Finite Group Databasebeta

← #760838

Subgroups — #760838

Order 512.

5478 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C212yesmaximal, complemented
2C2×#2405C212yesmaximal
3C22×#386C211yesmaximal, characteristic
4C22×#380C211yesmaximal, characteristic, complemented
5#16807C214yesmaximal, complemented
6#15069C212yesmaximal, complemented
7#14936C212yesmaximal, complemented
8#14839C214yesmaximal, complemented
9#14777C212yesmaximal, complemented
10#14734C212yesmaximal, complemented
11#13996C214yesmaximal, complemented
12#13965C212yesmaximal, complemented
13#13903C212yesmaximal, complemented
14C4×#379C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×#123C2214yes—
18C23×C4⋊C4C2211yescharacteristic, complemented
19C23×C4⋊C4C2211yescharacteristic, complemented
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2483—216no—
25#2405—216no—
26C2×#399C2214yescomplemented
27C2×#397C2218yescomplemented
28C2×#394C2218yes—
29C2×#386—28no—
30C2×#386C2214yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yescomplemented
38C2×#381C2214yes—
39C2×#381C2214yescomplemented
40C2×#380—28no—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yescomplemented
44C2×#380C2212yescomplemented
45C2×#379C2214yes—
46C2×#379C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yescomplemented
49C2×C4×C22⋊C4C2212yescomplemented
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52C22×#95C2212yescomplemented
53#1445C2218yescomplemented
54#1445C2214yescomplemented
55#1444C2214yescomplemented
56#1444C2214yescomplemented
57#1444C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2218yescomplemented
61#1439C2214yescomplemented
62#1437C2218yescomplemented
63C4⋊#95C2218yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2212yescomplemented
66C22⋊#95C2212yescomplemented
67C26C2311yescharacteristic
68C26C2311yescharacteristic
69C23×D4—24no—
70C23×D4—28no—
71C23×D4—28no—
72C23×D4—44no—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4—44no—
78C24×C4D418yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#123—216no—
91C2×#123—216no—
92C2×#123—48no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2318yes—
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2314yes—
126#399—48no—
127#397—232no—
128#394—232no—
129#386—28no—
130#386—28no—
131#386—28no—
132#386—28no—
133#386—216no—
134#385—216no—
135#385—216no—
136#385—216no—
137#381—216no—
138#381—216no—
139#380—28no—
140#380—28no—
141#380—28no—
142#380—28no—
143#379—216no—
144#379D4116yes—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—44no—
147C4×C22⋊C4—44no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—28no—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—24no—
168C25—22no—
169C25—24no—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—42no—
174C25—42no—
175C25—42no—
176C25—42no—
177C25—42no—
178C25C2×D414yes—
179C25C2×D418yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×D4—232no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—44no—
194C22×D4—464no—
195C22×D4—416no—
196C22×D4—416no—
197C22×D4#4118yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—232no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—416no—
227C23×C4—416no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—42no—
240C23×C4—464no—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4#4118yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2418yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258#123—464no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—48no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—416no—
304C2×C22⋊C4—432no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—416no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4C2×D4116yes—
310C2×C22⋊C4C2×D4116yes—
311C2×C22⋊C4C2×D4116yes—
312C2×C42—28no—
313C2×C42—44no—
314C2×C42—44no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—216no—
322#95—216no—
323#95—216no—
324#95—216no—
325#95—28no—
326#95—28no—
327#95—44no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—28no—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—232no—
341C24—232no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—28no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—44no—
358C24—44no—
359C24—416no—
360C24—416no—
361C24—416no—
362C24—416no—
363C24—416no—
364C24—416no—
365C24—416no—
366C24—416no—
367C24—44no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—416no—
374C24—88no—
