Finite Group Databasebeta

← #4335652

Subgroups — #4335652

Order 512.

3498 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2599C211yesmaximal, characteristic, complemented
2C2×#2309C211yesmaximal, characteristic
3#16693C212yesmaximal, complemented
4#16596C212yesmaximal, complemented
5#16556C212yesmaximal, complemented
6#16519C212yesmaximal, complemented
7#15480C212yesmaximal, complemented
8#15467C212yesmaximal, complemented
9#15297C214yesmaximal, complemented
10#14898C212yesmaximal, complemented
11#14892C212yesmaximal, complemented
12#14883C212yesmaximal, complemented
13#14724C212yesmaximal
14#14568C214yesmaximal, complemented
15#13957C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic, complemented
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2599—28no—
21C2×#398C2211yescharacteristic, complemented
22C2×#397C2211yescharacteristic
23C2×#397C2214yes—
24C2×#396C2214yescomplemented
25C2×#395C2211yescharacteristic, complemented
26C2×#394C2214yes—
27C2×#394C2212yes—
28C2×#394C2212yes—
29C2×#393C2211yescharacteristic
30C2×#393C2214yescomplemented
31#2309—28no—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yescomplemented
38C2×#387C2212yescomplemented
39C2×#385C2212yes—
40C2×#383C2211yescharacteristic, complemented
41C2×#382C2211yescharacteristic, complemented
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×#379C2214yes—
45C2×#379C2214yes—
46C2×#376C2211yescharacteristic, complemented
47#1444C2212yescomplemented
48#1442C2212yescomplemented
49#1441C2212yescomplemented
50#1441C2212yescomplemented
51#1441C2212yescomplemented
52#1440C2214yes—
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1439C2212yes—
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1438C2212yes—
65#1438C2212yes—
66#1438C2214yes—
67#1437C2212yes—
68#1437C2214yescomplemented
69#1437C2214yescomplemented
70#1437C2214yescomplemented
71#1437C2214yescomplemented
72#1436C2214yescomplemented
73#1436C2214yescomplemented
74#1436C2212yescomplemented
75#1434C2212yescomplemented
76C22⋊#95C2212yes—
77C26C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4—28no—
80C23×D4—24no—
81C23×D4—42no—
82C23×D4—42no—
83C23×D4—44no—
84C23×D4C2312yes—
85C23×D4C2312yes—
86C24×C4C2311yescharacteristic
87C2×C4×Q8—24no—
88C2×C4×Q8—42no—
89C2×C4×Q8—42no—
90C2×#117—24no—
91C2×#117—42no—
92C2×#117—42no—
93C2×#117—44no—
94C2×#117D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#398—24no—
128#397—24no—
129#397—48no—
130#396—48no—
131#395—24no—
132#394—216no—
133#394—44no—
134#394—44no—
135#393—24no—
136#393—48no—
137#390—28no—
138#390—28no—
139#390—28no—
140#390—28no—
141#388—28no—
142#387—28no—
143#387—28no—
144#385—28no—
145#383—24no—
146#382—42no—
147#379—28no—
148#379—28no—
149#379—48no—
150#379—48no—
151#376—42no—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25—24no—
175C25—28no—
176C25—28no—
177C25—42no—
178C25—42no—
179C25—44no—
180C25—44no—
181C25—44no—
182C25—44no—
183C25C2414yes—
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8—42no—
190C22×D4—232no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—432no—
203C22×D4—432no—
204C22×D4—416no—
205C22×D4—416no—
206C22×D4#4118yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4C2×D414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C4×Q8—416no—
226#117—416no—
227#117—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—24no—
346C24—216no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—416no—
362C24—416no—
363C24—48no—
364C24—416no—
365C24—416no—
366C24—48no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24—88no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24—84no—
376C24—84no—
377C24—84no—
378C24—84no—
379C24—88no—
380C24—88no—
381C24C22×D412yes—
382C24C2×#4118yes—
383C24H(5,𝔽2)14yes—
384C24C2511yescharacteristic
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—416no—
395C2×D4—416no—
396C2×D4—4128no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—28no—
507C42—28no—
508C42—48no—
509C42—48no—
510C42—48no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—432no—
527C23—44no—
528C23—44no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—832no—
563C23—832no—
564C23—832no—
565C23—832no—
566C23—832no—
567C23—832no—
568C23—816no—
569C23—816no—
570C23#53514yes—
571C23C23×D411yescharacteristic
572C23C22×#4112yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)12yes—
577C23#58512yes—
578C23#58514yes—
579Q8—416no—
580D4—432no—
581D4—432no—
582C2×C4—232no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4#52514yes—
653C2×C4#53414yes—
654C22—28no—
655C22—24no—
656C22—44no—
657C22—42no—
658C22—44no—
659C22—42no—
660C22—48no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—48no—
669C22—864no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22C2×#52511yescharacteristic
679C22C2×#53411yescharacteristic
680C22C2×#53511yescharacteristic
681C22#348812yes—
682C22#349012yes—
683C22#349012yes—
684C22#350012yes—
685C22#351414yes—
686C22#352114yes—
687C22#353314yes—
688C22#353711yescharacteristic
689C22#354012yes—
690C22#354011yescharacteristic
691C22#354214yes—
692C22#354212yes—
693C22#355112yes—
694C4—24no—
695C4—24no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—84no—
702C4—84no—
703C4—82no—
704C4—84no—
705C4—82no—
706C4—82no—
707C2—44no—
708C2—88no—
709C2#3505011yescharacteristic
710C2#3607512yes—
711C2#3726911yescharacteristic
712C2#3741911yescharacteristic
713C2#4193512yes—
714C2#4718012yes—
715C2#4765812yes—
716C2#5194314yes—
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×#2599C2semidirect
3. #16693C2semidirect
4. #16596C2semidirect
5. #16556C2semidirect
6. #16519C2semidirect
7. #15480C2semidirect
8. #15467C2semidirect
9. #15297C2semidirect
10. #14898C2semidirect
11. #14892C2semidirect
12. #14883C2semidirect
14. #14568C2semidirect
15. #13957C2semidirect
17. C22×C4×Q8C22semidirect
21. C2×#398C22semidirect
24. C2×#396C22semidirect
25. C2×#395C22semidirect
30. C2×#393C22semidirect
37. C2×#387C22semidirect
38. C2×#387C22semidirect
40. C2×#383C22semidirect
41. C2×#382C22semidirect
46. C2×#376C22semidirect
47. #1444C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
68. #1437C22semidirect
69. #1437C22semidirect
70. #1437C22semidirect
71. #1437C22semidirect
72. #1436C22semidirect
73. #1436C22semidirect
74. #1436C22semidirect
75. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25992
C2×#23092
#166932
#165962
#165562
#165192
#154802
#154672
#152972
#148982
#148922
#148832
#147242
#145682
#139572