Finite Group Databasebeta

← #4452776

Subgroups — #4452776

Order 512.

2962 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic, complemented
2C22×#396C211yesmaximal, characteristic
3#16847C214yesmaximal, complemented
4#16550C214yesmaximal, complemented
5#16547C214yesmaximal, complemented
6#16330C214yesmaximal, complemented
7#14822C211yesmaximal, characteristic, complemented
8#14821C211yesmaximal, characteristic, complemented
9#14820C211yesmaximal, characteristic
10#14734C211yesmaximal, characteristic
11#14728C212yesmaximal, complemented
12#14726C212yesmaximal
13#14043C212yesmaximal, complemented
14#13971C212yesmaximal
15#13778C211yesmaximal, characteristic, complemented
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2646—28no—
23C2×#400C2212yescomplemented
24C2×#400C2212yescomplemented
25C2×#400C2212yescomplemented
26C2×#399C2212yes—
27C2×#398C2212yescomplemented
28C2×#398C2212yescomplemented
29C2×#398C2214yescomplemented
30C2×#398C2212yescomplemented
31C2×#397C2212yes—
32C2×#396—28no—
33C2×#396C2214yes—
34C2×#396C2214yes—
35C2×#396C2214yes—
36C2×#389C2214yescomplemented
37C2×#388C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#379C2212yes—
46C2×#379C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2211yescharacteristic
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2212yescomplemented
56#1441C2214yescomplemented
57#1441C2212yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1437C2214yes—
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2314yes—
124#400—28no—
125#400—28no—
126#400—44no—
127#399—44no—
128#398—28no—
129#398—28no—
130#398—44no—
131#398—48no—
132#397—44no—
133#396—216no—
134#396—216no—
135#396—48no—
136#389—48no—
137#388—48no—
138#385—28no—
139#385—28no—
140#385—28no—
141#385—44no—
142#380—24no—
143#380—24no—
144#380—28no—
145#379—28no—
146#379—44no—
147C4×C22⋊C4—44no—
148C2×#95D418yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25—22no—
174C25—24no—
175C25—24no—
176C25—22no—
177C25—24no—
178C25—22no—
179C25—42no—
180C25—42no—
181C25—42no—
182C25C2412yes—
183C25C2411yescharacteristic
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—416no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4—416no—
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#4118yes—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95C2×D418yes—
328#95C2×D418yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—28no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—416no—
345C24—48no—
346C24—48no—
347C24—416no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—44no—
354C24—48no—
355C24—48no—
356C24—88no—
357C24—84no—
358C24—88no—
359C24—84no—
360C24C22×D412yes—
361C24C2×#4114yes—
362C24H(5,𝔽2)12yes—
363C242−1+414yes—
364C242−1+414yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—88no—
451C22×C4—84no—
452C22×C4—88no—
453C22×C4—84no—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D414yes—
458C22×C4C2×#4118yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C4H(5,𝔽2)14yes—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C22⋊C4—216no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—88no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—416no—
508C23—416no—
509C23—416no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—832no—
529C23—832no—
530C23—88no—
531C23—88no—
532C23—88no—
533C23—88no—
534C23#53214yes—
535C23#53614yes—
536C23#53714yes—
537C23#53918yes—
538C23#54118yes—
539C23C23×D411yescharacteristic
540C23C22×#4112yes—
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)12yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×H(5,𝔽2)12yes—
545C23C2×2−1+412yes—
546C23C2×2−1+411yescharacteristic
547C23C2×2−1+412yes—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4—88no—
623C22—216no—
624C22—24no—
625C22—24no—
626C22—216no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—48no—
632C22—44no—
633C22—48no—
634C22—44no—
635C22—44no—
636C22—832no—
637C22—82no—
638C22—84no—
639C22—82no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22#266418yes—
644C22C2×#53211yescharacteristic
645C22C2×#53611yescharacteristic
646C22C2×#53711yescharacteristic
647C22C2×#53912yes—
648C22C2×#54112yes—
649C22#348411yescharacteristic
650C22#348411yescharacteristic
651C22#348411yescharacteristic
652C22#348611yescharacteristic
653C22#351914yes—
654C22#352014yes—
655C22#353612yes—
656C22#353614yes—
657C22#353612yes—
658C22#353712yes—
659C22#353812yes—
660C22#353814yes—
661C4—44no—
662C4—44no—
663C4—48no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4—84no—
672C4—82no—
673C4—82no—
674C2—28no—
675C2—84no—
676C2C2×#266411yescharacteristic
677C2#3539511yescharacteristic
678C2#3539711yescharacteristic
679C2#3670611yescharacteristic
680C2#3670811yescharacteristic
681C2#3671811yescharacteristic
682C2#3671911yescharacteristic
683C2#4073514yes—
684C2#4079114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2646C2semidirect
3. #16847C2semidirect
4. #16550C2semidirect
5. #16547C2semidirect
6. #16330C2semidirect
7. #14822C2semidirect
8. #14821C2semidirect
11. #14728C2semidirect
13. #14043C2semidirect
15. #13778C2semidirect
23. C2×#400C22semidirect
24. C2×#400C22semidirect
25. C2×#400C22semidirect
27. C2×#398C22semidirect
28. C2×#398C22semidirect
29. C2×#398C22semidirect
30. C2×#398C22semidirect
36. C2×#389C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#3962
#168472
#165502
#165472
#163302
#148222
#148212
#148202
#147342
#147282
#147262
#140432
#139712
#137782