Finite Group Databasebeta

← #4388020

Subgroups — #4388020

Order 512.

3646 conjugacy classes of subgroups.

The subgroup lattice has 771 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2840C211yesmaximal, characteristic, complemented
2C2×#2405C211yesmaximal, characteristic
3#16368C214yesmaximal, complemented
4#15668C214yesmaximal, complemented
5#15468C214yesmaximal, complemented
6#15467C214yesmaximal, complemented
7#15371C214yesmaximal, complemented
8#14817C214yesmaximal, complemented
9#14341C211yesmaximal, characteristic, complemented
10#14335C211yesmaximal, characteristic
11#13869C211yesmaximal, characteristic, complemented
12#13867C211yesmaximal, characteristic
13#13775C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#123C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2840—28no—
19#2405—28no—
20C2×#394C2214yes—
21C2×#393C2214yes—
22C2×#393C2214yescomplemented
23C2×#393C2214yescomplemented
24C2×#393C2214yescomplemented
25C2×#393C2214yescomplemented
26C2×#391C2214yescomplemented
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#381C2212yes—
32C2×#381C2212yescomplemented
33C2×#379C2211yescharacteristic
34C2×#379C2211yescharacteristic
35C2×#379C2211yescharacteristic
36C2×#379C2212yes—
37C2×#379C2212yes—
38C2×#379C2212yes—
39C2×#379C2211yescharacteristic
40C2×C4×C22⋊C4C2212yescomplemented
41#1446C2214yescomplemented
42#1445C2214yescomplemented
43#1443C2214yescomplemented
44#1443C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1437C2214yes—
58#1437C2214yescomplemented
59#1437C2214yescomplemented
60#1437C2214yescomplemented
61#1434C2214yescomplemented
62C22⋊#95C2212yes—
63C22⋊#95C2212yescomplemented
64C22⋊#95C2212yescomplemented
65C22⋊#95C2212yescomplemented
66C22⋊#95C2212yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8C2314yes—
72C23×D4—24no—
73C23×D4—24no—
74C23×D4—24no—
75C23×D4—44no—
76C23×D4—44no—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—44no—
86C2×#123—44no—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2314yes—
116#394—216no—
117#393—216no—
118#393—216no—
119#393—216no—
120#393—48no—
121#393—48no—
122#391—48no—
123#386—28no—
124#386—44no—
125#386—44no—
126#385—28no—
127#381—44no—
128#381—44no—
129#379—24no—
130#379—24no—
131#379—24no—
132#379—28no—
133#379—44no—
134#379—44no—
135#379D418yes—
136C4×C22⋊C4—44no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25—22no—
157C25—24no—
158C25—22no—
159C25—22no—
160C25—42no—
161C25—42no—
162C25—42no—
163C25—42no—
164C25—42no—
165C25—42no—
166C25—42no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25—42no—
171C25C2×D414yes—
172C25#4114yes—
173C25#4114yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—28no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—416no—
200C22×D4—432no—
201C22×D4—432no—
202C22×D4—416no—
203C22×D4—416no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
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—42no—
226C23×C4—42no—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236#123—432no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—44no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—416no—
277C2×C22⋊C4—416no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4C2×D418yes—
307C2×C22⋊C4C2×D418yes—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—44no—
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—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—28no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—44no—
360C24—48no—
361C24—48no—
362C24—44no—
363C24—44no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—48no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—48no—
372C24—48no—
373C24—44no—
374C24—48no—
375C24—48no—
376C24—44no—
377C24—82no—
378C24—82no—
379C24—82no—
380C24—82no—
381C24—84no—
