Finite Group Databasebeta

← #4341463

Subgroups — #4341463

Order 512.

3214 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2708C214yesmaximal, complemented
2C2×#2535C211yesmaximal, characteristic
3C2×#2486C211yesmaximal, characteristic, complemented
4#17059C212yesmaximal, complemented
5#16964C212yesmaximal, complemented
6#16808C212yesmaximal, complemented
7#16807C212yesmaximal, complemented
8#15561C212yesmaximal, complemented
9#15539C212yesmaximal, complemented
10#15513C214yesmaximal, complemented
11#14839C212yesmaximal, complemented
12#14836C212yesmaximal, complemented
13#14807C212yesmaximal, complemented
14#14777C212yesmaximal, complemented
15#13784C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2708—232no—
22#2535—28no—
23#2486—28no—
24C2×#400C2214yes—
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#398C2214yes—
28C2×#398C2212yes—
29C2×#397C2211yescharacteristic
30C2×#397C2211yescharacteristic
31C2×#397C2214yes—
32C2×#397C2214yes—
33C2×#397C2212yes—
34C2×#394C2214yes—
35C2×#394C2214yes—
36C2×#392C2211yescharacteristic
37C2×#392C2211yescharacteristic
38C2×#389C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#385C2214yes—
47C2×#382C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2212yes—
50C2×#380C2214yes—
51C2×#380C2211yescharacteristic
52C2×#380C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C22×#95C2211yescharacteristic
55#1445C2214yes—
56#1445C2214yes—
57#1445C2212yes—
58#1445C2212yes—
59#1445C2214yes—
60#1445C2214yes—
61#1444C2212yes—
62#1444C2212yes—
63#1444C2214yes—
64#1444C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2212yes—
73#1436C2212yes—
74C4⋊#95C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#123—216no—
88C2×#123—28no—
89C2×#121—216no—
90C2×#121—28no—
91C2×C4×D4—24no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C2×C4×D4D418yes—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4—42no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#400—216no—
129#399—44no—
130#399—44no—
131#398—28no—
132#398—216no—
133#397—216no—
134#397—216no—
135#397—28no—
136#397—24no—
137#397—24no—
138#394—216no—
139#394—216no—
140#392—24no—
141#392—24no—
142#389—28no—
143#386—216no—
144#386—28no—
145#386—28no—
146#386—216no—
147#386D4116yescomplemented
148#385—216no—
149#385—216no—
150#385—216no—
151#382—24no—
152#381—28no—
153#380—216no—
154#380—28no—
155#380—24no—
156#380—28no—
157C4×C22⋊C4—42no—
158C2×#95—42no—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C25C2412yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4C2×D414yes—
203C22×D4C2×D418yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D418yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#123—432no—
240#121—432no—
241C4×D4—416no—
242C4×D4—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D4116yes—
314C2×C22⋊C4C2×D418yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42C2×D414yes—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95C2×D4116yes—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24C22×D412yes—
365C24C22×D414yes—
366C24C22×D414yes—
367C24C22×D412yes—
368C242−1+414yes—
369C24C2511yescharacteristic
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4#127116yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
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—22no—
446C22×C4—22no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—88no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D418yes—
471C22×C4C22×D418yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—416no—
484C4⋊C4—48no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—28no—
505C42—84no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23C2×#12714yes—
560C23C2×#12714yes—
561C23C23×D411yescharacteristic
562C23C23×D412yes—
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×2−1+412yes—
566C23#58512yes—
567C23#58514yes—
568C23#58512yes—
569D4—416no—
570D4—416no—
571D4—816no—
572D4—816no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—416no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
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—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—88no—
638C2×C4—816no—
639C2×C4—816no—
640C2×C4—816no—
641C2×C4C2×#12714yes—
642C2×C4#53414yes—
643C22—24no—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—42no—
648C22—42no—
649C22—48no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—44no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22C22×#12711yescharacteristic
671C22C2×#53411yescharacteristic
672C22#344911yescharacteristic
673C22#348512yes—
674C22#348514yes—
675C22#348914yes—
676C22#348912yes—
677C22#349612yes—
678C22#349612yes—
679C22#353312yes—
680C22#353312yes—
681C22#353811yescharacteristic
682C22#354011yescharacteristic
683C22#354112yes—
684C22#355314yes—
685C22#356614yes—
686C4—24no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—48no—
691C4—48no—
692C4—42no—
693C4—82no—
694C4—84no—
695C4—84no—
696C4—82no—
697C4—82no—
698C2—44no—
699C2—44no—
700C2—84no—
701C2#3500111yescharacteristic
702C2#3606012yes—
703C2#3695511yescharacteristic
704C2#3737411yescharacteristic
705C2#4142512yes—
706C2#4682912yes—
707C2#4683512yes—
708C2#4881314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#2708C2semidirect
3. C2×#2486C2semidirect
4. #17059C2semidirect
5. #16964C2semidirect
6. #16808C2semidirect
7. #16807C2semidirect
8. #15561C2semidirect
9. #15539C2semidirect
10. #15513C2semidirect
11. #14839C2semidirect
12. #14836C2semidirect
13. #14807C2semidirect
14. #14777C2semidirect
15. #13784C2semidirect
147. #386D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27082
C2×#25352
C2×#24862
#170592
#169642
#168082
#168072
#155612
#155392
#155132
#148392
#148362
#148072
#147772
#137842