Finite Group Databasebeta

← #900741

Subgroups — #900741

Order 512.

3838 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2679C211yesmaximal, characteristic
3#17077C212yesmaximal, complemented
4#16932C212yesmaximal, complemented
5#15391C214yesmaximal, complemented
6#15259C214yesmaximal, complemented
7#15107C212yesmaximal, complemented
8#14817C214yesmaximal, complemented
9#14708C212yesmaximal, complemented
10#14357C212yesmaximal, complemented
11#13995C214yesmaximal, complemented
12#13967C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×#120C2212yes—
16C22×#120C2214yes—
17C22×C4×D4C2211yescharacteristic
18C23×C22⋊C4C2212yescomplemented
19#2804—28no—
20#2679—28no—
21C2×#400C2212yescomplemented
22C2×#394C2212yes—
23C2×#394C2214yes—
24C2×#393C2214yescomplemented
25C2×#393C2214yescomplemented
26C2×#391C2212yescomplemented
27C2×#391C2212yescomplemented
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#386C2212yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#385C2214yescomplemented
37C2×#382C2212yescomplemented
38C2×#379C2214yescomplemented
39C2×#379C2212yescomplemented
40C2×C4×C4⋊C4C2211yescharacteristic, complemented
41C2×C4×C22⋊C4C2214yescomplemented
42C2×C43C2211yescharacteristic, complemented
43#1445C2212yescomplemented
44#1443C2214yescomplemented
45#1442C2214yescomplemented
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1438C2212yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1437C2214yescomplemented
59#1437C2214yescomplemented
60#1437C2214yescomplemented
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64C4⋊#95C2211yescharacteristic, complemented
65C4⋊#95C2211yescharacteristic, complemented
66#1434C2212yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C26C2312yes—
70C23×Q8C2312yes—
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C2×#124—28no—
78C2×#124—44no—
79C2×#124—44no—
80C2×#120—28no—
81C2×#120—216no—
82C2×#120—28no—
83C2×#120—28no—
84C2×#120—216no—
85C2×#120—48no—
86C2×C4×D4—24no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4—48no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#400—28no—
121#394—28no—
122#394—216no—
123#393—216no—
124#393—48no—
125#391—28no—
126#391—44no—
127#390—24no—
128#390—28no—
129#390—44no—
130#390—44no—
131#390D418yes—
132#386—28no—
133#386—48no—
134#386—48no—
135#385—216no—
136#382—44no—
137#379—44no—
138#379—48no—
139C4×C4⋊C4—42no—
140C4×C22⋊C4—48no—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C43—42no—
158C25—28no—
159C25—24no—
160C25—24no—
161C25—24no—
162C25—42no—
163C25—44no—
164C25—42no—
165C25—44no—
166C25—42no—
167C25—44no—
168C25—42no—
169C25C2414yes—
170C25C2412yes—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4C2×D414yes—
201C22×D4#4118yes—
202C22×D4#4114yes—
203C22×D4#4114yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225#124—432no—
226#120—464no—
227#120—432no—
228C4×D4—416no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
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—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—432no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—41no—
310C2×C42—42no—
311C2×C42—41no—
312C2×C42—41no—
313C2×C42—41no—
314C2×C42—44no—
315C2×C42C2×D414yes—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—416no—
355C24—48no—
356C24—416no—
357C24—416no—
358C24—48no—
359C24—416no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—84no—
364C24—84no—
365C24—84no—
366C24—84no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24—88no—
371C24—88no—
372C24—88no—
373C24—84no—
374C24—84no—
375C24—88no—
376C24—88no—
377C24C22×D414yes—
378C24C2×#4118yes—
379C24C2×#4114yes—
380C24C2×#4114yes—
381C24C2511yescharacteristic
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—84no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×D4—216no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—416no—
399C2×D4—416no—
