Finite Group Databasebeta

← #4452805

Subgroups — #4452805

Order 512.

4021 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C211yesmaximal, characteristic, complemented
2C22×#394C211yesmaximal, characteristic, complemented
3#16846C214yesmaximal, complemented
4#16541C214yesmaximal, complemented
5#16530C214yesmaximal, complemented
6#16276C214yesmaximal, complemented
7#14955C211yesmaximal, characteristic, complemented
8#14945C211yesmaximal, characteristic, complemented
9#14918C212yesmaximal, complemented
10#14911C212yesmaximal, complemented
11#14724C211yesmaximal, characteristic, complemented
12#14715C211yesmaximal, characteristic
13#13995C212yesmaximal, complemented
14#13971C212yesmaximal, complemented
15#13776C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#120C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22#2623—28no—
23C2×#400C2212yescomplemented
24C2×#399C2212yescomplemented
25C2×#398C2212yescomplemented
26C2×#397C2212yescomplemented
27C2×#394—28no—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#392C2214yes—
32C2×#390C2214yescomplemented
33C2×#388C2214yescomplemented
34C2×#386C2211yescharacteristic
35C2×#386C2211yescharacteristic
36C2×#386C2212yes—
37C2×#385C2211yescharacteristic
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2211yescharacteristic
47#1443C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1438C2214yes—
61#1438C2214yes—
62#1438C2214yescomplemented
63#1438C2214yescomplemented
64#1438C2214yescomplemented
65#1437C2214yescomplemented
66#1437C2214yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2211yescharacteristic
73C26C2312yes—
74C23×D4—24no—
75C23×D4—44no—
76C23×D4—48no—
77C23×D4C2311yescharacteristic
78C23×D4C2314yes—
79C23×D4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—24no—
85C2×#123—44no—
86C2×#120—28no—
87C2×#120—44no—
88C2×#120—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4D418yescomplemented
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—48no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2314yes—
125#400—28no—
126#399—28no—
127#398—28no—
128#397—28no—
129#394—216no—
130#394—216no—
131#394—48no—
132#392—48no—
133#390—48no—
134#388—48no—
135#386—24no—
136#386—24no—
137#386—44no—
138#385—24no—
139#385—24no—
140#385—44no—
141#381—24no—
142#381—28no—
143#379—28no—
144#379—44no—
145#379—44no—
146C4×C22⋊C4—44no—
147C2×#95—24no—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25—24no—
167C25—28no—
168C25—24no—
169C25—24no—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—24no—
174C25—44no—
175C25—44no—
176C25—44no—
177C25C2411yescharacteristic
178C25C2414yes—
179C25C2411yescharacteristic
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—22no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—416no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—432no—
201C22×D4—464no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230#123—416no—
231#120—432no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D418yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
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—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—432no—
290C2×C22⋊C4—432no—
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⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4#4118yes—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—44no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330C24—24no—
331C24—24no—
332C24—216no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—24no—
351C24—22no—
352C24—21no—
353C24—24no—
354C24—416no—
355C24—416no—
356C24—416no—
357C24—416no—
358C24—416no—
359C24—416no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—88no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24—88no—
376C24C22×D412yes—
377C24C2×#4114yes—
378C24H(5,𝔽2)18yes—
379C24H(5,𝔽2)14yes—
380C24H(5,𝔽2)12yes—
381C24H(5,𝔽2)12yes—
382C24C2511yescharacteristic
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—4128no—
392C2×D4—416no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C2×#4118yes—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C22⋊C4—216no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—88no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—432no—
526C23—432no—
527C23—432no—
528C23—44no—
529C23—44no—
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—41no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—832no—
570C23—832no—
571C23—832no—
572C23—832no—
573C23—816no—
574C23—816no—
575C23—816no—
576C23—816no—
577C23#53214yes—
578C23#53414yes—
579C23#53514yes—
580C23#53818yes—
581C23#54018yes—
582C23C23×D411yescharacteristic
583C23C22×#4112yes—
584C23C2×H(5,𝔽2)12yes—
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)14yes—
588C23#58514yes—
589D4—464no—
590D4—832no—
591D4—88no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
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—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
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—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—88no—
657C2×C4—816no—
658C22—28no—
659C22—24no—
660C22—28no—
661C22—24no—
662C22—216no—
663C22—216no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—48no—
671C22—864no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—81no—
681C22—82no—
682C22—81no—
683C22—82no—
684C22—81no—
685C22—84no—
686C22—84no—
687C22#266118yes—
688C22C2×#53211yescharacteristic
689C22C2×#53411yescharacteristic
690C22C2×#53511yescharacteristic
691C22C2×#53812yes—
692C22C2×#54012yes—
693C22#348412yes—
694C22#348712yes—
695C22#351314yes—
696C22#351314yes—
697C22#353214yes—
698C22#353414yes—
699C22#353414yes—
700C22#354514yes—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—44no—
705C4—48no—
706C4—81no—
707C4—82no—
708C4—84no—
709C4—81no—
710C4—82no—
711C4—81no—
712C4—84no—
713C2—28no—
714C2—88no—
715C2—161no—
716C2C2×#266111yescharacteristic
717C2#3539612yes—
718C2#3670211yescharacteristic
719C2#3718512yes—
720C2#3718711yescharacteristic
721C2#4080814yes—
722C2#4081014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14715C2
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×#2623C2semidirect
2. C22×#394C2semidirect
3. #16846C2semidirect
4. #16541C2semidirect
5. #16530C2semidirect
6. #16276C2semidirect
7. #14955C2semidirect
8. #14945C2semidirect
9. #14918C2semidirect
10. #14911C2semidirect
11. #14724C2semidirect
13. #13995C2semidirect
14. #13971C2semidirect
15. #13776C2semidirect
23. C2×#400C22semidirect
24. C2×#399C22semidirect
25. C2×#398C22semidirect
26. C2×#397C22semidirect
32. C2×#390C22semidirect
33. C2×#388C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
62. #1438C22semidirect
63. #1438C22semidirect
64. #1438C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
90. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C22×#3942
#168462
#165412
#165302
#162762
#149552
#149452
#149182
#149112
#147242
#147152
#139952
#139712
#137762