Finite Group Databasebeta

← #2923140

Subgroups — #2923140

Order 512.

4342 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic, complemented
2C2×#2678C214yesmaximal, complemented
3C2×#2501C211yesmaximal, characteristic, complemented
4#16514C212yesmaximal, complemented
5#16467C214yesmaximal, complemented
6#15611C214yesmaximal, complemented
7#15283C214yesmaximal, complemented
8#15281C212yesmaximal, complemented
9#15257C212yesmaximal, complemented
10#15136C212yesmaximal, complemented
11#14719C212yesmaximal, complemented
12#14228C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic, complemented
14C22×#123C2214yescomplemented
15C22×#121C2212yes—
16C22×#121C2211yescharacteristic
17C22×#120C2214yes—
18C22×#120C2212yes—
19C22×#120C2212yescomplemented
20C23×C4⋊C4C2211yescharacteristic, complemented
21#2679—28no—
22#2678—232no—
23#2501—28no—
24C2×#399C2214yescomplemented
25C2×#397C2214yescomplemented
26C2×#397C2214yescomplemented
27C2×#396C2214yescomplemented
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#394C2212yes—
33C2×#392C2214yes—
34C2×#392C2212yes—
35C2×#390C2214yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2214yescomplemented
40C2×#388C2214yescomplemented
41C2×#388C2214yescomplemented
42C2×#386C2214yescomplemented
43C2×#386C2214yescomplemented
44C2×#384C2211yescharacteristic, complemented
45C2×#384C2211yescharacteristic, complemented
46C2×#379C2214yescomplemented
47C2×#379C2212yescomplemented
48C2×C4×C22⋊C4C2211yescharacteristic, complemented
49#1444C2212yescomplemented
50#1441C2212yescomplemented
51#1441C2212yescomplemented
52#1441C2214yescomplemented
53#1440C2212yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2212yescomplemented
60#1438C2212yes—
61#1438C2214yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2212yescomplemented
65#1437C2214yescomplemented
66#1437C2214yescomplemented
67C4⋊#95C2214yescomplemented
68C22⋊#95C2212yescomplemented
69C4×#95C2212yescomplemented
70C26C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2314yes—
74C23×D4C2314yes—
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×#123—216no—
80C2×#123—216no—
81C2×#123—48no—
82C2×#121—216no—
83C2×#121—24no—
84C2×#121—42no—
85C2×#121—42no—
86C2×#121—44no—
87C2×#120—216no—
88C2×#120—28no—
89C2×#120—216no—
90C2×#120—216no—
91C2×#120—48no—
92C2×#120—48no—
93C2×#120—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128#399—216no—
129#397—216no—
130#397—48no—
131#396—216no—
132#394—28no—
133#394—216no—
134#394—216no—
135#394—216no—
136#394—44no—
137#392—216no—
138#392D4116yescomplemented
139#390—216no—
140#390—24no—
141#390—216no—
142#390—28no—
143#390D418yescomplemented
144#388—216no—
145#388—48no—
146#386—216no—
147#386—48no—
148#384—24no—
149#384—24no—
150#379—216no—
151#379—44no—
152C4×C22⋊C4—24no—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25—24no—
167C25—28no—
168C25—44no—
169C25—44no—
170C25—42no—
171C25—44no—
172C25—44no—
173C25—44no—
174C25—42no—
175C25C2412yes—
176C25C2414yes—
177C25C2411yescharacteristic
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—42no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—42no—
209C22×D4C2×D414yes—
210C22×D4C2×D418yes—
211C22×D4C2×D4116yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232#123—464no—
233#121—416no—
234#121—432no—
235#120—464no—
236#120—432no—
237#120—432no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—416no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
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—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
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—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D4116yes—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4C2×D4116yes—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95C2×D418yes—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—416no—
367C24—48no—
368C24—48no—
369C24—416no—
370C24—48no—
371C24—84no—
372C24—88no—
373C24—84no—
374C24—88no—
375C24—88no—
376C24—88no—
377C24—88no—
378C24—88no—
379C24—88no—
380C24—84no—
381C24—84no—
382C24—88no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24C22×D418yes—
388C24C22×D414yes—
