Finite Group Databasebeta

← #5024695

Subgroups — #5024695

Order 512.

4325 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C211yesmaximal, characteristic, complemented
2C2×#2405C212yesmaximal, complemented
3C2×#2622C212yesmaximal, complemented
4C2×#2613C211yesmaximal, characteristic
5#15561C214yesmaximal, complemented
6#15549C214yesmaximal, complemented
7#15539C212yesmaximal, complemented
8#15480C212yesmaximal, complemented
9#15384C212yesmaximal, complemented
10#15384C214yesmaximal, complemented
11#15370C214yesmaximal, complemented
12#15283C212yesmaximal, complemented
13#13827C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#123C2214yes—
16C22×#123C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×#120C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20#2622—216no—
21#2613—28no—
22#2483—28no—
23#2405—216no—
24C2×#399C2212yescomplemented
25C2×#399C2212yescomplemented
26C2×#397C2212yes—
27C2×#397C2214yes—
28C2×#397C2212yes—
29C2×#397C2214yescomplemented
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#394C2214yescomplemented
35C2×#392C2214yes—
36C2×#390C2212yescomplemented
37C2×#386C2214yes—
38C2×#386C2212yescomplemented
39C2×#386C2212yescomplemented
40C2×#385C2214yes—
41C2×#385C2214yescomplemented
42C2×#385C2212yescomplemented
43C2×#385C2212yescomplemented
44C2×#384C2212yescomplemented
45C2×#381C2211yescharacteristic
46C2×#381C2214yescomplemented
47C2×#379C2214yes—
48C2×#379C2211yescharacteristic
49C2×#379C2211yescharacteristic
50C2×#379C2212yes—
51#1445C2214yescomplemented
52#1445C2212yescomplemented
53#1445C2214yescomplemented
54#1444C2214yescomplemented
55#1444C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2214yescomplemented
62#1438C2212yescomplemented
63#1437C2212yes—
64#1437C2212yes—
65#1437C2214yescomplemented
66#1437C2214yescomplemented
67#1437C2214yescomplemented
68#1437C2212yescomplemented
69#1437C2212yescomplemented
70C4⋊#95C2214yescomplemented
71C26C2311yescharacteristic
72C26C2311yescharacteristic
73C23×D4—24no—
74C23×D4—28no—
75C23×D4—44no—
76C23×D4—44no—
77C23×D4C2314yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×#123—24no—
85C2×#123—216no—
86C2×#123—48no—
87C2×#123—44no—
88C2×#123—48no—
89C2×#120—24no—
90C2×#120—28no—
91C2×#120—24no—
92C2×#120—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125#399—44no—
126#399—44no—
127#397—216no—
128#397—216no—
129#397—28no—
130#397—28no—
131#394—216no—
132#394—216no—
133#394—216no—
134#394—216no—
135#394—216no—
136#392—216no—
137#390—44no—
138#386—28no—
139#386—28no—
140#386—216no—
141#385—28no—
142#385—28no—
143#385—216no—
144#385—216no—
145#384—44no—
146#381—24no—
147#381—216no—
148#379—24no—
149#379—216no—
150#379D418yes—
151#379D4116yescomplemented
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25—22no—
168C25—24no—
169C25—24no—
170C25—42no—
171C25—42no—
172C25—42no—
173C25—44no—
174C25—44no—
175C25—44no—
176C25—44no—
177C25C2×D414yes—
178C25C2×D418yes—
179C25C2411yescharacteristic
180C25C2412yes—
181C25C2411yescharacteristic
182C25C2412yes—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—416no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—464no—
206C22×D4—416no—
207C22×D4—416no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231#123—416no—
232#123—464no—
233#120—416no—
234#120—416no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
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—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—432no—
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⋊C4C2×D414yes—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D4116yes—
308C2×C22⋊C4C2×D4116yes—
309C2×C22⋊C4C2×D4116yes—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×D418yes—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—42no—
315C2×C42—44no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—42no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—28no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—44no—
357C24—416no—
358C24—44no—
359C24—416no—
360C24—48no—
361C24—48no—
362C24—416no—