400C2×D4—48no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—416no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—88no—
413C2×D4—84no—
414C2×D4#124116yescomplemented
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
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—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—88no—
479C22×C4—84no—
480C22×C4—88no—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)12yes—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C22⋊C4—232no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—88no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C42—28no—
516C42—216no—
517C42—44no—
518C42—48no—
519C42—44no—
520C42—84no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—432no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—832no—
580C23—816no—
581C23—816no—
582C23—816no—
583C23—832no—
584C23—832no—
585C23—832no—
586C23—816no—
587C23—816no—
588C23—816no—
589C23C2×#12418yes—
590C23C23×D411yescharacteristic
591C23C22×#4111yescharacteristic
592C23C22×#4111yescharacteristic
593C23C22×#4112yes—
594C23C2×H(5,𝔽2)11yescharacteristic
595C23C2×H(5,𝔽2)12yes—
596C23C2×H(5,𝔽2)11yescharacteristic
597C23C2×H(5,𝔽2)12yes—
598C23C2×H(5,𝔽2)12yes—
599C23#58511yescharacteristic
600C23#58511yescharacteristic
601Q8—816no—
602D4—416no—
603D4—816no—
604D4—816no—
605D4—832no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—816no—
671C2×C4—816no—
672C2×C4C2×#12414yes—
673C2×C4#54618yes—
674C2×C4H(3,𝔽4)14yes—
675C22—28no—
676C22—216no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—48no—
681C22—48no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—48no—
686C22—48no—
687C22—48no—
688C22—48no—
689C22—48no—
690C22—48no—
691C22—48no—
692C22—48no—
693C22—864no—
694C22—84no—
695C22—84no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22C22×#12411yescharacteristic
703C22#345912yes—
704C22#345911yescharacteristic
705C22#345911yescharacteristic
706C22C2×#54612yes—
707C22#348111yescharacteristic
708C22#348112yes—
709C22#348112yes—
710C22#349011yescharacteristic
711C22#349012yes—
712C22#349111yescharacteristic
713C22#349611yescharacteristic
714C22#349611yescharacteristic
715C22#349812yes—
716C22#351412yes—
717C22#351912yes—
718C22#351912yes—
719C22C2×H(3,𝔽4)11yescharacteristic
720C22#353712yes—
721C22#354311yescharacteristic
722C22#354911yescharacteristic
723C22#355812yes—
724C22#355812yes—
725C4—24no—
726C4—24no—
727C4—28no—
728C4—44no—
729C4—44no—
730C4—42no—
731C4—82no—
732C4—82no—
733C4—84no—
734C4—84no—
735C4—84no—
736C2—48no—
737C2—88no—
738C2#3499311yescharacteristic
739C2#3628912yes—
740C2#3724911yescharacteristic
741C2#3876012yes—
742C2#3876812yes—
743C2#3941212yes—
744C2#4599611yescharacteristic
745C2#4608911yescharacteristic
746C2#5230111yescharacteristic
747C2#5942011yescharacteristic
748C2#6004511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
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×#2804C2semidirect
3. #17077C2semidirect
4. #16932C2semidirect
5. #15391C2semidirect
6. #15259C2semidirect
7. #15107C2semidirect
8. #14817C2semidirect
9. #14708C2semidirect
10. #14357C2semidirect
11. #13995C2semidirect
12. #13967C2semidirect
13. C4×#390C2semidirect
18. C23×C22⋊C4C22semidirect
21. C2×#400C22semidirect
24. C2×#393C22semidirect
25. C2×#393C22semidirect
26. C2×#391C22semidirect
27. C2×#391C22semidirect
36. C2×#385C22semidirect
37. C2×#382C22semidirect
38. C2×#379C22semidirect
39. C2×#379C22semidirect
40. C2×C4×C4⋊C4C22semidirect
41. C2×C4×C22⋊C4C22semidirect
42. C2×C43C22semidirect
43. #1445C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
58. #1437C22semidirect
59. #1437C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. #1434C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
414. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#26792
#170772
#169322
#153912
#152592
#151072
#148172
#147082
#143572
#139952
#139672
C4×#3902