Finite Group Databasebeta

← #2036845

Subgroups — #2036845

Order 512.

4006 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic, complemented
3#17077C212yesmaximal, complemented
4#16442C214yesmaximal, complemented
5#15941C212yesmaximal, complemented
6#15453C212yesmaximal, complemented
7#15304C214yesmaximal, complemented
8#15259C212yesmaximal, complemented
9#15257C214yesmaximal, complemented
10#15257C212yesmaximal, complemented
11#14698C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic, complemented
13#14358C214yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#121C2212yes—
18C22×#120C2214yes—
19C22×#120C2212yes—
20C22×#120C2212yescomplemented
21C23×C42C2211yescharacteristic, complemented
22#2850—28no—
23#2679—28no—
24C2×#394C2214yes—
25C2×#394C2212yes—
26C2×#393C2214yescomplemented
27C2×#392C2212yes—
28C2×#392C2214yes—
29C2×#391C2211yescharacteristic, complemented
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#388C2214yescomplemented
35C2×#388C2214yescomplemented
36C2×#386C2214yescomplemented
37C2×#386C2214yescomplemented
38C2×#386C2214yescomplemented
39C2×#382C2211yescharacteristic, complemented
40C2×#380C2212yescomplemented
41C2×#379C2212yescomplemented
42C2×#379C2214yescomplemented
43#1445C2212yescomplemented
44#1443C2214yescomplemented
45#1442C2214yescomplemented
46#1440C2212yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2214yescomplemented
61#1438C2212yescomplemented
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65#1437C2212yescomplemented
66#1436C2212yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2212yescomplemented
69C26C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2314yes—
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×#124—24no—
80C2×#124—44no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—42no—
83C2×C22⋊Q8—42no—
84C2×#121—24no—
85C2×#121—28no—
86C2×#121—42no—
87C2×#121—44no—
88C2×#121—42no—
89C2×#121—44no—
90C2×#120—28no—
91C2×#120—216no—
92C2×#120—216no—
93C2×#120—48no—
94C2×#120—48no—
95C2×#120—48no—
96C2×#120—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C42—24no—
123C22×C42—44no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#394—28no—
129#394—216no—
130#393—48no—
131#392—44no—
132#392—48no—
133#391—24no—
134#390—28no—
135#390—44no—
136#390—44no—
137#390D418yescomplemented
138#388—216no—
139#388—48no—
140#386—216no—
141#386—48no—
142#386—48no—
143#382—24no—
144#380—44no—
145#379—48no—
146#379—44no—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25—24no—
162C25—28no—
163C25—44no—
164C25—44no—
165C25—42no—
166C25—44no—
167C25—44no—
168C25—42no—
169C25—44no—
170C25C2412yes—
171C25C2414yes—
172C25C2411yescharacteristic
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—42no—
178C22×D4—24no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—41no—
199C22×D4—41no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—44no—
212C22×D4C2×D414yes—
213C22×D4#4118yes—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236#124—416no—
237C22⋊Q8—416no—
238#121—416no—
239#121—432no—
240#120—464no—
241#120—432no—
242#120—432no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
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⋊C4C2×D4116yes—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—416no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42—42no—
324C2×C42C2×D414yes—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339C24—21no—
340C24—22no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—416no—
369C24—48no—
370C24—48no—
371C24—416no—
372C24—48no—
373C24—84no—
374C24—84no—
375C24—88no—
376C24—88no—
377C24—88no—
378C24—88no—
379C24—88no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24—88no—
385C24—84no—
386C24—88no—
387C24—84no—
388C24—88no—
389C24C22×D414yes—
390C24C2×#4118yes—
391C24H(5,𝔽2)12yes—
392C24H(5,𝔽2)14yes—
393C24C2511yescharacteristic
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×D4—28no—
400C2×D4—216no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—48no—
408C2×D4—416no—