363C24—416no—
364C24—44no—
365C24—416no—
366C24—44no—
367C24—84no—
368C24—84no—
369C24—88no—
370C24—84no—
371C24—84no—
372C24—88no—
373C24—84no—
374C24—84no—
375C24—84no—
376C24—84no—
377C24—84no—
378C24—84no—
379C24—88no—
380C24—88no—
381C24—88no—
382C24#127116yes—
383C24C22×D412yes—
384C24C22×D414yes—
385C24C22×D414yes—
386C24C22×D414yes—
387C24C22×D414yes—
388C24H(5,𝔽2)14yes—
389C24C2511yescharacteristic
390C2×D4—232no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—48no—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—4128no—
398C2×D4—416no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—88no—
407C2×D4—84no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—88no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
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—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—416no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—88no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—88no—
473C22×C4C22×D414yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C22×D418yes—
477C22×C4H(5,𝔽2)14yes—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—44no—
490C22⋊C4—416no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—48no—
506C42—84no—
507C23—22no—
508C23—28no—
509C23—22no—
510C23—28no—
511C23—24no—
512C23—28no—
513C23—24no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—416no—
532C23—432no—
533C23—416no—
534C23—44no—
535C23—42no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—41no—
565C23—41no—
566C23—44no—
567C23—41no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—44no—
577C23—44no—
578C23—816no—
579C23—816no—
580C23—816no—
581C23—832no—
582C23—832no—
583C23—816no—
584C23—832no—
585C23—832no—
586C23—816no—
587C23C2×#12714yes—
588C23C2×#12718yes—
589C23C23×D411yescharacteristic
590C23C23×D412yes—
591C23C2×H(5,𝔽2)12yes—
592C23C2×H(5,𝔽2)12yes—
593C23#58514yes—
594C23#58514yes—
595D4—464no—
596D4—816no—
597D4—88no—
598D4—816no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—44no—
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—42no—
651C2×C4—42no—
652C2×C4—816no—
653C2×C4—816no—
654C22—24no—
655C22—28no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22—48no—
660C22—44no—
661C22—48no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—832no—
675C22—832no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—81no—
681C22—82no—
682C22—84no—
683C22—82no—
684C22—84no—
685C22—82no—
686C22C22×#12711yescharacteristic
687C22#348411yescharacteristic
688C22#348414yes—
689C22#348411yescharacteristic
690C22#348514yes—
691C22#348512yes—
692C22#348514yes—
693C22#348512yes—
694C22#353312yes—
695C22#354012yes—
696C22#354512yes—
697C22#355112yes—
698C22#356014yes—
699C22#356214yes—
700C4—44no—
701C4—44no—
702C4—48no—
703C4—48no—
704C4—44no—
705C4—84no—
706C4—81no—
707C4—84no—
708C4—82no—
709C2—48no—
710C2—44no—
711C2—84no—
712C2—161no—
713C2#3539011yescharacteristic
714C2#3546312yes—
715C2#4782814yes—
716C2#4877312yes—
717C2#4877414yes—
718C2#5016512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
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
2. C2×#2405C2semidirect
3. C2×#2622C2semidirect
5. #15561C2semidirect
6. #15549C2semidirect
7. #15539C2semidirect
8. #15480C2semidirect
9. #15384C2semidirect
10. #15384C2semidirect
11. #15370C2semidirect
12. #15283C2semidirect
13. #13827C2semidirect
24. C2×#399C22semidirect
25. C2×#399C22semidirect
29. C2×#397C22semidirect
34. C2×#394C22semidirect
36. C2×#390C22semidirect
38. C2×#386C22semidirect
39. C2×#386C22semidirect
41. C2×#385C22semidirect
42. C2×#385C22semidirect
43. C2×#385C22semidirect
44. C2×#384C22semidirect
46. C2×#381C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1438C22semidirect
62. #1438C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. #1437C22semidirect
68. #1437C22semidirect
69. #1437C22semidirect
70. C4⋊#95C22semidirect
151. #379D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C2×#24052
C2×#26222
C2×#26132
#155612
#155492
#155392
#154802
#153842
#153702
#152832
#138272