Finite Group Databasebeta

← #641784

Subgroups — #641784

Order 512.

4890 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#386C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#16895C214yesmaximal, complemented
4#16398C214yesmaximal, complemented
5#15091C212yesmaximal, complemented
6#15015C212yesmaximal, complemented
7#14921C214yesmaximal, complemented
8#14777C212yesmaximal, complemented
9#14734C212yesmaximal, complemented
10#14680C211yesmaximal, characteristic, complemented
11#14338C212yesmaximal, complemented
12#14275C212yesmaximal, complemented
13#14127C212yesmaximal, complemented
14#13867C212yesmaximal
15C24×D4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic, complemented
18C23×C4⋊C4C2211yescharacteristic, complemented
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C2×#399C2214yescomplemented
24C2×#386—28no—
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yescomplemented
33C2×#380—28no—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yescomplemented
38C2×#380C2212yescomplemented
39C2×#380C2212yescomplemented
40C2×#380C2212yescomplemented
41C2×#379C2212yes—
42C2×#379C2214yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2212yescomplemented
46#1445C2214yescomplemented
47#1444C2214yescomplemented
48#1443C2218yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2218yescomplemented
52#1439C2218yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2218yescomplemented
56#1439C2214yescomplemented
57#1439C2218yescomplemented
58#1438C2218yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C22⋊#95C2212yes—
62C22⋊#95C2214yescomplemented
63C22⋊#95C2212yescomplemented
64C22⋊#95C2212yescomplemented
65C26C2311yescharacteristic
66C26C2311yescharacteristic
67C23×D4—24no—
68C23×D4—24no—
69C23×D4—44no—
70C23×D4—44no—
71C23×D4Q818yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4—28no—
75C24×C4—44no—
76C24×C4—44no—
77C24×C4D418yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2318yes—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—48no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2318yes—
120#399—48no—
121#386—28no—
122#386—28no—
123#386—28no—
124#386—28no—
125#386—48no—
126#386—48no—
127#385—48no—
128#385—48no—
129#380—28no—
130#380—28no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—44no—
135#380—44no—
136#379—28no—
137#379—48no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95—28no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2318yes—
150C2×#95C2314yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C25—24no—
156C25—24no—
157C25—24no—
158C25—24no—
159C25—22no—
160C25—22no—
161C25—24no—
162C25—24no—
163C25—42no—
164C25—42no—
165C25—42no—
166C25—42no—
167C25—42no—
168C25—42no—
169C25C2×D414yes—
170C25C2×Q814yes—
171C25C2×Q814yes—
172C25#4114yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×D4—232no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—432no—
187C22×D4—416no—
188C22×D4—416no—
189C22×D4—432no—
190C22×D4C2×Q818yes—
191C22×D4C2×Q818yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—232no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—416no—
214C23×C4—416no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—464no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×Q814yes—
240C23×C4#4118yes—
241C23×C4C2418yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—48no—
267C2×C22⋊C4—28no—
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—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4—432no—
294C2×C22⋊C4—416no—
295C2×C22⋊C4—416no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—48no—
313#95—44no—
314#95—48no—
315#95—48no—
316#95—48no—
317#95—48no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—28no—
327C24—28no—
328C24—28no—
329C24—28no—
330C24—216no—
331C24—216no—
332C24—216no—
333C24—216no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—28no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—216no—
350C24—44no—
351C24—44no—
352C24—48no—
353C24—48no—
354C24—48no—
355C24—48no—
356C24—416no—
357C24—416no—
358C24—416no—
359C24—416no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—48no—
364C24—44no—
365C24—48no—
366C24—44no—
367C24—44no—
368C24—48no—
369C24—88no—
370C24—88no—
371C24—88no—
372C24—88no—
373C24C22⋊Q8116yes—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×Q814yes—
