Finite Group Databasebeta

← #4307986

Subgroups — #4307986

Order 512.

6973 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic, complemented
2C2×#2853C211yesmaximal, characteristic, complemented
3C2×#2623C211yesmaximal, characteristic, complemented
4C2×#2613C211yesmaximal, characteristic
5#14860C214yesmaximal, complemented
6#14801C214yesmaximal, complemented
7#14796C214yesmaximal, complemented
8#14795C214yesmaximal, complemented
9#14694C212yesmaximal, complemented
10#14693C212yesmaximal, complemented
11#14684C211yesmaximal, characteristic, complemented
12#14342C211yesmaximal, characteristic, complemented
13#14335C211yesmaximal, characteristic, complemented
14#14262C212yesmaximal, complemented
15#14255C212yesmaximal, complemented
16C24×D4C2211yescharacteristic
17C25×C4C2212yes—
18C22×#123C2211yescharacteristic
19C22×#120C2212yes—
20C22×C4×D4—28no—
21C22×C4×D4—28no—
22C22×C4×D4—28no—
23C22×C4×D4C2214yes—
24C22×C4×D4C2214yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28#2853—28no—
29#2623—28no—
30#2613—28no—
31C2×#400C2212yescomplemented
32C2×#399C2212yescomplemented
33C2×#398C2212yescomplemented
34C2×#397C2212yescomplemented
35C2×#394C2212yes—
36C2×#394C2212yes—
37C2×#392C2212yes—
38C2×#392C2212yes—
39C2×#388C2212yescomplemented
40C2×#388C2212yescomplemented
41C2×#387C2212yescomplemented
42C2×#387C2212yescomplemented
43C2×#386C2211yescharacteristic
44C2×#386C2211yescharacteristic
45C2×#386C2212yes—
46C2×#386C2218yescomplemented
47C2×#385C2211yescharacteristic
48C2×#385C2211yescharacteristic
49C2×#385C2212yes—
50C2×#385C2218yescomplemented
51C2×#381C2211yescharacteristic
52C2×#381C2211yescharacteristic
53C2×#381C2211yescharacteristic
54C2×#379C2212yes—
55C2×#379C2212yes—
56C2×#379C2212yes—
57#1445C2214yescomplemented
58#1445C2214yescomplemented
59#1440C2218yescomplemented
60#1439C2218yescomplemented
61#1439C2218yescomplemented
62#1438C2214yes—
63#1438C2214yes—
64#1438C2218yescomplemented
65#1434C2214yescomplemented
66#1434C2214yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2218yescomplemented
72C22⋊#95C2218yescomplemented
73C26C2312yes—
74C23×D4—28no—
75C23×D4—28no—
76C23×D4—24no—
77C23×D4—24no—
78C23×D4D418yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2312yes—
82C23×D4C2314yes—
83C24×C4—28no—
84C24×C4—28no—
85C24×C4—28no—
86C24×C4—416no—
87C24×C4C2311yescharacteristic
88C24×C4C2318yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—24no—
95C2×#120—28no—
96C2×#120—28no—
97C2×#120—28no—
98C2×C4×D4—2128no—
99C2×C4×D4—216no—
100C2×C4×D4—216no—
101C2×C4×D4—216no—
102C2×C4×D4—216no—
103C2×C4×D4—216no—
104C2×C4×D4—216no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4D418yescomplemented
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—28no—
116C22×C22⋊C4—28no—
117C22×C22⋊C4—28no—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2318yes—
132C22×C22⋊C4C2318yes—
133C22×C22⋊C4C2312yes—
134C22×C42—24no—
135C22×C42—24no—
136C22×C42D418yescomplemented
137C22×C42C2312yes—
138C22×C42C2312yes—
139#400—28no—
140#399—28no—
141#398—28no—
142#397—28no—
143#394—28no—
144#394—28no—
145#392—28no—
146#392—28no—
147#388—28no—
148#388—28no—
149#387—28no—
150#387—28no—
151#386—24no—
152#386—24no—
153#386—28no—
154#386—416no—
155#385—24no—
156#385—24no—
157#385—28no—
158#385—416no—
159#381—24no—
160#381—24no—
161#381—24no—
162#379—28no—
163#379—28no—
164#379—28no—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2318yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2318yes—
179C2×#95C2318yes—
180C2×#95C2318yes—
181C25—24no—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—48no—
186C25—48no—
187C25C2×D418yes—
188C25#4118yes—
189C25#4118yes—
190C25C2411yescharacteristic
191C25C2414yes—
192C25C2411yescharacteristic
193C22×D4—232no—
194C22×D4—232no—
195C22×D4—264no—
196C22×D4—264no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—28no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—28no—
208C22×D4—28no—
209C22×D4—232no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—48no—
215C22×D4—48no—
216C22×D4C2×D418yes—
217C22×D4C2×D418yes—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—264no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—216no—
229C23×C4—216no—
230C23×C4—216no—
231C23×C4—216no—
232C23×C4—216no—
233C23×C4—48no—
234C23×C4—48no—
235C23×C4—48no—
236C23×C4—44no—
237C23×C4—464no—
238C23×C4—464no—
239C23×C4—464no—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D418yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2414yes—
253C23×C4C2412yes—
254C23×C4C2414yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258#123—416no—
259#120—432no—
260C4×D4—2128no—
261C4×D4—2128no—
262C2×C4⋊C4—232no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—264no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—216no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—48no—
323C2×C42—232no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42C2×D418yes—