382C24—84no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24—88no—
388C24—88no—
389C24—88no—
390C24—88no—
391C24#124116yes—
392C24C22×D412yes—
393C24C22×D412yes—
394C24C22×D412yes—
395C24C2×#4112yes—
396C24C2×#4112yes—
397C24C2×#4112yes—
398C24C2×#4112yes—
399C24C2×#4112yes—
400C24C2×#4112yes—
401C24H(5,𝔽2)14yes—
402C24C2511yescharacteristic
403C2×Q8—416no—
404C2×Q8—88no—
405C2×Q8—88no—
406C2×Q8—88no—
407C2×Q8—88no—
408C2×Q8—88no—
409C2×Q8—88no—
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—4128no—
423C2×D4—48no—
424C2×D4—48no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—88no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—84no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—84no—
499C22×C4—88no—
500C22×C4C22×D418yes—
501C22×C4H(5,𝔽2)14yes—
502C22×C4H(5,𝔽2)14yes—
503C4⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C42—88no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—28no—
568C23—416no—
569C23—416no—
570C23—416no—
571C23—416no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—44no—
586C23—42no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—88no—
603C23—88no—
604C23—88no—
605C23—88no—
606C23—88no—
607C23—88no—
608C23—832no—
609C23—88no—
610C23—832no—
611C23—88no—
612C23—832no—
613C23—832no—
614C23—88no—
615C23—88no—
616C23C2×#12414yes—
617C23C2×#12414yes—
618C23C2×#12414yes—
619C23C23×D411yescharacteristic
620C23C22×#4111yescharacteristic
621C23C22×#4111yescharacteristic
622C23C2×H(5,𝔽2)12yes—
623C23C2×H(5,𝔽2)12yes—
624C23C2×H(5,𝔽2)12yes—
625C23#58512yes—
626C23#58512yes—
627C23#58512yes—
628Q8—832no—
629D4—432no—
630D4—432no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—416no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
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—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—88no—
688C2×C4—88no—
689C2×C4—88no—
690C2×C4—88no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—48no—
707C22—48no—
708C22—48no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—48no—
713C22—48no—
714C22—48no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—44no—
719C22—44no—
720C22—48no—
721C22—48no—
722C22—44no—
723C22—832no—
724C22—832no—
725C22—82no—
726C22—84no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—84no—
731C22C22×#12411yescharacteristic
732C22#345912yes—
733C22#348412yes—
734C22#348512yes—
735C22#349011yescharacteristic
736C22#349011yescharacteristic
737C22#349012yes—
738C22#349011yescharacteristic
739C22#349111yescharacteristic
740C22#349111yescharacteristic
741C22#349112yes—
742C22#349111yescharacteristic
743C22#349612yes—
744C22#354014yes—
745C22#354514yes—
746C22#355814yes—
747C22#355814yes—
748C4—44no—
749C4—44no—
750C4—48no—
751C4—82no—
752C4—82no—
753C4—84no—
754C4—84no—
755C4—82no—
756C4—84no—
757C2—44no—
758C2—44no—
759C2—44no—
760C2—84no—
761C2#3499311yescharacteristic
762C2#3536711yescharacteristic
763C2#3545511yescharacteristic
764C2#4093214yes—
765C2#4638511yescharacteristic
766C2#4638511yescharacteristic
767C2#4683711yescharacteristic
768C2#4699811yescharacteristic
769C2#4793314yes—
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
1. C2×#2840C2semidirect
3. #16368C2semidirect
4. #15668C2semidirect
5. #15468C2semidirect
6. #15467C2semidirect
7. #15371C2semidirect
8. #14817C2semidirect
9. #14341C2semidirect
11. #13869C2semidirect
13. #13775C2semidirect
22. C2×#393C22semidirect
23. C2×#393C22semidirect
24. C2×#393C22semidirect
25. C2×#393C22semidirect
26. C2×#391C22semidirect
32. C2×#381C22semidirect
40. C2×C4×C22⋊C4C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
58. #1437C22semidirect
59. #1437C22semidirect
60. #1437C22semidirect
61. #1434C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28402
C2×#24052
#163682
#156682
#154682
#154672
#153712
#148172
#143412
#143352
#138692
#138672
#137752