Finite Group Databasebeta

← #641058

Subgroups — #641058

Order 512.

5154 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2405C212yesmaximal
2C22×#386C211yesmaximal, characteristic
3C22×#380C211yesmaximal, characteristic, complemented
4#16816C214yesmaximal, complemented
5#16401C214yesmaximal, complemented
6#15069C212yesmaximal, complemented
7#15014C212yesmaximal, complemented
8#14927C214yesmaximal, complemented
9#14820C212yesmaximal, complemented
10#14776C212yesmaximal, complemented
11#14680C211yesmaximal, characteristic, complemented
12#14338C212yesmaximal, complemented
13#14275C212yesmaximal, complemented
14#13869C212yesmaximal, complemented
15C24×D4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×#123C2214yes—
18C23×C4⋊C4C2211yescharacteristic, complemented
19C23×C4⋊C4C2211yescharacteristic, complemented
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2405—216no—
25C2×#397C2214yescomplemented
26C2×#394C2218yes—
27C2×#386—28no—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#381C2214yescomplemented
35C2×#380—28no—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yescomplemented
40C2×#380C2212yescomplemented
41C2×#380C2212yescomplemented
42C2×#380C2212yescomplemented
43C2×#379C2214yes—
44C2×#379C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2212yescomplemented
48#1445C2214yescomplemented
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1442C2218yescomplemented
52#1442C2214yescomplemented
53#1441C2218yescomplemented
54#1440C2218yescomplemented
55#1440C2218yescomplemented
56#1439C2218yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60C22⋊#95C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2212yes—
63C22⋊#95C2214yescomplemented
64C22⋊#95C2212yescomplemented
65C22⋊#95C2212yescomplemented
66C26C2311yescharacteristic
67C26C2311yescharacteristic
68C23×D4—24no—
69C23×D4—24no—
70C23×D4—44no—
71C23×D4—44no—
72C23×D4D418yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4—44no—
78C24×C4D418yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#123—216no—
91C2×#123—48no—
92C2×#123—48no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2318yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125#397—48no—
126#394—232no—
127#386—216no—
128#386—28no—
129#386—28no—
130#386—28no—
131#386—28no—
132#385—216no—
133#381—48no—
134#380—28no—
135#380—28no—
136#380—28no—
137#380—28no—
138#380—28no—
139#380—44no—
140#380—44no—
141#379—216no—
142#379—28no—
143C2×#95—24no—
144C2×#95—24no—
145C2×#95—28no—
146C2×#95C2318yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C25—24no—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—22no—
165C25—22no—
166C25—24no—
167C25—24no—
168C25—42no—
169C25—42no—
170C25—42no—
171C25—42no—
172C25—42no—
173C25—42no—
174C25C2×D414yes—
175C25C2×D414yes—
176C25C2×D414yes—
177C25#4114yes—
178C25C2412yes—
179C25C2411yescharacteristic
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C22×D4—232no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—432no—
192C22×D4—416no—
193C22×D4—416no—
194C22×D4—432no—
195C22×D4C2×D418yes—
196C22×D4C2×D418yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—232no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—416no—
223C23×C4—416no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—44no—
237C23×C4—42no—
238C23×C4—44no—
239C23×C4—464no—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4#4118yes—
248C23×C4C2418yes—
249C23×C4C2414yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258#123—464no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—432no—
307C2×C22⋊C4—416no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—48no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—216no—
319#95—28no—
320#95—28no—
321#95—48no—
322#95—44no—
323#95—48no—
324#95—48no—
325#95—48no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—28no—
335C24—28no—
336C24—216no—
337C24—28no—
338C24—28no—
339C24—216no—
340C24—216no—
341C24—216no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—28no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—216no—
358C24—44no—
359C24—44no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—416no—
365C24—416no—
366C24—416no—
367C24—416no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—48no—
372C24—44no—
373C24—48no—
374C24—44no—
375C24—44no—
376C24—48no—
377C24—88no—
378C24—88no—
379C24—88no—
380C24—88no—
381C24#121116yes—
382C24C22×D414yes—
383C24C22×D412yes—
384C24C22×D412yes—
385C24C22×D412yes—
386C24C22×D412yes—
387C24C22×D412yes—
388C24C22×D412yes—
389C24C2×#4112yes—
390C24C2×#4112yes—
391C24C2×#4112yes—