375C24—88no—
376C24—88no—
377C24#127116yes—
378C24C22×D414yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C22×D414yes—
383C24C22×D414yes—
384C24C2×#4114yes—
385C24C2×#4114yes—
386C24C2×#4114yes—
387C24H(5,𝔽2)14yes—
388C242−1+414yes—
389C24C2511yescharacteristic
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—232no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—4128no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—416no—
401C22×C4—232no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—4128no—
463C22×C4—416no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—416no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4#123116yescomplemented
485C22×C4C22×D412yes—
486C22×C4C22×D4116yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)14yes—
494C4⋊C4—432no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—816no—
500C22⋊C4—232no—
501C22⋊C4—232no—
502C22⋊C4—416no—
503C22⋊C4—432no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—816no—
513C42—88no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—232no—
544C23—232no—
545C23—24no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—416no—
550C23—416no—
551C23—416no—
552C23—416no—
553C23—4128no—
554C23—416no—
555C23—416no—
556C23—416no—
557C23—416no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—48no—
570C23—44no—
571C23—48no—
572C23—48no—
573C23—44no—
574C23—48no—
575C23—864no—
576C23—88no—
577C23—88no—
578C23—88no—
579C23—88no—
580C23C2×#12314yes—
581C23#52514yes—
582C23C2×#12714yes—
583C23C2×#12714yes—
584C23C2×#12714yes—
585C23#53414yes—
586C23#53614yes—
587C23#546116yes—
588C23#547116yes—
589C23C23×D411yescharacteristic
590C23C23×D412yes—
591C23C22×#4112yes—
592C23C22×#4112yes—
593C23C2×H(5,𝔽2)12yes—
594C23C2×H(5,𝔽2)12yes—
595C23C2×2−1+412yes—
596C23#58512yes—
597D4—432no—
598D4—432no—
599C2×C4—232no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—216no—
614C2×C4—28no—
615C2×C4—216no—
616C2×C4—216no—
617C2×C4—416no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—416no—
621C2×C4—416no—
622C2×C4—416no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—816no—
650C2×C4—88no—
651C2×C4—88no—
652C2×C4—816no—
653C2×C4C2×#12314yes—
654C2×C4C2×#12314yes—
655C2×C4#53414yes—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—232no—
662C22—232no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—464no—
674C22—42no—
675C22—42no—
676C22—416no—
677C22—42no—
678C22—42no—
679C22—44no—
680C22—44no—
681C22—416no—
682C22—44no—
683C22—832no—
684C22—82no—
685C22—88no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22#250118yes—
690C22#253918yes—
691C22C22×#12311yescharacteristic
692C22C2×#52511yescharacteristic
693C22C22×#12711yescharacteristic
694C22C2×#53411yescharacteristic
695C22C2×#53411yescharacteristic
696C22C2×#53611yescharacteristic
697C22#344911yescharacteristic
698C22C2×#54614yes—
699C22C2×#54714yes—
700C22#347514yes—
701C22#349014yes—
702C22#349614yes—
703C22#352014yes—
704C22#353514yes—
705C4—24no—
706C4—24no—
707C4—24no—
708C4—44no—
709C4—416no—
710C4—42no—
711C4—84no—
712C4—82no—
713C4—84no—
714C4—82no—
715C2—28no—
716C2—28no—
717C2—44no—
718C2—84no—
719C2C2×#250111yescharacteristic
720C2#3480911yescharacteristic
721C2C2×#253911yescharacteristic
722C2#3499211yescharacteristic
723C2#3500111yescharacteristic
724C2#3508411yescharacteristic
725C2#3744111yescharacteristic
726C2#5931118yes—
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
1. C2×#2483C2semidirect
4. C22×#380C2semidirect
5. #16807C2semidirect
6. #15069C2semidirect
7. #14936C2semidirect
8. #14839C2semidirect
9. #14777C2semidirect
10. #14734C2semidirect
11. #13996C2semidirect
12. #13965C2semidirect
13. #13903C2semidirect
14. C4×#379C2semidirect
18. C23×C4⋊C4C22semidirect
19. C23×C4⋊C4C22semidirect
26. C2×#399C22semidirect
27. C2×#397C22semidirect
37. C2×#385C22semidirect
39. C2×#381C22semidirect
43. C2×#380C22semidirect
44. C2×#380C22semidirect
48. C2×C4×C22⋊C4C22semidirect
49. C2×C4×C22⋊C4C22semidirect
52. C22×#95C22semidirect
53. #1445C22semidirect
54. #1445C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
57. #1444C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1437C22semidirect
63. C4⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
484. C22×C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C2×#24052
C22×#3862
C22×#3802
#168072
#150692
#149362
#148392
#147772
#147342
#139962
#139652
#139032
C4×#3792