Finite Group Databasebeta

← #4314279

Subgroups — #4314279

Order 512.

3874 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2684C212yesmaximal, complemented
2C2×#2679C211yesmaximal, characteristic
3C2×#2326C211yesmaximal, characteristic, complemented
4#16887C212yesmaximal, complemented
5#16514C212yesmaximal, complemented
6#16285C214yesmaximal, complemented
7#15566C214yesmaximal, complemented
8#15268C214yesmaximal, complemented
9#15120C212yesmaximal, complemented
10#14883C212yesmaximal, complemented
11#14718C214yesmaximal, complemented
12#14486C211yesmaximal, characteristic, complemented
13#13967C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×#120C2212yes—
16C22×#120C2214yes—
17C22×C4×Q8C2211yescharacteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C23×C22⋊C4C2212yescomplemented
20#2684—216no—
21#2679—28no—
22C2×#397C2214yescomplemented
23C2×#396C2214yescomplemented
24C2×#394C2212yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#393C2214yescomplemented
30#2326—28no—
31C2×#391C2211yescharacteristic, complemented
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#388C2214yescomplemented
36C2×#388C2214yescomplemented
37C2×#386C2214yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#385C2214yescomplemented
41C2×#384C2211yescharacteristic, complemented
42C2×#379C2214yescomplemented
43C2×#379C2212yescomplemented
44C2×#376C2211yescharacteristic, complemented
45#1445C2214yescomplemented
46#1444C2212yescomplemented
47#1442C2212yescomplemented
48#1441C2212yescomplemented
49#1441C2212yescomplemented
50#1441C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1438C2212yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2214yes—
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65#1434C2212yescomplemented
66#1434C2212yescomplemented
67C22⋊#95C2214yescomplemented
68C22⋊#95C2212yescomplemented
69C22⋊#95C2212yescomplemented
70C26C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#124—24no—
79C2×#124—24no—
80C2×#124D418yescomplemented
81C2×#120—28no—
82C2×#120—28no—
83C2×#120—28no—
84C2×#120—216no—
85C2×#120—48no—
86C2×#120—48no—
87C2×C4×Q8—24no—
88C2×C4×Q8—42no—
89C2×C4×Q8—42no—
90C2×C4×D4—24no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—44no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
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⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#397—216no—
124#396—48no—
125#394—28no—
126#394—216no—
127#394—216no—
128#394—44no—
129#394—48no—
130#393—216no—
131#391—24no—
132#390—28no—
133#390—44no—
134#390D418yes—
135#388—216no—
136#388—216no—
137#386—28no—
138#386—216no—
139#386—48no—
140#385—216no—
141#384—24no—
142#379—216no—
143#379—44no—
144#376—24no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C25—24no—
163C25—28no—
164C25—24no—
165C25—24no—
166C25—42no—
167C25—44no—
168C25—42no—
169C25—44no—
170C25—44no—
171C25—42no—
172C25—42no—
173C25C2414yes—
174C25C2412yes—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—41no—
178C22×Q8—41no—
179C22×Q8—41no—
180C22×Q8—41no—
181C22×Q8C2×D414yes—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4C2×D414yes—
204C22×D4C2×D414yes—
205C22×D4#4114yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#124—232no—
228#120—464no—
229#120—432no—
230C4×Q8—416no—
231C4×D4—416no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
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—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—432no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
302C2×C22⋊C4C2×D4116yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—42no—
311C2×C42—41no—
312C2×C42—41no—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—24no—
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—48no—
355C24—416no—
356C24—416no—
357C24—416no—
358C24—48no—
359C24—416no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—84no—
364C24—84no—
365C24—88no—
366C24—84no—
367C24—84no—
368C24—88no—
369C24—88no—
370C24—88no—
371C24—88no—
372C24—84no—
373C24—88no—
374C24—84no—
375C24—88no—
376C24—88no—
377C24C22×D414yes—
378C24C22×D414yes—
379C24C2×#4114yes—
380C24H(5,𝔽2)18yes—
381C24C2511yescharacteristic
382C2×Q8—28no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×D4—216no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—84no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4—88no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—84no—
412C2×D4#121116yes—
413C22×C4—21no—
414C22×C4—21no—
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—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
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—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—88no—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C22⋊C4—232no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C42—28no—
511C42—28no—
512C42—44no—
513C42—48no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—432no—
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—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—832no—
573C23—816no—
574C23—816no—
575C23—816no—
576C23—832no—
577C23—832no—
578C23—832no—
579C23—816no—
580C23—816no—
581C23—816no—
582C23C2×#12118yes—
583C23C23×D411yescharacteristic
584C23C23×D411yescharacteristic
585C23C22×#4111yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)14yes—
589C23#58511yescharacteristic
590C23#58511yescharacteristic
591C23#58514yes—
592Q8—416no—
593D4—416no—
594D4—816no—
595D4—816no—
596D4—832no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
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—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—816no—
660C2×C4—816no—
661C2×C4C2×#12114yes—
662C22—216no—
663C22—28no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—48no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—48no—
676C22—48no—
677C22—48no—
678C22—48no—
679C22—48no—
680C22—864no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22C22×#12111yescharacteristic
690C22#345911yescharacteristic
691C22#345911yescharacteristic
692C22#348411yescharacteristic
693C22#348412yes—
694C22#348511yescharacteristic
695C22#348814yes—
696C22#349611yescharacteristic
697C22#349611yescharacteristic
698C22#350012yes—
699C22#351314yes—
700C22#353212yes—
701C22#353314yes—
702C22#353712yes—
703C22#354012yes—
704C22#354512yes—
705C22#354514yes—
706C4—24no—
707C4—42no—
708C4—44no—
709C4—42no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—82no—
714C4—84no—
715C4—84no—
716C4—84no—
717C2—48no—
718C2—88no—
719C2#3498411yescharacteristic
720C2#3572812yes—
721C2#3687112yes—
722C2#4241412yes—
723C2#4681412yes—
724C2#4767812yes—
725C2#5174911yescharacteristic
726C2#5178512yes—
727C2#5200411yescharacteristic
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×#2684C2semidirect
3. C2×#2326C2semidirect
4. #16887C2semidirect
5. #16514C2semidirect
6. #16285C2semidirect
7. #15566C2semidirect
8. #15268C2semidirect
9. #15120C2semidirect
10. #14883C2semidirect
11. #14718C2semidirect
12. #14486C2semidirect
13. #13967C2semidirect
17. C22×C4×Q8C22semidirect
19. C23×C22⋊C4C22semidirect
22. C2×#397C22semidirect
23. C2×#396C22semidirect
29. C2×#393C22semidirect
31. C2×#391C22semidirect
35. C2×#388C22semidirect
36. C2×#388C22semidirect
40. C2×#385C22semidirect
41. C2×#384C22semidirect
42. C2×#379C22semidirect
43. C2×#379C22semidirect
44. C2×#376C22semidirect
45. #1445C22semidirect
46. #1444C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
69. C22⋊#95C22semidirect
80. C2×#124D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26842
C2×#26792
C2×#23262
#168872
#165142
#162852
#155662
#152682
#151202
#148832
#147182
#144862
#139672