Finite Group Databasebeta

← #556912

Subgroups — #556912

Order 512.

3648 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#120C211yesmaximal, characteristic
2#14673C212yesmaximal, complemented
3#14657C212yesmaximal
4#14351C212yesmaximal, complemented
5#14345C211yesmaximal, characteristic, complemented
6#14344C211yesmaximal, characteristic, complemented
7#14269C214yesmaximal, complemented
8#14267C214yesmaximal, complemented
9#14145C212yesmaximal, complemented
10#14143C212yesmaximal, complemented
11#13892C212yesmaximal, complemented
12#13890C212yesmaximal, complemented
13#13888C212yesmaximal, complemented
14#13886C212yesmaximal, complemented
15C4×#379C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C4×#120C4116yescomplemented
21C2×#388C2214yescomplemented
22C2×#387C2214yescomplemented
23C2×#386C2211yescharacteristic
24C2×#386C2212yes—
25C2×#385C2211yescharacteristic
26C2×#385C2212yes—
27C2×#383C2211yescharacteristic, complemented
28C2×#382C2211yescharacteristic, complemented
29C2×#381C2211yescharacteristic, complemented
30C2×#380C2211yescharacteristic, complemented
31C2×#379C2214yes—
32C2×#379C2211yescharacteristic
33C2×C4×C22⋊C4C2212yes—
34C2×C4×C22⋊C4C2214yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2214yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39#1445C2212yescomplemented
40#1443C2214yescomplemented
41#1442C2214yescomplemented
42#1441C2212yescomplemented
43#1440C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1440C2212yescomplemented
47#1439C2212yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2212yescomplemented
55#1438C2212yes—
56#1438C2212yes—
57#1438C2214yes—
58#1438C2214yes—
59#1437C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63C4⋊#95C2212yescomplemented
64#1434C2212yescomplemented
65#1434C2212yescomplemented
66#1434C2212yescomplemented
67#1434C2212yescomplemented
68#1434C2214yescomplemented
69#1434C2214yescomplemented
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yescomplemented
76C26C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C24×C4—48no—
80C24×C4—42no—
81C24×C4—42no—
82C24×C4C2×C418yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×#120—28no—
89C2×#120C2×C418yes—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4—48no—
93C2×C4×D4C2×C418yes—
94C2×C4×D4C2×C4116yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
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⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121#388—48no—
122#387—48no—
123#386C2×C418yes—
124#386C2×C4116yes—
125#385C2×C418yes—
126#385C2×C4116yes—
127#383—42no—
128#382—42no—
129#381—42no—
130#380—42no—
131#379—216no—
132#379C2×C418yes—
133C4×C22⋊C4—44no—
134C4×C22⋊C4—44no—
135C4×C22⋊C4—48no—
136C4×C22⋊C4—48no—
137C4×C22⋊C4C2×C4116yes—
138C4×C22⋊C4C2×C418yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C25—41no—
165C25—44no—
166C25—41no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25C22×C414yes—
171C25#4118yes—
172C25C2411yescharacteristic
173C25C2412yes—
174C25C2411yescharacteristic
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—44no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4C22×C414yes—
184C22×D4C22×C418yes—
185C23×C4—28no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—432no—
195C23×C4—416no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—432no—
205C23×C4—432no—
206C23×C4C22×C418yes—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C22×C414yes—
210C23×C4C22×C418yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223#120—232no—
224C4×D4—416no—
225C4×D4—432no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C414yes—
277C2×C22⋊C4C22×C414yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C414yes—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C22×C414yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C22×C414yes—
292C2×C42C22×C418yes—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95C22×C418yes—
316#95C22×C418yes—
317#95C22×C418yes—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—22no—
322C24—21no—
323C24—24no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—44no—
341C24—48no—
342C24—48no—
343C24—44no—
344C24—48no—
345C24—416no—
346C24—416no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—416no—
352C24—44no—
353C24—44no—
354C24—416no—
355C24#117116yes—
356C24C23×C412yes—
357C24C23×C414yes—
358C24C23×C412yes—
359C24C2×#4118yes—
360C24C2×#4114yes—
361C24C2511yescharacteristic
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—44no—
368C2×D4—48no—
369C2×D4—416no—
370C2×D4—48no—
371C2×D4—416no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
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—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—4128no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4C23×C412yes—
450C22×C4C23×C414yes—
451C22×C4C23×C412yes—
452C22×C4C23×C414yes—
453C22×C4C23×C412yes—
454C22×C4C23×C412yes—
455C22×C4C23×C414yes—
456C22×C4C23×C412yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)18yes—
460C4⋊C4—216no—
461C4⋊C4—44no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C22⋊C4—216no—
466C22⋊C4—28no—
467C22⋊C4—216no—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C22⋊C4—84no—
482C42—28no—
483C42—48no—
484C42—88no—
485C42—88no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—416no—
512C23—432no—
513C23—416no—
514C23—416no—
515C23—432no—
516C23—464no—
517C23—432no—
518C23—432no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—41no—
545C23—44no—
546C23—41no—
547C23C2×#11718yes—
548C23C2×#11714yes—
549C23C24×C411yescharacteristic
550C23C22×#4112yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)14yes—
553C23C2×H(5,𝔽2)11yescharacteristic
554C23#58512yes—
555C23#58512yes—
556C23#58512yes—
557D4—416no—
558D4—432no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—432no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—432no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
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—44no—
626C2×C4—88no—
627C2×C4#51814yes—
628C2×C4#51818yes—
629C22—24no—
630C22—28no—
631C22—28no—
632C22—28no—
633C22—28no—
634C22—464no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—84no—
655C22—82no—
656C22—82no—
657C22C22×#11711yescharacteristic
658C22C2×#51811yescharacteristic
659C22C2×#51812yes—
660C22#342812yes—
661C22#342811yescharacteristic
662C22#345912yes—
663C22#345914yes—
664C22#345914yes—
665C22#345912yes—
666C22#353212yes—
667C22#353212yes—
668C22#353912yes—
669C22#353914yes—
670C22#354011yescharacteristic
671C22#354011yescharacteristic
672C22#354014yes—
673C4—24no—
674C4—28no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—84no—
682C4—84no—
683C4—84no—
684C4—84no—
685C2—44no—
686C2—48no—
687C2—82no—
688C2#3405711yescharacteristic
689C2#3405712yes—
690C2#3435611yescharacteristic
691C2#3436111yescharacteristic
692C2#3439112yes—
693C2#3752714yes—
694C2#3752712yes—
695C2#3752812yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14673C2semidirect
4. #14351C2semidirect
5. #14345C2semidirect
6. #14344C2semidirect
7. #14269C2semidirect
8. #14267C2semidirect
9. #14145C2semidirect
10. #14143C2semidirect
11. #13892C2semidirect
12. #13890C2semidirect
13. #13888C2semidirect
14. #13886C2semidirect
15. C4×#379C2semidirect
20. C4×#120C4semidirect
21. C2×#388C22semidirect
22. C2×#387C22semidirect
27. C2×#383C22semidirect
28. C2×#382C22semidirect
29. C2×#381C22semidirect
30. C2×#380C22semidirect
39. #1445C22semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
46. #1440C22semidirect
54. #1439C22semidirect
59. #1437C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. C4⋊#95C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
69. #1434C22semidirect
75. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1202
#146732
#146572
#143512
#143452
#143442
#142692
#142672
#141452
#141432
#138922
#138902
#138882
#138862
C4×#3792