
136C2×#95—42no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C25—24no—
159C25—28no—
160C25—28no—
161C25—28no—
162C25—44no—
163C25—44no—
164C25—44no—
165C25—42no—
166C25—42no—
167C25C2412yes—
168C25C2411yescharacteristic
169C25C2414yes—
170C2×#41—464no—
171C2×#41—432no—
172C2×#41—432no—
173C2×#41—464no—
174C2×#41—432no—
175C22×Q8—28no—
176C22×Q8—24no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—44no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—28no—
194C22×D4—24no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4C2×D418yes—
210C22×D4C2×D4116yes—
211C22×D4#4118yescomplemented
212C22×D4#4114yescomplemented
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—28no—
226C23×C4—28no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246#121—416no—
247#120—432no—
248#120—464no—
249#120—432no—
250C4×D4—432no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
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—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
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—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—464no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×D4116yes—
304C2×C22⋊C4C2×D418yes—
305C2×C42—24no—
306C2×C42—44no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—42no—
316#95—41no—
317#95—41no—
318#95—42no—
319#95—44no—
320#95—41no—
321#95—41no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95C2×D418yes—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—21no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—21no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—216no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—416no—
356C24—416no—
357C24—416no—
358C24—416no—
359C24—48no—
360C24—416no—
361C24—48no—
362C24—416no—
363C24—416no—
364C24—88no—
365C24—88no—
366C24—88no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24—84no—
371C24—84no—
372C24—88no—
373C24—84no—
374C24—84no—
375C24C22×D414yes—
376C24C22×D418yes—
377C24C2×#4114yes—
378C24C2×#4118yes—
379C24H(5,𝔽2)12yes—
380C24H(5,𝔽2)18yes—
381C24C2511yescharacteristic
382#41—4256no—
383C2×Q8—48no—
384C2×Q8—416no—
385C2×Q8—416no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×D4—232no—
389C2×D4—232no—
390C2×D4—232no—
391C2×D4—232no—
392C2×D4—48no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—44no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—48no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—416no—
404C2×D4—416no—
405C2×D4—416no—
406C2×D4—48no—
407C2×D4—416no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—48no—
413C2×D4—416no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—84no—
421C22×C4—232no—
422C22×C4—216no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—48no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—88no—
484C22×C4—84no—
485C22×C4—88no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—88no—
489C22×C4—88no—
490C22×C4—88no—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D418yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4114yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)12yes—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—84no—
517C42—48no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—24no—
532C23—22no—
533C23—28no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—24no—
541C23—28no—
542C23—28no—
543C23—22no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—464no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—41no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—41no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—42no—
588C23—42no—
589C23—832no—
590C23—832no—
591C23—816no—
592C23—816no—
593C23—832no—
594C23—816no—
595C23—832no—
596C23—832no—
597C23D4214yes—
598C23#546116yes—
599C23C23×D412yes—
600C23C22×#4112yes—
601C23C22×#4111yescharacteristic
602C23C2×H(5,𝔽2)11yescharacteristic
603C23C2×H(5,𝔽2)14yes—
604C23C2×H(5,𝔽2)12yes—
605C23C2×H(5,𝔽2)11yescharacteristic
606C23#58512yes—
607Q8—432no—
608D4—432no—
609D4—432no—
610D4—464no—
611D4—88no—
612D4—832no—
613D4—816no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—464no—
630C2×C4—432no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—88no—
671C2×C4—832no—
672C22—216no—
673C22—28no—
674C22—216no—
675C22—216no—
676C22—24no—
677C22—44no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—42no—
682C22—48no—
683C22—42no—
684C22—48no—
685C22—48no—
686C22—48no—
687C22—48no—
688C22—48no—
689C22—44no—
690C22—48no—
691C22—48no—
692C22—48no—
693C22—44no—
694C22—48no—
695C22—44no—
696C22—44no—
697C22—48no—
698C22—48no—
699C22—44no—
700C22—864no—
701C22—84no—
702C22—84no—
703C22—84no—
704C22—84no—
705C22C2×D4211yescharacteristic
706C22C2×#54614yes—
707C22D4×#4112yes—
708C22#348112yes—
709C22#348111yescharacteristic
710C22#348414yes—
711C22#349012yes—
712C22#349011yescharacteristic
713C22#349012yes—
714C22#349612yes—
715C22#351314yes—
716C22#353212yes—
717C22#353212yes—
718C22#354214yes—
719C22#354511yescharacteristic
720C22#354911yescharacteristic
721C4—44no—
722C4—48no—
723C4—44no—
724C4—42no—
725C4—42no—
726C4—44no—
727C4—44no—
728C4—84no—
729C4—82no—
730C4—84no—
731C2—44no—
732C2—416no—
733C2—84no—
734C2#4699611yescharacteristic
735C2#4717012yes—
736C2#4779412yes—
737C2#5899112yes—
738C2#5902914yes—
739C2#5980312yes—
740C2#5981411yescharacteristic
741C2#6004511yescharacteristic
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. C2×#2404C2semidirect
2. #17137C2semidirect
3. #15952C2semidirect
4. #15947C2semidirect
5. #15506C2semidirect
6. #15468C2semidirect
7. #15369C2semidirect
8. #15257C2semidirect
9. #14175C2semidirect
10. #14152C2semidirect
11. #14134C2semidirect
12. #13967C2semidirect
13. #13956C2semidirect
18. C22×#120C22semidirect
20. C23×C22⋊C4C22semidirect
22. C2×#393C22semidirect
23. C2×#393C22semidirect
30. C2×#386C22semidirect
31. C2×#386C22semidirect
32. C2×#386C22semidirect
33. C2×#386C22semidirect
34. C2×#381C22semidirect
35. C2×#381C22semidirect
36. C2×#379C22semidirect
37. C2×C4×C22⋊C4C22semidirect
38. C22×#95C22semidirect
39. #1446C22semidirect
40. #1446C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
60. #1438C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1434C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
123. #392D4semidirect
211. C22×D4#41semidirect
212. C22×D4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
#171372
#159522
#159472
#155062
#154682
#153692
#152572
#141752
#141522
#141342
#139672
#139562