389C24C22×D414yes—
390C24H(5,𝔽2)12yes—
391C24C2511yescharacteristic
392C2×D4—232no—
393C2×D4—28no—
394C2×D4—216no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—416no—
398C2×D4—416no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—416no—
405C2×D4—416no—
406C2×D4—48no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—88no—
412C2×D4—84no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—88no—
425C2×D4—88no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
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—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—416no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—88no—
482C22×C4—84no—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)14yes—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—416no—
499C4⋊C4—84no—
500C4⋊C4—88no—
501C22⋊C4—232no—
502C22⋊C4—216no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—84no—
520C42—44no—
521C42—44no—
522C42—88no—
523C23—24no—
524C23—22no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—42no—
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—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—816no—
595C23—832no—
596C23—832no—
597C23—832no—
598C23—816no—
599C23—816no—
600C23—832no—
601C23—832no—
602C23—832no—
603C23D4214yes—
604C23C23×D411yescharacteristic
605C23C23×D412yes—
606C23C2×H(5,𝔽2)12yes—
607C23C2×H(5,𝔽2)11yescharacteristic
608C23C2×H(5,𝔽2)12yes—
609C23C2×H(5,𝔽2)11yescharacteristic
610C23#58512yes—
611C23#58514yes—
612D4—88no—
613D4—816no—
614D4—832no—
615D4—816no—
616D4—816no—
617D4—832no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—416no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
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—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—816no—
674C2×C4#53514yes—
675C2×C4D4218yes—
676C22—28no—
677C22—24no—
678C22—42no—
679C22—42no—
680C22—48no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—48no—
685C22—48no—
686C22—44no—
687C22—44no—
688C22—48no—
689C22—48no—
690C22—48no—
691C22—48no—
692C22—48no—
693C22—48no—
694C22—48no—
695C22—44no—
696C22—864no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22—84no—
703C22—84no—
704C22—84no—
705C22—84no—
706C22—84no—
707C22—84no—
708C22—84no—
709C22C2×#53511yescharacteristic
710C22C2×D4211yescharacteristic
711C22C2×D4212yes—
712C22#348411yescharacteristic
713C22#348514yes—
714C22#348511yescharacteristic
715C22#349614yes—
716C22#349812yes—
717C22#350012yes—
718C22#353412yes—
719C22#353412yes—
720C22#354012yes—
721C22#354311yescharacteristic
722C22#354512yes—
723C22#354614yes—
724C22#356014yes—
725C4—24no—
726C4—28no—
727C4—48no—
728C4—42no—
729C4—44no—
730C4—44no—
731C4—84no—
732C4—84no—
733C4—82no—
734C4—82no—
735C2—44no—
736C2—88no—
737C2—84no—
738C2#3672111yescharacteristic
739C2#3936212yes—
740C2#4301712yes—
741C2#4349212yes—
742C2#4956914yes—
743C2#5174911yescharacteristic
744C2#5348912yes—
745C2#5912611yescharacteristic
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×#2678C2semidirect
3. C2×#2501C2semidirect
4. #16514C2semidirect
5. #16467C2semidirect
6. #15611C2semidirect
7. #15283C2semidirect
8. #15281C2semidirect
9. #15257C2semidirect
10. #15136C2semidirect
11. #14719C2semidirect
12. #14228C2semidirect
13. #14157C2semidirect
14. C22×#123C22semidirect
19. C22×#120C22semidirect
20. C23×C4⋊C4C22semidirect
24. C2×#399C22semidirect
25. C2×#397C22semidirect
26. C2×#397C22semidirect
27. C2×#396C22semidirect
39. C2×#390C22semidirect
40. C2×#388C22semidirect
41. C2×#388C22semidirect
42. C2×#386C22semidirect
43. C2×#386C22semidirect
44. C2×#384C22semidirect
45. C2×#384C22semidirect
46. C2×#379C22semidirect
47. C2×#379C22semidirect
48. C2×C4×C22⋊C4C22semidirect
49. #1444C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1440C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
64. #1438C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. C4⋊#95C22semidirect
68. C22⋊#95C22semidirect
69. C4×#95C22semidirect
138. #392D4semidirect
143. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#26782
C2×#25012
#165142
#164672
#156112
#152832
#152812
#152572
#151362
#147192
#142282
#141572