378C24C22×Q812yes—
379C24C22×Q812yes—
380C24C22×Q812yes—
381C24C2×#4112yes—
382C24C2×#4112yes—
383C24C2×#4112yes—
384C24C2×#4114yes—
385C24H(5,𝔽2)14yes—
386C24H(5,𝔽2)14yes—
387C24H(5,𝔽2)14yes—
388C24C2511yescharacteristic
389C2×D4—216no—
390C2×D4—216no—
391C2×D4—232no—
392C2×D4—416no—
393C2×D4—4128no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—416no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—416no—
400C22×C4—232no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—4128no—
463C22×C4—416no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—44no—
479C22×C4—88no—
480C22×C4—816no—
481C22×C4—816no—
482C22×C4—88no—
483C22×C4#123116yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×Q814yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4H(5,𝔽2)14yes—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—88no—
497C4⋊C4—816no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—88no—
509C22⋊C4—816no—
510C22⋊C4—816no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—232no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—232no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—416no—
554C23—416no—
555C23—464no—
556C23—416no—
557C23—416no—
558C23—416no—
559C23—464no—
560C23—416no—
561C23—416no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—48no—
575C23—48no—
576C23—44no—
577C23—416no—
578C23—48no—
579C23—48no—
580C23—832no—
581C23—832no—
582C23—88no—
583C23—88no—
584C23—88no—
585C23—88no—
586C23C2×C22⋊Q814yes—
587C23C2×C22⋊Q814yes—
588C23C2×C22⋊Q814yes—
589C23C2×#12314yes—
590C23#53414yes—
591C23#53514yes—
592C23#53514yes—
593C23#53818yes—
594C23#54318yes—
595C23#55318yes—
596C23#55418yes—
597C23C23×D411yescharacteristic
598C23C23×Q811yescharacteristic
599C23C22×#4111yescharacteristic
600C23C22×#4112yes—
601C23C2×H(5,𝔽2)12yes—
602C23C2×H(5,𝔽2)12yes—
603C23C2×H(5,𝔽2)12yes—
604C23C2×H(5,𝔽2)12yes—
605C23#58512yes—
606D4—432no—
607D4—432no—
608C2×C4—232no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—28no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—48no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—48no—
657C2×C4—48no—
658C2×C4—816no—
659C2×C4—816no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4C2×#12314yes—
663C2×C4C2×#12314yes—
664C2×C4#53414yes—
665C22—216no—
666C22—216no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—216no—
679C22—216no—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—464no—
685C22—48no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—48no—
691C22—44no—
692C22—44no—
693C22—416no—
694C22—44no—
695C22—832no—
696C22—88no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22#248718yes—
702C22#262618yes—
703C22C22×C22⋊Q811yescharacteristic
704C22C22×#12311yescharacteristic
705C22C2×#53411yescharacteristic
706C22C2×#53411yescharacteristic
707C22C2×#53511yescharacteristic
708C22C2×#53511yescharacteristic
709C22#344911yescharacteristic
710C22C2×#53812yes—
711C22C2×#54312yes—
712C22C2×#55312yes—
713C22C2×#55412yes—
714C22#349014yes—
715C22#349414yes—
716C22#350112yes—
717C22#350512yes—
718C22#353414yes—
719C22#353714yes—
720C4—24no—
721C4—24no—
722C4—24no—
723C4—42no—
724C4—44no—
725C4—88no—
726C4—84no—
727C4—82no—
728C4—84no—
729C4—82no—
730C2—28no—
731C2—28no—
732C2—44no—
733C2—84no—
734C2C2×#262611yescharacteristic
735C2C2×#248711yescharacteristic
736C2#3499211yescharacteristic
737C2#3501011yescharacteristic
738C2#3526211yescharacteristic
739C2#3744011yescharacteristic
740C2#3745011yescharacteristic
741C2#3787214yes—
742C2#3789114yes—
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
2. C22×#380C2semidirect
3. #16895C2semidirect
4. #16398C2semidirect
5. #15091C2semidirect
6. #15015C2semidirect
7. #14921C2semidirect
8. #14777C2semidirect
9. #14734C2semidirect
10. #14680C2semidirect
11. #14338C2semidirect
12. #14275C2semidirect
13. #14127C2semidirect
17. C23×C4⋊C4C22semidirect
18. C23×C4⋊C4C22semidirect
23. C2×#399C22semidirect
32. C2×#385C22semidirect
37. C2×#380C22semidirect
38. C2×#380C22semidirect
39. C2×#380C22semidirect
40. C2×#380C22semidirect
45. C22×#95C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
62. C22⋊#95C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3862
C22×#3802
#168952
#163982
#150912
#150152
#149212
#147772
#147342
#146802
#143382
#142752
#141272
#138672