Finite Group Databasebeta

← #640985

Subgroups — #640985

Order 512.

4256 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic, complemented
2C2×#2416C212yesmaximal, complemented
3C2×#2619C211yesmaximal, characteristic, complemented
4C2×#2290C211yesmaximal, characteristic, complemented
5C22×#382C211yesmaximal, characteristic, complemented
6#16519C214yesmaximal, complemented
7#16516C214yesmaximal, complemented
8#15222C212yesmaximal, complemented
9#15219C212yesmaximal, complemented
10#14757C214yesmaximal, complemented
11#14747C214yesmaximal, complemented
12#14340C211yesmaximal, characteristic, complemented
13#14277C211yesmaximal, characteristic, complemented
14#14272C211yesmaximal, characteristic, complemented
15#14137C212yesmaximal, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2214yes—
20C22×#121C2214yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C42C2211yescharacteristic
25#2856—28no—
26#2619—28no—
27#2416—216no—
28C2×#396C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#392C2214yes—
32C2×#390C2212yes—
33C2×#390C2214yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#389C2212yes—
38#2290—28no—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#386C2214yes—
42C2×#386C2212yes—
43C2×#386C2218yes—
44C2×#385C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382—28no—
47C2×#382C2211yescharacteristic
48C2×#382C2212yes—
49C2×#382C2214yes—
50C2×#382C2211yescharacteristic
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×#376C2211yescharacteristic
56C22×#95C2214yes—
57#1444C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1440C2218yes—
61#1439C2214yes—
62#1439C2218yes—
63#1439C2218yes—
64#1439C2218yes—
65#1438C2218yes—
66#1438C2214yes—
67#1438C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2314yes—
74C23×D4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2314yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×#127—24no—
80C2×#127—24no—
81C2×#127—24no—
82C2×#121—216no—
83C2×#121—216no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—28no—
89C2×#121—44no—
90C2×#121—416no—
91C2×#121—416no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4D418yescomplemented
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2318yes—
123C22×C42—24no—
124C22×C42—24no—
125C22×C42D418yescomplemented
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#396—216no—
133#394—216no—
134#394—216no—
135#392—216no—
136#390—28no—
137#390—28no—
138#390—216no—
139#390—28no—
140#390—44no—
141#389—28no—
142#388—28no—
143#387—28no—
144#386—28no—
145#386—216no—
146#386—416no—
147#385—28no—
148#384—24no—
149#382—28no—
150#382—24no—
151#382—216no—
152#382D418yescomplemented
153#380—28no—
154C4×C4⋊C4—24no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—24no—
157#376—24no—
158C2×#95—216no—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2318yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2314yes—
171C25C2414yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—216no—
181C22×D4—216no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—28no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—48no—
195C22×D4—48no—
196C22×D4—44no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4C2×D414yes—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—28no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—216no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—48no—
219C23×C4C2×D414yes—
220C23×C4C2×D418yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236#127—416no—
237#121—432no—
238#121—464no—
239#121—464no—
240#121—432no—
241C4×D4—232no—
242#117—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—432no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—232no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42#4114yes—
321#95—216no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—48no—
331#95—48no—
332#95C2×D4116yes—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—24no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—24no—
343C24—21no—
344C24—21no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—28no—
352C24—24no—
353C24—28no—
354C24—24no—
355C24—28no—
356C24—28no—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24H(5,𝔽2)18yes—
362C24C2511yescharacteristic
363C2×D4—28no—
364C2×D4—216no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—432no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—432no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—816no—
385C2×D4—816no—
386C2×D4—816no—
387C2×D4—88no—
388C2×D4—816no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—216no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—432no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—416no—
456C22×C4—88no—
457C22×C4—816no—
458C22×C4#121116yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D418yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+412yes—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—432no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—816no—
497C22⋊C4—88no—
498C22⋊C4—816no—
499C42—216no—
500C42—28no—
501C42—28no—
502C42—44no—
503C42—44no—
504C42#121116yescomplemented
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—48no—
539C23—44no—
540C23—44no—
541C23—48no—
542C23—48no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—44no—
552C23—44no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23C2×#12114yes—
559C23#53518yes—
560C23#53518yes—
561C23#54018yes—
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C2×H(5,𝔽2)14yes—
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×2−1+411yescharacteristic
570C23#58514yes—
571D4—416no—
572D4—832no—
573D4—832no—
574D4—832no—
575D4—816no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—232no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—216no—
598C2×C4—24no—
599C2×C4—432no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—832no—
635C2×C4C2×#12118yes—
636C2×C4C2×#12114yes—
637C2×C4C2×#12114yes—
638C2×C4C2×#12114yes—
639C2×C4#53714yes—
640C2×C4D4214yes—
641C2×C4#54714yes—
642C2×C4#54814yes—
643C2×C4#55314yes—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—24no—
649C22—28no—
650C22—28no—
651C22—28no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—84no—
662C22—88no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—88no—
667C22—84no—
668C22—88no—
669C22—88no—
670C22—84no—
671C22—84no—
672C22#261918yescomplemented
673C22C22×#12111yescharacteristic
674C22C22×#12111yescharacteristic
675C22C2×#53512yes—
676C22C2×#53512yes—
677C22C2×#53711yescharacteristic
678C22#344912yes—
679C22C2×#54012yes—
680C22C2×D4211yescharacteristic
681C22C2×#54711yescharacteristic
682C22C2×#54811yescharacteristic
683C22C2×#55311yescharacteristic
684C22#348814yes—
685C22#349414yes—
686C22#350012yes—
687C22#351212yes—
688C22#355814yes—
689C22#355914yes—
690C4—24no—
691C4—28no—
692C4—42no—
693C4—48no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—84no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4#241618yes—
702C2—28no—
703C2—82no—
704C2—164no—
705C2C2×#261911yescharacteristic
706C2C2×#241611yescharacteristic
707C2#3525212yes—
708C2#3532811yescharacteristic
709C2#3755412yes—
710C2#3767712yes—
711C2#3770612yes—
712C2#3786412yes—
713C2#3789312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2856C2semidirect
2. C2×#2416C2semidirect
3. C2×#2619C2semidirect
4. C2×#2290C2semidirect
5. C22×#382C2semidirect
6. #16519C2semidirect
7. #16516C2semidirect
8. #15222C2semidirect
9. #15219C2semidirect
10. #14757C2semidirect
11. #14747C2semidirect
12. #14340C2semidirect
13. #14277C2semidirect
14. #14272C2semidirect
15. #14137C2semidirect
94. C2×C4×D4D4semidirect
125. C22×C42D4semidirect
152. #382D4semidirect
504. C42#121semidirect
672. C22#2619semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#24162
C2×#26192
C2×#22902
C22×#3822
#165192
#165162
#152222
#152192
#147572
#147472
#143402
#142772
#142722
#141372