Finite Group Databasebeta

← #639830

Subgroups — #639830

Order 512.

3977 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2849C211yesmaximal, characteristic
2C2×#2622C211yesmaximal, characteristic
3C2×#2280C211yesmaximal, characteristic, complemented
4C22×#381C211yesmaximal, characteristic, complemented
5#16397C214yesmaximal, complemented
6#16396C214yesmaximal, complemented
7#14919C212yesmaximal, complemented
8#14912C212yesmaximal, complemented
9#14838C214yesmaximal, complemented
10#14837C214yesmaximal, complemented
11#14341C211yesmaximal, characteristic, complemented
12#14262C211yesmaximal, characteristic, complemented
13#14255C211yesmaximal, characteristic
14#14135C212yesmaximal, complemented
15#13962C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#120C2211yescharacteristic
20C22×#117C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2849—28no—
24#2622—28no—
25C2×#397C2214yes—
26C2×#394C2214yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35#2280—28no—
36C2×#381—28no—
37C2×#381C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#381C2214yes—
40C2×#381C2212yes—
41C2×#381C2214yes—
42C2×#380C2214yes—
43C2×#379C2211yescharacteristic
44C2×#379C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×C4×C22⋊C4C2214yes—
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1445C2214yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yescomplemented
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1438C2214yes—
67#1437C2214yescomplemented
68#1436C2214yescomplemented
69C4⋊#95C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C26C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—28no—
81C24×C4—44no—
82C24×C4D418yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C2×#123—24no—
90C2×#123—24no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#120—24no—
94C2×#120—24no—
95C2×#120—24no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2314yes—
128#397—216no—
129#394—216no—
130#386—28no—
131#386—28no—
132#386—28no—
133#386—44no—
134#385—28no—
135#385—28no—
136#385—28no—
137#385—44no—
138#381—24no—
139#381—24no—
140#381—216no—
141#381—216no—
142#381—28no—
143#380—216no—
144#379—24no—
145#379—24no—
146#379D418yes—
147C4×C22⋊C4—216no—
148C2×#95—28no—
149C2×#95—28no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25—24no—
173C25—24no—
174C25—44no—
175C25—44no—
176C25C2×D414yes—
177C25C2×D414yes—
178C25#4114yes—
179C25C2411yescharacteristic
180C25C2412yes—
181C25C2411yescharacteristic
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—44no—
191C22×D4—44no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—232no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—232no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—416no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—432no—
225C23×C4—432no—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×D418yes—
229C23×C4C2×D414yes—
230C23×C4#4118yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243#123—416no—
244#123—416no—
245#120—416no—
246#117—432no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—416no—
302C2×C22⋊C4—416no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×D418yes—
309C2×C22⋊C4C2×D4116yes—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—44no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—24no—
336C24—24no—
337C24—28no—
338C24—28no—
339C24—216no—
340C24—216no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—21no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—48no—
357C24—48no—
358C24—416no—
359C24—416no—
360C24—416no—
361C24—416no—
362C24—416no—
363C24—416no—
364C24#121116yes—
365C24C22×D414yes—
366C24C22×D414yes—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C2×#4114yes—
370C24C2×#4112yes—
371C24C2×#4114yes—
372C24H(5,𝔽2)14yes—
373C24C2511yescharacteristic
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—88no—
383C2×D4—88no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
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—24no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—4128no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4#123116yes—
479C22×C4C22×D414yes—
480C22×C4C22×D418yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C4⋊C4—416no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—416no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—232no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—432no—
533C23—464no—
534C23—432no—
535C23—416no—
536C23—416no—
537C23—464no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23C2×#12118yes—
560C23C2×#12114yes—
561C23C2×#12314yes—
562C23#53414yes—
563C23#54618yes—
564C23#55118yes—
565C23C23×D411yescharacteristic
566C23C23×D411yescharacteristic
567C23C22×#4111yescharacteristic
568C23C22×#4112yes—
569C23C2×H(5,𝔽2)12yes—
570C23#58514yes—
571C23#58514yes—
572D4—816no—
573D4—88no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—232no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—432no—
590C2×C4—432no—
591C2×C4—416no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
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—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—816no—
643C2×C4—816no—
644C2×C4C2×#12314yes—
645C2×C4C2×#12314yes—
646C22—28no—
647C22—216no—
648C22—28no—
649C22—24no—
650C22—24no—
651C22—216no—
652C22—28no—
653C22—28no—
654C22—28no—
655C22—464no—
656C22—48no—
657C22—48no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—84no—
671C22#241218yes—
672C22C22×#12111yescharacteristic
673C22C22×#12311yescharacteristic
674C22C2×#53411yescharacteristic
675C22#344912yes—
676C22#344912yes—
677C22#345514yes—
678C22C2×#54612yes—
679C22C2×#55112yes—
680C22#348514yes—
681C22#349114yes—
682C22#349314yes—
683C22#353314yes—
684C22#353314yes—
685C4—24no—
686C4—24no—
687C4—48no—
688C4—44no—
689C4—84no—
690C4—82no—
691C4—84no—
692C4—82no—
693C4—84no—
694C4—81no—
695C4—81no—
696C4—81no—
697C2—28no—
698C2—48no—
699C2—161no—
700C2#3463511yescharacteristic
701C2C2×#241211yescharacteristic
702C2#3508912yes—
703C2#3515711yescharacteristic
704C2#3744312yes—
705C2#3766714yes—
706C2#3785514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2280C2semidirect
4. C22×#381C2semidirect
5. #16397C2semidirect
6. #16396C2semidirect
7. #14919C2semidirect
8. #14912C2semidirect
9. #14838C2semidirect
10. #14837C2semidirect
11. #14341C2semidirect
12. #14262C2semidirect
14. #14135C2semidirect
15. #13962C2semidirect
49. #1445C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
61. #1440C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
67. #1437C22semidirect
68. #1436C22semidirect
69. C4⋊#95C22semidirect
70. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
C2×#26222
C2×#22802
C22×#3812
#163972
#163962
#149192
#149122
#148382
#148372
#143412
#142622
#142552
#141352
#139622