409C2×D4—48no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—416no—
415C2×D4—48no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—84no—
431C2×D4—88no—
432C2×D4—84no—
433C2×D4—88no—
434C2×D4—88no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
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—22no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—48no—
497C22×C4—44no—
498C22×C4—48no—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D414yes—
502C22×C4C2×#4114yes—
503C22×C4H(5,𝔽2)14yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)14yes—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C22⋊C4—232no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C22⋊C4—84no—
531C22⋊C4—88no—
532C42—28no—
533C42—44no—
534C42—48no—
535C42—84no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—24no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—42no—
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—42no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—42no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—816no—
613C23—832no—
614C23—832no—
615C23—832no—
616C23—832no—
617C23—832no—
618C23—816no—
619C23—816no—
620C23—832no—
621C23H(3,𝔽4)14yes—
622C23C23×D411yescharacteristic
623C23C22×#4112yes—
624C23C2×H(5,𝔽2)12yes—
625C23C2×H(5,𝔽2)12yes—
626C23C2×H(5,𝔽2)12yes—
627C23C2×H(5,𝔽2)11yescharacteristic
628C23C2×H(5,𝔽2)11yescharacteristic
629C23C2×H(5,𝔽2)12yes—
630C23#58512yes—
631Q8—88no—
632D4—88no—
633D4—816no—
634D4—816no—
635D4—832no—
636D4—816no—
637D4—832no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—416no—
656C2×C4—432no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
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—42no—
682C2×C4—44no—
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—42no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4#53514yes—
699C2×C4#54618yes—
700C22—28no—
701C22—24no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—48no—
710C22—48no—
711C22—48no—
712C22—48no—
713C22—44no—
714C22—48no—
715C22—48no—
716C22—44no—
717C22—48no—
718C22—44no—
719C22—48no—
720C22—48no—
721C22—864no—
722C22—84no—
723C22—84no—
724C22—84no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22C2×#53511yescharacteristic
735C22#345912yes—
736C22C2×#54612yes—
737C22#348111yescharacteristic
738C22#348411yescharacteristic
739C22#348411yescharacteristic
740C22#348812yes—
741C22#349012yes—
742C22#351412yes—
743C22#351912yes—
744C22#351912yes—
745C22C2×H(3,𝔽4)11yescharacteristic
746C22#353412yes—
747C22#353712yes—
748C22#353712yes—
749C22#353712yes—
750C22#354312yes—
751C22#354511yescharacteristic
752C22#354612yes—
753C22#354911yescharacteristic
754C22#355812yes—
755C4—24no—
756C4—28no—
757C4—42no—
758C4—44no—
759C4—44no—
760C4—84no—
761C4—82no—
762C4—82no—
763C4—84no—
764C4—84no—
765C2—44no—
766C2—88no—
767C2—84no—
768C2#3671711yescharacteristic
769C2#3715412yes—
770C2#3774412yes—
771C2#3876012yes—
772C2#3891012yes—
773C2#4597711yescharacteristic
774C2#4608912yes—
775C2#5318112yes—
776C2#5904111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2679C2semidirect
2. C2×#2850C2semidirect
3. #17077C2semidirect
4. #16442C2semidirect
5. #15941C2semidirect
6. #15453C2semidirect
7. #15304C2semidirect
8. #15259C2semidirect
9. #15257C2semidirect
10. #15257C2semidirect
11. #14698C2semidirect
12. #14445C2semidirect
13. #14358C2semidirect
15. C22×C22⋊Q8C22semidirect
20. C22×#120C22semidirect
21. C23×C42C22semidirect
26. C2×#393C22semidirect
29. C2×#391C22semidirect
34. C2×#388C22semidirect
35. C2×#388C22semidirect
36. C2×#386C22semidirect
37. C2×#386C22semidirect
38. C2×#386C22semidirect
39. C2×#382C22semidirect
40. C2×#380C22semidirect
41. C2×#379C22semidirect
42. C2×#379C22semidirect
43. #1445C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
60. #1438C22semidirect
61. #1438C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1437C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
137. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#28502
#170772
#164422
#159412
#154532
#153042
#152592
#152572
#146982
#144452
#143582