Finite Group Databasebeta

← #4328652

Subgroups — #4328652

Order 512.

3018 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2406C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3#16699C212yesmaximal, complemented
4#16611C214yesmaximal, complemented
5#16394C212yesmaximal, complemented
6#16014C212yesmaximal, complemented
7#15494C214yesmaximal, complemented
8#15463C212yesmaximal, complemented
9#14892C212yesmaximal, complemented
10#14843C212yesmaximal, complemented
11#14826C212yesmaximal, complemented
12#14822C212yesmaximal, complemented
13#14513C212yesmaximal, complemented
14#14505C212yesmaximal, complemented
15#13956C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2406—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#397C2214yes—
26C2×#395C2212yes—
27C2×#394C2212yes—
28#2311—28no—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2211yescharacteristic
35C2×#386C2211yescharacteristic
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2211yescharacteristic
40C2×#381C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C22×#95C2211yescharacteristic
44#1446C2212yes—
45#1445C2212yes—
46#1445C2212yes—
47#1444C2212yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22×#41—28no—
78C22×#41—42no—
79C22×#41—42no—
80C22×#41—44no—
81C22×#41D418yes—
82C23×Q8C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—28no—
96C2×#123—44no—
97C2×#117—24no—
98C2×#117—44no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—42no—
102C22×C4⋊C4—42no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4—42no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2312yes—
126#400—44no—
127#400—44no—
128#397—48no—
129#395—28no—
130#394—28no—
131#391—28no—
132#390—28no—
133#386—28no—
134#386—28no—
135#386—24no—
136#386—24no—
137#386—28no—
138#386—44no—
139#385—28no—
140#385—28no—
141#381—24no—
142#381—42no—
143#380—28no—
144#380—28no—
145C2×#95—42no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C2×#41—232no—
177C2×#41—416no—
178C2×#41—416no—
179C2×#41—432no—
180C2×#41—432no—
181C22×Q8—24no—
182C22×Q8—42no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8C2×D414yes—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—42no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4C2×D414yes—
198C22×D4C2×D414yes—
199C22×D4C2×D414yes—
200C22×D4#4118yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—41no—
218C23×C4—41no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—41no—
223C23×C4—41no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—42no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248#123—416no—
249#123—416no—
250#117—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
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—42no—
304C2×C22⋊C4—42no—
305C2×C42—24no—
306C2×C42—42no—
307C2×C42—42no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C2×#4114yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)14yes—
359C24C2511yescharacteristic
360#41—4128no—
361C2×Q8—28no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4#123116yes—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
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—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
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—24no—
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—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—88no—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4114yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+414yes—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—88no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—48no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—44no—
532C23—44no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23C2×#12314yes—
553C23C2×#12314yes—
554C23#53414yes—
555C23C23×D411yescharacteristic
556C23C22×#4112yes—
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)12yes—
560C23C2×2−1+412yes—
561C23#58512yes—
562C23#58514yes—
563Q8—416no—
564D4—416no—
565D4—416no—
566D4—416no—
567C2×C4—232no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—416no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
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—42no—
614C2×C4—42no—
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—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—816no—
641C2×C4—816no—
642C2×C4C2×#12314yes—
643C2×C4#53414yes—
644C22—28no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22—28no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22C22×#12311yescharacteristic
669C22C2×#53411yescharacteristic
670C22C2×#53411yescharacteristic
671C22#348912yes—
672C22#349012yes—
673C22#349012yes—
674C22#350012yes—
675C22#351514yes—
676C22#352114yes—
677C22#353811yescharacteristic
678C22#354012yes—
679C22#354011yescharacteristic
680C22#354112yes—
681C22#355112yes—
682C22#355914yes—
683C22#356014yes—
684C4—24no—
685C4—24no—
686C4—42no—
687C4—42no—
688C4—44no—
689C4—84no—
690C4—84no—
691C4—84no—
692C4—84no—
693C4—82no—
694C4—82no—
695C4—82no—
696C2—44no—
697C2—44no—
698C2—44no—
699C2#3499211yescharacteristic
700C2#3606712yes—
701C2#3737411yescharacteristic
702C2#3741911yescharacteristic
703C2#4214912yes—
704C2#4662214yes—
705C2#5164612yes—
706C2#5180312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16699C2semidirect
4. #16611C2semidirect
5. #16394C2semidirect
6. #16014C2semidirect
7. #15494C2semidirect
8. #15463C2semidirect
9. #14892C2semidirect
10. #14843C2semidirect
11. #14826C2semidirect
12. #14822C2semidirect
13. #14513C2semidirect
14. #14505C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
C2×#23112
#166992
#166112
#163942
#160142
#154942
#154632
#148922
#148432
#148262
#148222
#145132
#145052
#139562