Finite Group Databasebeta

← #4308952

Subgroups — #4308952

Order 512.

3004 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C212yesmaximal, complemented
2C2×#2618C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4#16595C214yesmaximal, complemented
5#16594C214yesmaximal, complemented
6#16550C214yesmaximal, complemented
7#16549C214yesmaximal, complemented
8#15014C211yesmaximal, characteristic, complemented
9#14921C212yesmaximal, complemented
10#14913C211yesmaximal, characteristic
11#14821C211yesmaximal, characteristic, complemented
12#14729C212yesmaximal, complemented
13#14719C211yesmaximal, characteristic
14#14706C212yesmaximal, complemented
15#14339C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2623—216no—
24#2618—28no—
25C2×#400C2214yes—
26C2×#399C2214yes—
27C2×#398C2214yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#397C2214yes—
31C2×#396C2212yes—
32C2×#396C2212yes—
33C2×#396C2214yes—
34C2×#394C2212yes—
35C2×#394C2212yes—
36C2×#389—28no—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1445C2212yes—
52#1444C2212yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#121—24no—
91C2×#121—44no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4D418yes—
100C22×C4⋊C4D418yescomplemented
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42—24no—
128C22×C42—44no—
129C22×C42C2314yes—
130#400—216no—
131#399—216no—
132#398—28no—
133#398—216no—
134#398—44no—
135#397—216no—
136#396—28no—
137#396—28no—
138#396—48no—
139#394—28no—
140#394—44no—
141#389—216no—
142#389—216no—
143#386—28no—
144#386—28no—
145#386—28no—
146#386—44no—
147#386—44no—
148#385—28no—
149#385—28no—
150#381—28no—
151#380—24no—
152#380—44no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C25C2412yes—
179C25C2411yescharacteristic
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4#4114yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235#123—432no—
236#121—416no—
237C2×C4⋊C4—232no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D4116yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
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⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C42—28no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—416no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—42no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C2×#4112yes—
359C24H(5,𝔽2)14yes—
360C242−1+414yes—
361C24C2511yescharacteristic
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—88no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
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—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4#121116yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C42−1+414yes—
477C4⋊C4—232no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C42—416no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23C2×#12114yes—
540C23#53414yes—
541C23#53614yes—
542C23#53718yes—
543C23#53918yes—
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C22×#4111yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+412yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552C23#58512yes—
553D4—88no—
554D4—816no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
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—88no—
633C2×C4—88no—
634C2×C4C2×#12118yes—
635C22—24no—
636C22—216no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—216no—
642C22—24no—
643C22—82no—
644C22—84no—
645C22—84no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—84no—
652C22—82no—
653C22—82no—
654C22#261818yes—
655C22C22×#12111yescharacteristic
656C22C2×#53411yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#53712yes—
659C22C2×#53912yes—
660C22#346012yes—
661C22#346112yes—
662C22#348411yescharacteristic
663C22#348511yescharacteristic
664C22#348611yescharacteristic
665C22#348711yescharacteristic
666C22#349612yes—
667C22#349712yes—
668C22#353414yes—
669C22#353514yes—
670C22#355214yes—
671C22#355314yes—
672C4—28no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—84no—
681C4—84no—
682C4—82no—
683C4—82no—
684C4—82no—
685C2—28no—
686C2—162no—
687C2C2×#261811yescharacteristic
688C2#3515311yescharacteristic
689C2#3516811yescharacteristic
690C2#3711711yescharacteristic
691C2#3712011yescharacteristic
692C2#3713014yes—
693C2#3718611yescharacteristic
694C2#3723311yescharacteristic
695C2#3752514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2623C2semidirect
3. C22×#389C2semidirect
4. #16595C2semidirect
5. #16594C2semidirect
6. #16550C2semidirect
7. #16549C2semidirect
8. #15014C2semidirect
9. #14921C2semidirect
11. #14821C2semidirect
12. #14729C2semidirect
14. #14706C2semidirect
15. #14339C2semidirect
100. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C2×#26182
C22×#3892
#165952
#165942
#165502
#165492
#150142
#149212
#149132
#148212
#147292
#147192
#147062
#143392