Finite Group Databasebeta

← #640437

Subgroups — #640437

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3#16705C212yesmaximal, complemented
4#16703C214yesmaximal, complemented
5#16700C212yesmaximal, complemented
6#15193C211yesmaximal, characteristic, complemented
7#15192C211yesmaximal, characteristic, complemented
8#15156C211yesmaximal, characteristic, complemented
9#15149C211yesmaximal, characteristic, complemented
10#15019C214yesmaximal
11#14826C212yesmaximal
12#14766C212yesmaximal
13#14266C212yesmaximal, complemented
14#14261C212yesmaximal, complemented
15#14139C212yesmaximal
16#13907C212yesmaximal
17C2×C4⋊#95C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C42C2211yescharacteristic
21#2855—28no—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#385C2212yes—
35#2283—28no—
36C2×#383C2212yes—
37C2×#382C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2212yes—
49C22×#95C2212yes—
50C22×#95C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42—28no—
115C22×C42D418yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#400—44no—
123#398—44no—
124#391—28no—
125#391—24no—
126#391—24no—
127#391—44no—
128#389—28no—
129#389—24no—
130#389—24no—
131#389—44no—
132#387—28no—
133#387—28no—
134#385—28no—
135#383—28no—
136#382—24no—
137#382—24no—
138#381—28no—
139#381—28no—
140#380—24no—
141#380—24no—
142#380—24no—
143#380—48no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—28no—
147#376—28no—
148C2×#95—44no—
149C2×#95—44no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4118yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C22⋊Q8—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4#4118yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C42—232no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
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—24no—
315#95—24no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24C22×D412yes—
338C24C2×#4114yes—
339C24H(5,𝔽2)12yes—
340C242−1+414yes—
341C242−1+412yes—
342C24C2511yescharacteristic
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—48no—
347C2×Q8—48no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
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—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4#123116yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C4⋊C4—216no—
465C4⋊C4—416no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C42—28no—
484C42—28no—
485C42—28no—
486C42—28no—
487C42—48no—
488C42—48no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23C2×#12314yes—
508C23#53614yes—
509C23#53714yes—
510C23#54118yes—
511C23#54218yes—
512C23C23×D411yescharacteristic
513C23C22×#4112yes—
514C23C22×#4112yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×2−1+412yes—
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23#58512yes—
523Q8—88no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—232no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—416no—
546C2×C4—416no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
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—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—832no—
615C2×C4C2×#12314yes—
616C2×C4C2×#12314yes—
617C2×C4#53214yes—
618C2×C4#53714yes—
619C2×C4#53714yes—
620C2×C4#55118yes—
621C2×C4#55218yes—
622C22—24no—
623C22—28no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—24no—
628C22—28no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22#262518yes—
633C22C22×#12311yescharacteristic
634C22C2×#53211yescharacteristic
635C22C2×#53611yescharacteristic
636C22C2×#53711yescharacteristic
637C22C2×#53711yescharacteristic
638C22C2×#53711yescharacteristic
639C22#344911yescharacteristic
640C22C2×#54112yes—
641C22C2×#54212yes—
642C22C2×#55112yes—
643C22C2×#55212yes—
644C22#349412yes—
645C22#349512yes—
646C22#351312yes—
647C22#351612yes—
648C22#352012yes—
649C22#352312yes—
650C22#354311yescharacteristic
651C22#354411yescharacteristic
652C22#355211yescharacteristic
653C22#355211yescharacteristic
654C22#356112yes—
655C22#356312yes—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—48no—
661C4—42no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#249818yes—
672C2—28no—
673C2C2×#262511yescharacteristic
674C2C2×#249811yescharacteristic
675C2#3523411yescharacteristic
676C2#3524111yescharacteristic
677C2#3526311yescharacteristic
678C2#3755511yescharacteristic
679C2#3762011yescharacteristic
680C2#3805412yes—
681C2#3805712yes—
682C2#3806612yes—
683C2#3806912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16705C2semidirect
4. #16703C2semidirect
5. #16700C2semidirect
6. #15193C2semidirect
7. #15192C2semidirect
8. #15156C2semidirect
9. #15149C2semidirect
13. #14266C2semidirect
14. #14261C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22832
#167052
#167032
#167002
#151932
#151922
#151562
#151492
#150192
#148262
#147662
#142662
#142612
#141392
#139072
C2×C4⋊#952