Finite Group Databasebeta

← #4308937

Subgroups — #4308937

Order 512.

4109 conjugacy classes of subgroups.

The subgroup lattice has 781 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2613C211yesmaximal, characteristic
2C22×#390C211yesmaximal, characteristic, complemented
3#16573C214yesmaximal, complemented
4#16530C214yesmaximal, complemented
5#16527C214yesmaximal, complemented
6#16526C214yesmaximal, complemented
7#14727C212yesmaximal, complemented
8#14727C212yesmaximal, complemented
9#14718C212yesmaximal, complemented
10#14718C212yesmaximal, complemented
11#14706C211yesmaximal, characteristic, complemented
12#14705C211yesmaximal, characteristic, complemented
13#14335C211yesmaximal, characteristic, complemented
14#14263C211yesmaximal, characteristic, complemented
15#14256C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#120C2211yescharacteristic
19C22×#120C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24#2613—28no—
25C2×#397C2214yescomplemented
26C2×#397C2214yescomplemented
27C2×#394C2212yes—
28C2×#394C2212yes—
29C2×#394C2214yescomplemented
30C2×#394C2214yescomplemented
31C2×#392C2212yes—
32C2×#392C2212yes—
33C2×#390—28no—
34C2×#390C2214yes—
35C2×#390C2214yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2211yescharacteristic
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#379C2211yescharacteristic
44C2×#379C2211yescharacteristic
45C2×#379C2212yes—
46C2×#379C2212yes—
47#1444C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2214yes—
62#1438C2214yes—
63#1438C2214yescomplemented
64#1438C2214yescomplemented
65#1437C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C26C2311yescharacteristic
74C26C2311yescharacteristic
75C23×D4—24no—
76C23×D4—24no—
77C23×D4—44no—
78C23×D4—44no—
79C23×D4D418yes—
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#123—24no—
90C2×#123—24no—
91C2×#123—24no—
92C2×#120—24no—
93C2×#120—24no—
94C2×#120—44no—
95C2×#120—44no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—48no—
105C22×C22⋊C4—48no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42—24no—
130C22×C42—44no—
131C22×C42C2314yes—
132#397—48no—
133#397—48no—
134#394—28no—
135#394—28no—
136#394—48no—
137#394—48no—
138#392—28no—
139#392—28no—
140#390—216no—
141#390—216no—
142#386—44no—
143#386—44no—
144#385—44no—
145#385—44no—
146#381—24no—
147#381—24no—
148#381—24no—
149#379—24no—
150#379—24no—
151#379—44no—
152#379—44no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25—22no—
170C25—22no—
171C25—22no—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—24no—
177C25—44no—
178C25—44no—
179C25—44no—
180C25—44no—
181C25C2×D414yes—
182C25C2×D414yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C25C2412yes—
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—232no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—416no—
202C22×D4—416no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—432no—
212C22×D4—432no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4C2×D418yes—
222C22×D4C2×D418yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4#4114yes—
237C23×C4#4114yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2414yes—
247C23×C4C2412yes—
248#123—416no—
249#120—416no—
250#120—416no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—416no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
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—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—432no—
305C2×C22⋊C4—432no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C42—28no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42—416no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347C24—24no—
348C24—24no—
349C24—28no—
350C24—28no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—21no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—28no—
374C24—28no—
375C24—28no—
376C24—28no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—48no—
385C24—48no—
386C24—48no—
387C24—48no—
388C24—416no—
389C24—416no—
390C24—416no—
