Finite Group Databasebeta

← #4386443

Subgroups — #4386443

Order 512.

3782 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2607C211yesmaximal, characteristic, complemented
2C2×#2308C211yesmaximal, characteristic
3#16298C212yesmaximal, complemented
4#16297C212yesmaximal, complemented
5#15355C214yesmaximal, complemented
6#15334C214yesmaximal, complemented
7#15331C214yesmaximal, complemented
8#15280C214yesmaximal, complemented
9#14734C212yesmaximal, complemented
10#14717C212yesmaximal, complemented
11#14674C211yesmaximal, characteristic, complemented
12#14297C212yesmaximal, complemented
13#13902C212yesmaximal
14C24×D4C2211yescharacteristic
15C22×C22⋊Q8C2212yescomplemented
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2607—28no—
19C2×#399C2212yes—
20C2×#399C2212yescomplemented
21C2×#396C2214yescomplemented
22C2×#393C2212yes—
23C2×#393C2212yescomplemented
24C2×#393C2214yescomplemented
25#2308—28no—
26C2×#391C2214yescomplemented
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#387C2214yescomplemented
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#385C2214yescomplemented
35C2×#381C2211yescharacteristic
36C2×#381C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2212yescomplemented
39C2×#379C2211yescharacteristic
40C2×#379C2211yescharacteristic
41C2×#379C2214yes—
42C2×#379C2214yes—
43C2×C4×C22⋊C4C2212yes—
44#1444C2214yescomplemented
45#1442C2214yescomplemented
46#1441C2214yescomplemented
47#1441C2214yescomplemented
48#1439C2212yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yescomplemented
53#1439C2212yescomplemented
54#1439C2212yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1438C2214yes—
63#1438C2214yes—
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C22⋊#95C2214yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4—24no—
73C23×D4—24no—
74C23×D4—42no—
75C23×D4—42no—
76C23×D4—42no—
77C23×D4—42no—
78C23×D4Q818yescomplemented
79C23×D4C2312yes—
80C23×D4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—48no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—42no—
97C22×C22⋊C4—42no—
98C22×C22⋊C4—42no—
99C22×C22⋊C4—42no—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120#399—28no—
121#399—44no—
122#396—48no—
123#393—28no—
124#393—48no—
125#393—44no—
126#391—216no—
127#390—44no—
128#390—44no—
129#390—44no—
130#390—44no—
131#387—216no—
132#386—216no—
133#386—216no—
134#385—48no—
135#381—24no—
136#381D418yes—
137#380—28no—
138#380—44no—
139#379—24no—
140#379—24no—
141#379—48no—
142#379—48no—
143C4×C22⋊C4—28no—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25—22no—
162C25—22no—
163C25—22no—
164C25—22no—
165C25—22no—
166C25—22no—
167C25—22no—
168C25—22no—
169C25—41no—
170C25—41no—
171C25—41no—
172C25—41no—
173C25—44no—
174C25—44no—
175C25—44no—
176C25—44no—
177C25C2×Q814yes—
178C25C2×Q814yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C22×Q8—24no—
184C22×Q8—28no—
185C22×Q8—42no—
186C22×Q8—44no—
187C22×Q8—42no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—232no—
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—416no—
202C22×D4—416no—
203C22×D4—416no—
204C22×D4—416no—
205C22×D4—416no—
206C22×D4—416no—
207C22×D4C2×Q818yes—
208C22×D4C2×Q818yes—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—41no—
214C23×C4—41no—
215C23×C4—41no—
216C23×C4—41no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D414yes—
224C23×C4C2×Q814yes—
225C23×C4#4114yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C22⋊Q8—432no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
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—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—416no—
291C2×C22⋊C4—416no—
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⋊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⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—24no—
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#95C2×D4116yes—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—28no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—28no—
360C24—28no—
361C24—28no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—28no—
373C24—44no—
374C24—44no—
375C24—416no—
376C24—416no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—88no—
382C24—88no—
383C24—84no—
384C24—84no—
385C24—84no—
386C24—84no—
387C24—84no—
