Finite Group Databasebeta

← #4311785

Subgroups — #4311785

Order 512.

3902 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2309C211yesmaximal, characteristic
3#17007C212yesmaximal, complemented
4#16556C212yesmaximal, complemented
5#16298C214yesmaximal, complemented
6#15508C214yesmaximal, complemented
7#15280C214yesmaximal, complemented
8#14893C212yesmaximal, complemented
9#14887C212yesmaximal, complemented
10#14717C214yesmaximal, complemented
11#14513C212yesmaximal, complemented
12#14486C211yesmaximal, characteristic, complemented
13#13867C212yesmaximal
14C24×D4C2211yescharacteristic
15C22×C22⋊Q8C2214yescomplemented
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19#2683—28no—
20C2×#400C2212yescomplemented
21C2×#399C2212yes—
22C2×#396C2214yescomplemented
23C2×#396C2214yescomplemented
24C2×#395C2212yescomplemented
25C2×#393C2212yes—
26C2×#393C2214yescomplemented
27#2309—28no—
28C2×#391C2212yescomplemented
29C2×#390C2211yescharacteristic
30C2×#390C2214yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2212yescomplemented
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#379C2214yes—
40C2×#379C2214yes—
41C2×#379C2212yes—
42#1444C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2212yescomplemented
45#1441C2212yescomplemented
46#1439C2214yes—
47#1439C2212yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1438C2214yes—
60#1438C2214yes—
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1434C2212yescomplemented
64#1434C2212yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yescomplemented
67C22⋊#95C2214yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2312yes—
70C23×Q8C2312yes—
71C23×D4—24no—
72C23×D4—24no—
73C23×D4—42no—
74C23×D4—42no—
75C23×D4—42no—
76C23×D4—42no—
77C23×D4Q818yescomplemented
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—48no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—42no—
90C2×#117—42no—
91C2×#117—42no—
92C2×#117—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—48no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#400—44no—
124#399—44no—
125#396—216no—
126#396—48no—
127#395—28no—
128#393—48no—
129#393—44no—
130#391—44no—
131#390—24no—
132#390—24no—
133#390—216no—
134#390—24no—
135#390D418yes—
136#389—28no—
137#386—28no—
138#386—28no—
139#386—44no—
140#386—44no—
141#379—216no—
142#379—48no—
143#379—44no—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—24no—
165C25—44no—
166C25—42no—
167C25—42no—
168C25—44no—
169C25—44no—
170C25—44no—
171C25C2×Q818yes—
172C25C2412yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—28no—
177C22×Q8—44no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—232no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—416no—
188C22×D4—416no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—42no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—42no—
197C22×D4—41no—
198C22×D4—41no—
199C22×D4—416no—
200C22×D4—416no—
201C22×D4—416no—
202C22×D4—416no—
203C22×D4C2×D414yes—
204C22×D4C2×Q818yes—
205C22×D4C2×Q814yes—
206C22×D4C2×Q814yes—
207C22×D4#4114yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C2×Q814yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C22⋊Q8—464no—
236#117—416no—
237#117—416no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D414yes—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—432no—
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—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D4116yes—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—41no—
316C2×C42—41no—
317C2×C42—42no—
318C2×C42—41no—
319C2×C42—41no—
320C2×C42C2×D414yes—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—42no—
337#95—44no—
338C24—21no—
339C24—21no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—216no—
363C24—216no—
364C24—48no—
365C24—416no—
366C24—416no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—48no—
372C24—48no—
373C24—416no—
374C24—88no—
375C24—88no—
376C24—88no—
377C24—88no—
378C24—84no—
379C24—84no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24—88no—
385C24C22×D414yes—
386C24C22×Q814yes—
387C24C22×Q814yes—
388C24C22×Q812yes—
389C24C2×#4114yes—
390C24H(5,𝔽2)12yes—
391C24H(5,𝔽2)12yes—
392C24C2511yescharacteristic
393C2×Q8—416no—
394C2×Q8—48no—
395C2×Q8—84no—
396C2×Q8—88no—
397C2×Q8—84no—
398C2×D4—216no—
399C2×D4—28no—
400C2×D4—216no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—4128no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4C22⋊Q8116yes—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
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—24no—
480C22×C4—48no—
481C22×C4—416no—
482C22×C4—48no—
483C22×C4—88no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—88no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—88no—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×Q812yes—
494C22×C4C22×Q812yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)18yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C4⋊C4—216no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—416no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—88no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C42—44no—
525C42—48no—
526C42—44no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—28no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—24no—
548C23—28no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—432no—
555C23—432no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—832no—
591C23—816no—
592C23—832no—
593C23—816no—
594C23—816no—
595C23—832no—
596C23—832no—
597C23—816no—
598C23—816no—
599C23C2×C22⋊Q818yes—
600C23#54314yes—
601C23C23×D411yescharacteristic
602C23C23×Q811yescharacteristic
603C23C22×#4111yescharacteristic
604C23C2×H(5,𝔽2)11yescharacteristic
605C23C2×H(5,𝔽2)11yescharacteristic
606C23C2×H(5,𝔽2)11yescharacteristic
607C23C2×H(5,𝔽2)11yescharacteristic
608C23C2×H(5,𝔽2)14yes—
609C23#58514yes—
610Q8—816no—
611D4—416no—
612D4—432no—
613D4—416no—
614C2×C4—232no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
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—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—816no—
676C2×C4—816no—
677C2×C4C2×C22⋊Q814yes—
678C2×C4#54314yes—
679C22—216no—
680C22—28no—
681C22—24no—
682C22—28no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—48no—
688C22—44no—
689C22—48no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—48no—
698C22—44no—
699C22—48no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—48no—
704C22—864no—
705C22—82no—
706C22—82no—
707C22—84no—
708C22—84no—
709C22—82no—
710C22—82no—
711C22C22×C22⋊Q811yescharacteristic
712C22C2×#54311yescharacteristic
713C22C2×#54311yescharacteristic
714C22#348411yescharacteristic
715C22#348412yes—
716C22#348411yescharacteristic
717C22#349011yescharacteristic
718C22#349011yescharacteristic
719C22#350012yes—
720C22#350114yes—
721C22#351414yes—
722C22#353212yes—
723C22#353312yes—
724C22#353712yes—
725C22#354214yes—
726C22#354614yes—
727C22#355112yes—
728C4—24no—
729C4—24no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—82no—
734C4—84no—
735C4—82no—
736C4—84no—
737C4—84no—
738C2—48no—
739C2—44no—
740C2—84no—
741C2#3498211yescharacteristic
742C2#3570212yes—
743C2#3674712yes—
744C2#4249012yes—
745C2#4642812yes—
746C2#4765812yes—
747C2#5194512yes—
748C2#5197811yescharacteristic
749C2#5200411yescharacteristic
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×#2683C2semidirect
3. #17007C2semidirect
4. #16556C2semidirect
5. #16298C2semidirect
6. #15508C2semidirect
7. #15280C2semidirect
8. #14893C2semidirect
9. #14887C2semidirect
10. #14717C2semidirect
11. #14513C2semidirect
12. #14486C2semidirect
15. C22×C22⋊Q8C22semidirect
20. C2×#400C22semidirect
22. C2×#396C22semidirect
23. C2×#396C22semidirect
24. C2×#395C22semidirect
26. C2×#393C22semidirect
28. C2×#391C22semidirect
34. C2×#389C22semidirect
42. #1444C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
77. C23×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#23092
#170072
#165562
#162982
#155082
#152802
#148932
#148872
#147172
#145132
#144862
#138672