391C24—416no—
392C24—48no—
393C24—48no—
394C24—48no—
395C24—48no—
396C24—88no—
397C24—88no—
398C24—88no—
399C24—88no—
400C24—88no—
401C24—88no—
402C24—88no—
403C24—88no—
404C24C22×D412yes—
405C24C22×D414yes—
406C24C22×D412yes—
407C24C22×D412yes—
408C24C2×#4112yes—
409C24C2×#4112yes—
410C24H(5,𝔽2)14yes—
411C24H(5,𝔽2)14yes—
412C24C2511yescharacteristic
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—4128no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—44no—
429C2×D4—48no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—88no—
441C2×D4—88no—
442C2×D4—88no—
443C2×D4—88no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—216no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—416no—
502C22×C4—416no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—88no—
509C22×C4—88no—
510C22×C4#123116yes—
511C22×C4C22×D414yes—
512C22×C4C2×#4114yes—
513C22×C4C2×#4114yes—
514C22×C4H(5,𝔽2)18yes—
515C4⋊C4—416no—
516C4⋊C4—48no—
517C4⋊C4—88no—
518C4⋊C4—88no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—88no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—88no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C42—416no—
543C23—28no—
544C23—24no—
545C23—24no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—432no—
569C23—432no—
570C23—416no—
571C23—416no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—42no—
608C23—42no—
609C23—44no—
610C23—42no—
611C23—42no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—88no—
618C23—88no—
619C23—816no—
620C23—816no—
621C23—88no—
622C23—88no—
623C23—832no—
624C23—832no—
625C23—832no—
626C23—832no—
627C23C2×#12314yes—
628C23#53414yes—
629C23#53414yes—
630C23#53818yes—
631C23#53818yes—
632C23C23×D411yescharacteristic
633C23C22×#4111yescharacteristic
634C23C22×#4111yescharacteristic
635C23C2×H(5,𝔽2)12yes—
636C23C2×H(5,𝔽2)12yes—
637C23C2×H(5,𝔽2)14yes—
638C23#58514yes—
639D4—432no—
640D4—432no—
641D4—88no—
642D4—816no—
643D4—88no—
644D4—88no—
645C2×C4—232no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—432no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—816no—
703C2×C4C2×#12318yes—
704C22—216no—
705C22—28no—
706C22—24no—
707C22—216no—
708C22—28no—
709C22—24no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—42no—
715C22—42no—
716C22—44no—
717C22—42no—
718C22—44no—
719C22—42no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—42no—
725C22—42no—
726C22—44no—
727C22—832no—
728C22—832no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22—84no—
736C22—84no—
737C22—84no—
738C22—82no—
739C22—81no—
740C22—84no—
741C22—81no—
742C22—82no—
743C22—81no—
744C22#262218yes—
745C22C22×#12311yescharacteristic
746C22C2×#53411yescharacteristic
747C22C2×#53411yescharacteristic
748C22C2×#53812yes—
749C22C2×#53812yes—
750C22#346014yes—
751C22#348412yes—
752C22#348512yes—
753C22#350614yes—
754C22#353414yes—
755C22#353414yes—
756C22#354314yes—
757C22#354514yes—
758C4—28no—
759C4—48no—
760C4—84no—
761C4—82no—
762C4—84no—
763C4—81no—
764C4—84no—
765C4—81no—
766C4—82no—
767C4—81no—
768C2—28no—
769C2—82no—
770C2—84no—
771C2—82no—
772C2—161no—
773C2C2×#262211yescharacteristic
774C2#3515212yes—
775C2#3711712yes—
776C2#3711911yescharacteristic
777C2#3719814yes—
778C2#3723211yescharacteristic
779C2#3751714yes—
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
2. C22×#390C2semidirect
3. #16573C2semidirect
4. #16530C2semidirect
5. #16527C2semidirect
6. #16526C2semidirect
7. #14727C2semidirect
8. #14727C2semidirect
9. #14718C2semidirect
10. #14718C2semidirect
11. #14706C2semidirect
12. #14705C2semidirect
13. #14335C2semidirect
14. #14263C2semidirect
15. #14256C2semidirect
25. C2×#397C22semidirect
26. C2×#397C22semidirect
29. C2×#394C22semidirect
30. C2×#394C22semidirect
47. #1444C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
63. #1438C22semidirect
64. #1438C22semidirect
65. #1437C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26132
C22×#3902
#165732
#165302
#165272
#165262
#147272
#147182
#147062
#147052
#143352
#142632
#142562