Finite Group Databasebeta

← #615503

Subgroups — #615503

Order 512.

4573 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C22×#381C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊C4C211yesmaximal, characteristic, complemented
5#14974C214yesmaximal, complemented
6#14965C214yesmaximal, complemented
7#14919C212yesmaximal, complemented
8#14912C212yesmaximal, complemented
9#14706C211yesmaximal, characteristic, complemented
10#14705C211yesmaximal, characteristic, complemented
11#14684C211yesmaximal, characteristic, complemented
12#14134C212yesmaximal
13#14047C214yesmaximal, complemented
14#13999C214yesmaximal, complemented
15#13877C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#120C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2853—28no—
25#2849—28no—
26C2×#388C2214yescomplemented
27C2×#387C2214yescomplemented
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2214yes—
32C2×#386C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#381—28no—
38C2×#381C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#381C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2214yes—
43C2×#381C2211yescharacteristic
44C2×#380C2214yes—
45C2×#379C2211yescharacteristic
46C2×C4×C22⋊C4—28no—
47C2×C4×C22⋊C4C2214yes—
48C2×C4×C22⋊C4C2214yes—
49C22×#95C2212yes—
50C22×#95C2214yes—
51#1445C2214yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2214yes—
58#1440C2214yescomplemented
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yescomplemented
62#1438C2214yes—
63#1437C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1434C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C26C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—28no—
80C24×C4—24no—
81C24×C4—24no—
82C24×C4—44no—
83C24×C4D418yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2314yes—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—44no—
96C2×#120—24no—
97C2×#120—28no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—216no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C42—24no—
124C22×C42—24no—
125C22×C42—24no—
126C22×C42—24no—
127C22×C42—44no—
128C22×C42C2314yes—
129C22×C42C2314yes—
130#388—216no—
131#387—216no—
132#386—28no—
133#386—28no—
134#386—28no—
135#386—216no—
136#386—44no—
137#385—28no—
138#385—28no—
139#385—28no—
140#385—44no—
141#381—24no—
142#381—24no—
143#381—24no—
144#381—28no—
145#381—216no—
146#381D418yescomplemented
147#380—216no—
148#379—24no—
149C4×C22⋊C4—216no—
150C4×C22⋊C4—216no—
151C2×#95—28no—
152C2×#95—216no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25—22no—
176C25—28no—
177C25—22no—
178C25—24no—
179C25—44no—
180C25C2×D414yes—
181C25#4118yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C25C2411yescharacteristic
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—28no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—44no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—232no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—232no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—416no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—432no—
236C23×C4—432no—
237C23×C4C2×D414yes—
238C23×C4C2×D418yes—
239C23×C4C2×D418yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4#4114yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2414yes—
252C23×C4C2414yes—
253C23×C4C2412yes—
254#123—416no—
255#123—416no—
256#120—416no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C22⋊C4—264no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—432no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4#41116yes—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—416no—
320C2×C42—416no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C2×D4116yes—
343C24—21no—
344C24—21no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—28no—
350C24—216no—
351C24—232no—
352C24—28no—
353C24—216no—
354C24—216no—
355C24—21no—
356C24—24no—
357C24—21no—
358C24—22no—
359C24—24no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—48no—
368C24—416no—
369C24—416no—
370C24—416no—
371C24—416no—
372C24—416no—
373C24#123116yes—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2×#4114yes—
379C24C2×#4112yes—
380C24C2×#4114yes—
381C24C2×#4114yes—
382C24C2×#4118yes—
383C24H(5,𝔽2)14yes—
384C24C2511yescharacteristic
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—416no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—88no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—4128no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—416no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4—88no—
493C22×C4#121116yes—
494C22×C4C22×D414yes—
495C22×C4C22×D418yes—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C2×#4118yes—
499C22×C4C2×#4118yes—
500C22×C4C2×#4114yes—
501C4⋊C4—416no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—88no—
505C22⋊C4—232no—
506C22⋊C4—232no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C42—416no—
522C42—416no—
523C23—22no—
524C23—24no—
525C23—28no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—232no—
542C23—22no—
543C23—24no—
544C23—232no—
545C23—28no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—416no—
552C23—464no—
553C23—464no—
554C23—432no—
555C23—432no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—44no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23C2×#12114yes—
574C23C2×#12314yes—
575C23C2×#12314yes—
576C23C2×#12314yes—
577C23#52518yes—
578C23#529116yes—
579C23#53414yes—
580C23#54618yes—
581C23#55118yes—
582C23C23×D411yescharacteristic
583C23C23×D411yescharacteristic
584C23C22×#4111yescharacteristic
585C23C22×#4112yes—
586C23C22×#4114yes—
587C23C2×H(5,𝔽2)12yes—
588C23#58514yes—
589D4—88no—
590D4—88no—
591D4—88no—
592C2×C4—24no—
593C2×C4—232no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—432no—
612C2×C4—432no—
613C2×C4—432no—
614C2×C4—432no—
615C2×C4—44no—
616C2×C4—416no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—832no—
661C2×C4C2×#12118yes—
662C22—24no—
663C22—216no—
664C22—216no—
665C22—24no—
666C22—24no—
667C22—216no—
668C22—216no—
669C22—24no—
670C22—28no—
671C22—28no—
672C22—28no—
673C22—28no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—464no—
678C22—48no—
679C22—48no—
680C22—44no—
681C22—44no—
682C22—81no—
683C22—84no—
684C22—81no—
685C22—84no—
686C22—81no—
687C22—82no—
688C22—82no—
689C22#241218yes—
690C22#285318yescomplemented
691C22C22×#12111yescharacteristic
692C22C22×#12311yescharacteristic
693C22C2×#52512yes—
694C22C2×#52914yes—
695C22C2×#53411yescharacteristic
696C22#344912yes—
697C22C2×#54612yes—
698C22D4×#4114yes—
699C22C2×#55112yes—
700C22#347514yes—
701C22#349114yes—
702C22#351314yes—
703C22#351314yes—
704C4—28no—
705C4—48no—
706C4—44no—
707C4—48no—
708C4—48no—
709C4—81no—
710C4—84no—
711C4—81no—
712C4—81no—
713C4—82no—
714C4—82no—
715C2—28no—
716C2—28no—
717C2—44no—
718C2—161no—
719C2C2×#241211yescharacteristic
720C2#3473712yes—
721C2#3506312yes—
722C2#3509711yescharacteristic
723C2#3610914yes—
724C2#3616914yes—
725C2C2×#285311yescharacteristic
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
1. C2×#2853C2semidirect
3. C22×#381C2semidirect
4. C22×C4×C22⋊C4C2semidirect
5. #14974C2semidirect
6. #14965C2semidirect
7. #14919C2semidirect
8. #14912C2semidirect
9. #14706C2semidirect
10. #14705C2semidirect
11. #14684C2semidirect
13. #14047C2semidirect
14. #13999C2semidirect
15. #13877C2semidirect
26. C2×#388C22semidirect
27. C2×#387C22semidirect
51. #1445C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
58. #1440C22semidirect
61. #1439C22semidirect
63. #1437C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
146. #381D4semidirect
690. C22#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#28492
C22×#3812
C22×C4×C22⋊C42
#149742
#149652
#149192
#149122
#147062
#147052
#146842
#141342
#140472
#139992
#138772