Finite Group Databasebeta

← #725970

Subgroups — #725970

Order 512.

4118 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#394C211yesmaximal, characteristic
2C2×D4×C22⋊C4C211yesmaximal, characteristic
3#16573C214yesmaximal, complemented
4#15702C214yesmaximal, complemented
5#15502C214yesmaximal, complemented
6#14918C212yesmaximal, complemented
7#14823C214yesmaximal, complemented
8#14727C212yesmaximal, complemented
9#14339C211yesmaximal, characteristic, complemented
10#14149C211yesmaximal, characteristic, complemented
11#13966C211yesmaximal, characteristic, complemented
12#13902C211yesmaximal, characteristic
13#13869C212yesmaximal, complemented
14#13869C212yesmaximal, complemented
15#13776C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×C4×D4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C2×#397C2214yescomplemented
23C2×#394—28no—
24C2×#394C2214yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27D4×C22⋊C4—28no—
28C2×#390C2212yes—
29C2×#390C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#381C2211yescharacteristic
34C2×#381C2211yescharacteristic
35C2×#381C2212yes—
36C2×#381C2214yescomplemented
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#379C2212yes—
40C2×#379C2212yes—
41C2×#379C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1442C2214yescomplemented
46#1441C2212yescomplemented
47#1441C2212yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1440C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2212yes—
62#1438C2212yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1434C2212yescomplemented
67#1434C2212yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2214yescomplemented
75C22⋊#95C2214yescomplemented
76C22⋊#95C2214yescomplemented
77C26C2312yes—
78C23×D4—24no—
79C23×D4—24no—
80C23×D4—48no—
81C23×D4D418yes—
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4×D4—28no—
92C2×C4×D4—48no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4—48no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131#397—48no—
132#394—28no—
133#394—216no—
134#394—28no—
135#390—28no—
136#390—48no—
137#386—28no—
138#386—28no—
139#385—44no—
140#381—24no—
141#381—24no—
142#381—44no—
143#381—48no—
144#380—24no—
145#380—24no—
146#379—28no—
147#379—44no—
148#379—44no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C2×#95—24no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25—28no—
175C25—24no—
176C25—24no—
177C25—24no—
178C25—24no—
179C25—24no—
180C25—24no—
181C25—44no—
182C25—44no—
183C25—44no—
184C25C2×D418yes—
185C25C2412yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—232no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—432no—
200C22×D4—432no—
201C22×D4—432no—
202C22×D4C2×D414yes—
203C22×D4C2×D418yes—
204C22×D4C2×D414yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D414yes—
232C23×C4#4114yes—
233C23×C4#4114yes—
234C23×C4#4114yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C4×D4—432no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—432no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—432no—
307C2×C22⋊C4—432no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4#4114yes—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—44no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—216no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—216no—
370C24—216no—
371C24—216no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—416no—
376C24—416no—
377C24—416no—
378C24—416no—
379C24—416no—
380C24—416no—
381C24—48no—
382C24—48no—
383C24—48no—
384C24—48no—
385C24—88no—
386C24—88no—
387C24—88no—
388C24—88no—
389C24—88no—
390C24—88no—
391C24C22×D414yes—
392C24C22×D414yes—
393C24C22×D412yes—
394C24C2×#4112yes—
395C24C2×#4112yes—
396C24C2×#4112yes—
397C24C2×#4112yes—
398C24H(5,𝔽2)18yes—
399C24H(5,𝔽2)12yes—
400C24H(5,𝔽2)12yes—
401C24C2511yescharacteristic
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—216no—
406C2×D4—28no—
407C2×D4—216no—
408C2×D4—28no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—4128no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—88no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4—88no—
496C22×C4—84no—
497C22×C4—84no—
498C22×C4—88no—
499C22×C4—88no—
500C22×C4—88no—
501C22×C4#123116yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C2×#4114yes—
505C22×C4C2×#4114yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—84no—
514C4⋊C4—88no—
515C22⋊C4—28no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—88no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C22⋊C4—88no—
535C42—48no—
536C42—84no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—28no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—432no—
567C23—432no—
568C23—432no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—432no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—816no—
597C23—816no—
598C23—816no—
599C23—832no—
600C23—832no—
601C23—832no—
602C23—832no—
603C23C2×#12314yes—
604C23#52914yes—
605C23#53418yes—
606C23#54014yes—
607C23#54614yes—
608C23#54614yes—
609C23#55314yes—
610C23#55514yes—
611C23C23×D411yescharacteristic
612C23C22×#4111yescharacteristic
613C23C22×#4111yescharacteristic
614C23C22×#4111yescharacteristic
615C23C22×#4111yescharacteristic
616C23C2×H(5,𝔽2)14yes—
617C23C2×H(5,𝔽2)11yescharacteristic
618C23C2×H(5,𝔽2)11yescharacteristic
619C23#58514yes—
620D4—232no—
621D4—432no—
622D4—416no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—416no—
635C2×C4—42no—
636C2×C4—416no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—816no—
688C2×C4—816no—
689C2×C4—88no—
690C2×C4—816no—
691C2×C4C2×#12314yes—
692C2×C4C2×#12314yes—
693C22—24no—
694C22—28no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—24no—
699C22—28no—
700C22—28no—
701C22—28no—
702C22—28no—
703C22—28no—
704C22—44no—
705C22—48no—
706C22—48no—
707C22—44no—
708C22—44no—
709C22—48no—
710C22—48no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—864no—
720C22—84no—
721C22—84no—
722C22—84no—
723C22—84no—
724C22#242018yes—
725C22C22×#12311yescharacteristic
726C22C2×#52911yescharacteristic
727C22C2×#53412yes—
728C22C2×#54011yescharacteristic
729C22C2×#54611yescharacteristic
730C22C2×#54611yescharacteristic
731C22C2×#55311yescharacteristic
732C22C2×#55511yescharacteristic
733C22#348512yes—
734C22#349012yes—
735C22#349012yes—
736C22#349414yes—
737C22#351012yes—
738C22#351012yes—
739C22#351314yes—
740C22#353414yes—
741C22#353414yes—
742C4—24no—
743C4—24no—
744C4—44no—
745C4—44no—
746C4—82no—
747C4—84no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—84no—
754C2—28no—
755C2—48no—
756C2—84no—
757C2C2×#242011yescharacteristic
758C2#3499211yescharacteristic
759C2#3562611yescharacteristic
760C2#3573112yes—
761C2#3783412yes—
762C2#3802212yes—
763C2#4054612yes—
764C2#4308312yes—
765C2#4396012yes—
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
3. #16573C2semidirect
4. #15702C2semidirect
5. #15502C2semidirect
6. #14918C2semidirect
7. #14823C2semidirect
8. #14727C2semidirect
9. #14339C2semidirect
10. #14149C2semidirect
11. #13966C2semidirect
13. #13869C2semidirect
14. #13869C2semidirect
15. #13776C2semidirect
22. C2×#397C22semidirect
36. C2×#381C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
74. C22⋊#95C22semidirect
75. C22⋊#95C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3942
C2×D4×C22⋊C42
#165732
#157022
#155022
#149182
#148232
#147272
#143392
#141492
#139662
#139022
#138692
#137762