388C24—84no—
389C24—88no—
390C24—84no—
391C24—84no—
392C24—88no—
393C24—84no—
394C24—84no—
395C24—88no—
396C24—84no—
397C24—84no—
398C24—88no—
399C24C22×D412yes—
400C24C22×Q814yes—
401C24C22×Q812yes—
402C24C22×Q812yes—
403C24C22×Q812yes—
404C24C2×#4112yes—
405C24H(5,𝔽2)12yes—
406C24H(5,𝔽2)12yes—
407C24H(5,𝔽2)12yes—
408C24H(5,𝔽2)12yes—
409C24C2511yescharacteristic
410C2×Q8—48no—
411C2×Q8—416no—
412C2×Q8—84no—
413C2×Q8—88no—
414C2×Q8—84no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—48no—
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—48no—
429C2×D4—4128no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
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—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—88no—
490C22×C4—88no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4—88no—
495C22×C4C22⋊Q8116yescomplemented
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×Q814yes—
500C22×C4C2×#4114yes—
501C22×C4H(5,𝔽2)18yes—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—88no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C42—84no—
529C42—84no—
530C23—232no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—28no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—28no—
558C23—24no—
559C23—24no—
560C23—28no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—28no—
565C23—28no—
566C23—416no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—44no—
588C23—44no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—816no—
604C23—816no—
605C23—816no—
606C23—816no—
607C23—816no—
608C23—816no—
609C23—88no—
610C23—88no—
611C23—816no—
612C23—816no—
613C23—816no—
614C23—816no—
615C23—816no—
616C23—88no—
617C23—88no—
618C23—816no—
619C23C2×C22⋊Q814yes—
620C23#54314yes—
621C23#54314yes—
622C23C23×D411yescharacteristic
623C23C23×Q811yescharacteristic
624C23C22×#4111yescharacteristic
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)11yescharacteristic
627C23C2×H(5,𝔽2)11yescharacteristic
628C23C2×H(5,𝔽2)11yescharacteristic
629C23C2×H(5,𝔽2)14yes—
630C23#58514yes—
631Q8—816no—
632D4—432no—
633D4—432no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—416no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
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—44no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—816no—
690C2×C4—816no—
691C2×C4C2×C22⋊Q818yes—
692C22—28no—
693C22—28no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—28no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—44no—
712C22—44no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—48no—
720C22—48no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—832no—
725C22—832no—
726C22—82no—
727C22—82no—
728C22—84no—
729C22—82no—
730C22—82no—
731C22—84no—
732C22C22×C22⋊Q811yescharacteristic
733C22C2×#54311yescharacteristic
734C22C2×#54311yescharacteristic
735C22#348411yescharacteristic
736C22#348411yescharacteristic
737C22#348814yes—
738C22#349011yescharacteristic
739C22#349011yescharacteristic
740C22#350114yes—
741C22#350812yes—
742C22#351012yes—
743C22#353212yes—
744C22#353312yes—
745C22#353414yes—
746C22#354312yes—
747C22#354512yes—
748C22#355814yes—
749C4—28no—
750C4—44no—
751C4—48no—
752C4—44no—
753C4—84no—
754C4—82no—
755C4—84no—
756C4—84no—
757C4—82no—
758C2—44no—
759C2—44no—
760C2—44no—
761C2—84no—
762C2#3498211yescharacteristic
763C2#3551512yes—
764C2#4060812yes—
765C2#4061212yes—
766C2#4645212yes—
767C2#4703111yescharacteristic
768C2#4703511yescharacteristic
769C2#5063512yes—
770C2#5085612yes—
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×#2607C2semidirect
3. #16298C2semidirect
4. #16297C2semidirect
5. #15355C2semidirect
6. #15334C2semidirect
7. #15331C2semidirect
8. #15280C2semidirect
9. #14734C2semidirect
10. #14717C2semidirect
11. #14674C2semidirect
12. #14297C2semidirect
15. C22×C22⋊Q8C22semidirect
20. C2×#399C22semidirect
21. C2×#396C22semidirect
23. C2×#393C22semidirect
24. C2×#393C22semidirect
26. C2×#391C22semidirect
31. C2×#387C22semidirect
34. C2×#385C22semidirect
38. C2×#380C22semidirect
44. #1444C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
78. C23×D4Q8semidirect
495. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26072
C2×#23082
#162982
#162972
#153552
#153342
#153312
#152802
#147342
#147172
#146742
#142972
#139022