392C24C2×#4114yes—
393C24H(5,𝔽2)14yes—
394C24H(5,𝔽2)14yes—
395C242−1+414yes—
396C24C2511yescharacteristic
397C2×D4—216no—
398C2×D4—216no—
399C2×D4—232no—
400C2×D4—416no—
401C2×D4—4128no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—416no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—416no—
408C22×C4—232no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—416no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—4128no—
471C22×C4—416no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—416no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—416no—
489C22×C4—88no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4—88no—
493C22×C4#123116yescomplemented
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C2×#4114yes—
500C22×C4C2×#4114yes—
501C22×C4H(5,𝔽2)14yes—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—432no—
506C4⋊C4—48no—
507C4⋊C4—88no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—432no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—88no—
520C22⋊C4—816no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—232no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—232no—
559C23—24no—
560C23—28no—
561C23—28no—
562C23—28no—
563C23—416no—
564C23—416no—
565C23—416no—
566C23—416no—
567C23—416no—
568C23—464no—
569C23—464no—
570C23—416no—
571C23—416no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—48no—
586C23—44no—
587C23—416no—
588C23—48no—
589C23—48no—
590C23—832no—
591C23—88no—
592C23—88no—
593C23—88no—
594C23—88no—
595C23—832no—
596C23C2×#12114yes—
597C23C2×#12114yes—
598C23C2×#12114yes—
599C23C2×#12314yes—
600C23#53418yes—
601C23#53414yes—
602C23#53514yes—
603C23#53614yes—
604C23#53818yes—
605C23#54718yes—
606C23#55218yes—
607C23C23×D411yescharacteristic
608C23C23×D411yescharacteristic
609C23C22×#4111yescharacteristic
610C23C22×#4112yes—
611C23C2×H(5,𝔽2)12yes—
612C23C2×H(5,𝔽2)12yes—
613C23C2×H(5,𝔽2)12yes—
614C23C2×2−1+412yes—
615C23#58512yes—
616D4—432no—
617D4—432no—
618C2×C4—232no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—28no—
632C2×C4—22no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—416no—
638C2×C4—416no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—48no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—48no—
658C2×C4—48no—
659C2×C4—48no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—48no—
663C2×C4—48no—
664C2×C4—48no—
665C2×C4—48no—
666C2×C4—48no—
667C2×C4—48no—
668C2×C4—88no—
669C2×C4—816no—
670C2×C4—88no—
671C2×C4—816no—
672C2×C4C2×#12314yes—
673C2×C4C2×#12314yes—
674C2×C4#53414yes—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—216no—
687C22—216no—
688C22—216no—
689C22—216no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—464no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—48no—
700C22—48no—
701C22—44no—
702C22—44no—
703C22—416no—
704C22—44no—
705C22—832no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—88no—
711C22#248618yes—
712C22#261718yes—
713C22C22×#12111yescharacteristic
714C22C22×#12311yescharacteristic
715C22C2×#53411yescharacteristic
716C22C2×#53411yescharacteristic
717C22C2×#53412yes—
718C22C2×#53511yescharacteristic
719C22C2×#53611yescharacteristic
720C22#344911yescharacteristic
721C22C2×#53812yes—
722C22C2×#54712yes—
723C22C2×#55212yes—
724C22#349014yes—
725C22#349414yes—
726C22#349612yes—
727C22#350512yes—
728C22#353514yes—
729C22#353614yes—
730C4—24no—
731C4—24no—
732C4—24no—
733C4—42no—
734C4—44no—
735C4—82no—
736C4—84no—
737C4—82no—
738C4—88no—
739C4—84no—
740C2—28no—
741C2—28no—
742C2—44no—
743C2—84no—
744C2C2×#261711yescharacteristic
745C2C2×#248611yescharacteristic
746C2#3499211yescharacteristic
747C2#3500011yescharacteristic
748C2#3526211yescharacteristic
749C2#3744111yescharacteristic
750C2#3745111yescharacteristic
751C2#3786014yes—
752C2#3788114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#380C2semidirect
4. #16816C2semidirect
5. #16401C2semidirect
6. #15069C2semidirect
7. #15014C2semidirect
8. #14927C2semidirect
9. #14820C2semidirect
10. #14776C2semidirect
11. #14680C2semidirect
12. #14338C2semidirect
13. #14275C2semidirect
14. #13869C2semidirect
18. C23×C4⋊C4C22semidirect
19. C23×C4⋊C4C22semidirect
25. C2×#397C22semidirect
34. C2×#381C22semidirect
39. C2×#380C22semidirect
40. C2×#380C22semidirect
41. C2×#380C22semidirect
42. C2×#380C22semidirect
47. C22×#95C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
493. C22×C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24052
C22×#3862
C22×#3802
#168162
#164012
#150692
#150142
#149272
#148202
#147762
#146802
#143382
#142752
#138692