Finite Group Databasebeta

← #4341772

Subgroups — #4341772

Order 512.

2918 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2709C212yesmaximal
2C2×#2486C211yesmaximal, characteristic, complemented
3C2×#2328C211yesmaximal, characteristic
4#16816C212yesmaximal, complemented
5#16688C212yesmaximal, complemented
6#16596C212yesmaximal, complemented
7#16401C212yesmaximal, complemented
8#15538C214yesmaximal, complemented
9#15494C214yesmaximal, complemented
10#15122C212yesmaximal, complemented
11#14868C212yesmaximal, complemented
12#14776C212yesmaximal, complemented
13#14735C212yesmaximal, complemented
14#14505C212yesmaximal, complemented
15#13960C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2709—216no—
22#2486—28no—
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30#2328—28no—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#387C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#382C2211yescharacteristic
43C2×#381C2212yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#376C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1445C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2212yes—
70#1436C2214yes—
71#1434C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2214yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C2×#127—24no—
88C2×#127—24no—
89C2×#127D418yes—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×C4×D4—24no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4D418yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4—42no—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#398—28no—
129#397—28no—
130#397—44no—
131#397—44no—
132#394—216no—
133#394—216no—
134#394—216no—
135#390—24no—
136#390—24no—
137#389—28no—
138#387—216no—
139#386—216no—
140#386—216no—
141#386—28no—
142#386—28no—
143#386—28no—
144#386—28no—
145#386—28no—
146#382—24no—
147#381—28no—
148#380—28no—
149#380—24no—
150#380—42no—
151#376—24no—
152C2×#95—42no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4C2×D418yes—
199C22×D4C2×D418yes—
200C22×D4C2×D414yes—
201C22×D4C2×D414yes—
202C22×D4C2×D414yes—
203C22×D4#4114yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—42no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#127—232no—
242#123—432no—
243C4×D4—416no—
244C4×D4—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
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—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D414yes—
308C2×C22⋊C4C2×D414yes—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—41no—
315C2×C42—41no—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24C22×D414yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C2×#4112yes—
370C24C2×#4112yes—
371C24C2511yescharacteristic
372C2×D4—216no—
373C2×D4—216no—
374C2×D4—28no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4#121116yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
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—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—88no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—88no—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—84no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C42—28no—
511C42—28no—
512C42—44no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—28no—
535C23—28no—
536C23—24no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23C2×#12114yes—
569C23C2×#12114yes—
570C23C23×D411yescharacteristic
571C23C23×D411yescharacteristic
572C23C22×#4111yescharacteristic
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23#58511yescharacteristic
576C23#58511yescharacteristic
577C23#58514yes—
578C23#58514yes—
579D4—416no—
580D4—416no—
581D4—816no—
582D4—816no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—416no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
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—42no—
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—44no—
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—88no—
649C2×C4—816no—
650C2×C4—816no—
651C2×C4—816no—
652C2×C4C2×#12114yes—
653C2×C4#53414yes—
654C22—28no—
655C22—28no—
656C22—24no—
657C22—24no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—48no—
677C22—84no—
678C22—82no—
679C22—82no—
680C22—84no—
681C22—82no—
682C22—82no—
683C22C22×#12111yescharacteristic
684C22C2×#53411yescharacteristic
685C22#344911yescharacteristic
686C22#348512yes—
687C22#348514yes—
688C22#349611yescharacteristic
689C22#349611yescharacteristic
690C22#350012yes—
691C22#350511yescharacteristic
692C22#350511yescharacteristic
693C22#352114yes—
694C22#353312yes—
695C22#353312yes—
696C22#354012yes—
697C22#355112yes—
698C22#356614yes—
699C22#356714yes—
700C4—24no—
701C4—24no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—44no—
706C4—82no—
707C4—84no—
708C4—82no—
709C4—84no—
710C4—84no—
711C4—84no—
712C2—44no—
713C2—44no—
714C2—84no—
715C2#3500011yescharacteristic
716C2#3600912yes—
717C2#3695511yescharacteristic
718C2#3741911yescharacteristic
719C2#4215112yes—
720C2#4697412yes—
721C2#4780012yes—
722C2#5200212yes—
723C2#5206412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2486C2semidirect
4. #16816C2semidirect
5. #16688C2semidirect
6. #16596C2semidirect
7. #16401C2semidirect
8. #15538C2semidirect
9. #15494C2semidirect
10. #15122C2semidirect
11. #14868C2semidirect
12. #14776C2semidirect
13. #14735C2semidirect
14. #14505C2semidirect
15. #13960C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27092
C2×#24862
C2×#23282
#168162
#166882
#165962
#164012
#155382
#154942
#151222
#148682
#147762
#147352
#145052
#139602