Finite Group Databasebeta

← #640983

Subgroups — #640983

Order 512.

4020 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×#2856C211yesmaximal, characteristic, complemented
2C2×#2414C212yesmaximal, complemented
3C2×#2619C211yesmaximal, characteristic, complemented
4C2×#2290C211yesmaximal, characteristic, complemented
5C2×#2285C211yesmaximal, characteristic, complemented
6#16261C214yesmaximal, complemented
7#16260C214yesmaximal, complemented
8#15134C212yesmaximal, complemented
9#15130C212yesmaximal, complemented
10#14815C214yesmaximal, complemented
11#14813C214yesmaximal, complemented
12#14340C211yesmaximal, characteristic, complemented
13#14277C211yesmaximal, characteristic, complemented
14#14272C211yesmaximal, characteristic, complemented
15#14136C212yesmaximal, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2214yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2212yes—
23C23×C42C2211yescharacteristic
24#2856—28no—
25#2619—28no—
26#2414—216no—
27C2×#396C2218yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2218yes—
31C2×#392C2214yes—
32C2×#392C2214yes—
33C2×#390C2212yes—
34C2×#390C2214yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2214yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40#2290—28no—
41C2×#388C2214yes—
42C2×#388C2212yes—
43C2×#387C2212yes—
44C2×#386C2212yes—
45C2×#385C2212yes—
46#2285—28no—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2214yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2214yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×#376C2211yescharacteristic
58#1444C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1439C2214yes—
62#1439C2218yes—
63#1439C2218yes—
64#1438C2218yes—
65#1438C2214yes—
66#1438C2214yes—
67#1437C2218yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2314yes—
73C23×D4C2314yes—
74C23×D4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C2×#127—24no—
79C2×#127—24no—
80C2×#127—24no—
81C2×#121—216no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—416no—
88C2×#121—44no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4D418yescomplemented
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2318yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2318yes—
121C22×C42—24no—
122C22×C42—24no—
123C22×C42D418yescomplemented
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#396—416no—
132#394—216no—
133#394—216no—
134#394—416no—
135#392—216no—
136#392—216no—
137#390—28no—
138#390—216no—
139#390—28no—
140#390—216no—
141#390—28no—
142#390—44no—
143#389—28no—
144#388—216no—
145#388—28no—
146#387—28no—
147#386—28no—
148#385—28no—
149#384—28no—
150#384—24no—
151#382—24no—
152#382D418yescomplemented
153#380—28no—
154#380—216no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—216no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159#376—24no—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
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—24no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—44no—
192C22×D4—48no—
193C22×D4—44no—
194C22×D4—48no—
195C22×D4—48no—
196C22×D4—48no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4C2×D414yes—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2×D418yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4#4114yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231#127—416no—
232#121—432no—
233#121—464no—
234#121—432no—
235C4×D4—232no—
236#117—432no—
237#117—416no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—232no—
306C2×C42—22no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—44no—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42#4114yes—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—48no—
330#95—48no—
331#95C2×D4116yes—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—24no—
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—28no—
353C24—24no—
354C24—24no—
355C24—28no—
356C24—28no—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24C2511yescharacteristic
362C2×D4—28no—
363C2×D4—48no—
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—432no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—816no—
381C2×D4—816no—
382C2×D4—816no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—816no—
387C2×D4—816no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—216no—
420C22×C4—28no—
421C22×C4—216no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4#121116yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D418yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)18yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—432no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—88no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—432no—
490C22⋊C4—816no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—816no—
494C42—216no—
495C42—28no—
496C42—28no—
497C42—416no—
498C42—44no—
499C42—44no—
500C42#121116yes—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—48no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—44no—
551C23—44no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23C2×#12114yes—
558C23C23×D411yescharacteristic
559C23C23×D411yescharacteristic
560C23C23×D411yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4111yescharacteristic
563C23C2×H(5,𝔽2)14yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23#58514yes—
567D4—416no—
568D4—832no—
569D4—832no—
570D4—832no—
571D4—816no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—232no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—216no—
595C2×C4—24no—
596C2×C4—432no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4C2×#12118yes—
630C2×C4C2×#12114yes—
631C2×C4C2×#12114yes—
632C2×C4C2×#12114yes—
633C2×C4#53518yes—
634C2×C4#53714yes—
635C2×C4D4214yes—
636C2×C4#54714yes—
637C2×C4#54814yes—
638C2×C4#55314yes—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—88no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—88no—
662C22—84no—
663C22—88no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—88no—
668C22C22×#12111yescharacteristic
669C22C22×#12111yescharacteristic
670C22C2×#53512yes—
671C22C2×#53711yescharacteristic
672C22#344912yes—
673C22#344912yes—
674C22#345512yes—
675C22C2×D4211yescharacteristic
676C22C2×#54711yescharacteristic
677C22C2×#54811yescharacteristic
678C22C2×#55311yescharacteristic
679C22#348814yes—
680C22#349414yes—
681C22#349812yes—
682C22#350812yes—
683C22#354614yes—
684C22#354714yes—
685C4—24no—
686C4—28no—
687C4—28no—
688C4—48no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—84no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4#241618yes—
697C2—44no—
698C2—82no—
699C2—164no—
700C2#3463311yescharacteristic
701C2C2×#241611yescharacteristic
702C2#3525212yes—
703C2#3527211yescharacteristic
704C2#3762112yes—
705C2#3768112yes—
706C2#3771012yes—
707C2#3786912yes—
708C2#3789812yes—
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×#2414C2semidirect
3. C2×#2619C2semidirect
4. C2×#2290C2semidirect
5. C2×#2285C2semidirect
6. #16261C2semidirect
7. #16260C2semidirect
8. #15134C2semidirect
9. #15130C2semidirect
10. #14815C2semidirect
11. #14813C2semidirect
12. #14340C2semidirect
13. #14277C2semidirect
14. #14272C2semidirect
15. #14136C2semidirect
91. C2×C4×D4D4semidirect
123. C22×C42D4semidirect
152. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#24142
C2×#26192
C2×#22902
C2×#22852
#162612
#162602
#151342
#151302
#148152
#148132
#143402
#142772
#142722
#141362