329C2×C42C2×D418yes—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—216no—
343#95—48no—
344#95—48no—
345#95—48no—
346C24—24no—
347C24—24no—
348C24—216no—
349C24—216no—
350C24—216no—
351C24—216no—
352C24—216no—
353C24—216no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—28no—
359C24—28no—
360C24—28no—
361C24—28no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—416no—
370C24—416no—
371C24—432no—
372C24—432no—
373C24—416no—
374C24—416no—
375C24—416no—
376C24—432no—
377C24—416no—
378C24—432no—
379C24#123132yes—
380C24C22×D418yes—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C2×#4118yes—
385C24C2×#4118yes—
386C24C2×#4114yes—
387C24C2×#4114yes—
388C24C2511yescharacteristic
389C2×D4—2256no—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—216no—
393C2×D4—216no—
394C2×D4—232no—
395C2×D4—232no—
396C2×D4—232no—
397C2×D4—232no—
398C2×D4—216no—
399C2×D4—216no—
400C2×D4—432no—
401C2×D4—44no—
402C2×D4—416no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—816no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—28no—
413C22×C4—216no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—216no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—264no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—416no—
468C22×C4—48no—
469C22×C4—432no—
470C22×C4—432no—
471C22×C4—432no—
472C22×C4—432no—
473C22×C4—4256no—
474C22×C4—432no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—432no—
480C22×C4#120116yes—
481C22×C4C22×D414yes—
482C22×C4C22×D418yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4H(5,𝔽2)18yes—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C22⋊C4—264no—
499C22⋊C4—48no—
500C22⋊C4—432no—
501C22⋊C4—44no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—816no—
514C42—216no—
515C42—216no—
516C23—24no—
517C23—24no—
518C23—216no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—216no—
524C23—264no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—216no—
532C23—216no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—216no—
538C23—216no—
539C23—464no—
540C23—432no—
541C23—464no—
542C23—464no—
543C23—432no—
544C23—464no—
545C23—464no—
546C23—464no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—41no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—48no—
568C23C2×#12014yes—
569C23C2×#123116yes—
570C23C2×#12318yes—
571C23#52514yes—
572C23#52514yes—
573C23#52518yes—
574C23C23×D411yescharacteristic
575C23C23×D411yescharacteristic
576C23C23×D411yescharacteristic
577C23C22×#4112yes—
578C23C22×#4112yes—
579C23C2×H(5,𝔽2)14yes—
580C23#58514yes—
581D4—264no—
582D4—264no—
583D4—832no—
584D4—88no—
585C2×C4—232no—
586C2×C4—232no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—4128no—
606C2×C4—416no—
607C2×C4—432no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4C2×#12014yes—
644C2×C4C2×#12014yes—
645C2×C4#53518yes—
646C22—28no—
647C22—28no—
648C22—24no—
649C22—24no—
650C22—216no—
651C22—216no—
652C22—216no—
653C22—216no—
654C22—216no—
655C22—4128no—
656C22—416no—
657C22—416no—
658C22—48no—
659C22—48no—
660C22—48no—
661C22—48no—
662C22—416no—
663C22—416no—
664C22—81no—
665C22—82no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—84no—
670C22—84no—
671C22#284918yes—
672C22C22×#12011yescharacteristic
673C22C22×#12312yes—
674C22C2×#52511yescharacteristic
675C22C2×#52511yescharacteristic
676C22C2×#52512yes—
677C22C2×#53512yes—
678C22#344912yes—
679C22#345918yes—
680C22#348412yes—
681C22#348412yes—
682C22#348512yes—
683C22#348712yes—
684C22#350518yes—
685C4—24no—
686C4—24no—
687C4—28no—
688C4—44no—
689C4—48no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4—81no—
694C4—81no—
695C4—84no—
696C4—84no—
697C2—28no—
698C2—416no—
699C2—161no—
700C2#3459312yes—
701C2#3498314yes—
702C2#3498714yes—
703C2#3510212yes—
704C2#3514911yescharacteristic
705C2#3522211yescharacteristic
706C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×D4C2semidirect
2. C2×#2853C2semidirect
3. C2×#2623C2semidirect
5. #14860C2semidirect
6. #14801C2semidirect
7. #14796C2semidirect
8. #14795C2semidirect
9. #14694C2semidirect
10. #14693C2semidirect
11. #14684C2semidirect
12. #14342C2semidirect
13. #14335C2semidirect
14. #14262C2semidirect
15. #14255C2semidirect
31. C2×#400C22semidirect
32. C2×#399C22semidirect
33. C2×#398C22semidirect
34. C2×#397C22semidirect
39. C2×#388C22semidirect
40. C2×#388C22semidirect
41. C2×#387C22semidirect
42. C2×#387C22semidirect
46. C2×#386C22semidirect
50. C2×#385C22semidirect
57. #1445C22semidirect
58. #1445C22semidirect
59. #1440C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
64. #1438C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
71. C22⋊#95C22semidirect
72. C22⋊#95C22semidirect
107. C22×C4⋊C4D4semidirect
136. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D42
C2×#28532
C2×#26232
C2×#26132
#148602
#148012
#147962
#147952
#146942
#146932
#146842
#143422
#143352
#142622
#142552