Finite Group Databasebeta

← #4940665

Subgroups — #4940665

Order 512.

4922 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
1C22×#563C211yesmaximal, characteristic, complemented
2C2×#2435C211yesmaximal, characteristic
3C2×#2405C212yesmaximal
4#16633C212yesmaximal, complemented
5#16251C214yesmaximal, complemented
6#15986C214yesmaximal, complemented
7#15908C214yesmaximal, complemented
8#15761C212yesmaximal, complemented
9#15265C212yesmaximal, complemented
10#14916C212yesmaximal, complemented
11#14717C214yesmaximal, complemented
12#14505C212yesmaximal, complemented
13#14473C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C2×#563—28no—
16C2×#563—216no—
17C22×C4⋊Q8C2212yescomplemented
18C22×#126C2214yescomplemented
19C22×#125C2212yescomplemented
20C22×#124C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C22×#123C2212yes—
23C22×#123C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2214yescomplemented
26C23×C22⋊C4C2212yes—
27#2435—28no—
28#2405—216no—
29C2×#395C2212yescomplemented
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#393C2212yes—
34C2×#393C2214yescomplemented
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2214yes—
40C2×#379C2212yes—
41C2×#379C2214yes—
42C2×#379C2214yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44#1446C2212yescomplemented
45#1444C2214yescomplemented
46#1444C2212yescomplemented
47#1442C2212yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1439C2212yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1437C2212yes—
64#1437C2214yescomplemented
65#1436C2212yescomplemented
66#1436C2214yescomplemented
67#1434C2212yescomplemented
68C22⋊#95C2214yescomplemented
69C22⋊#95C2214yescomplemented
70C22⋊#95C2214yescomplemented
71C26C2312yes—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C23×D4—28no—
75C23×D4—24no—
76C23×D4—42no—
77C23×D4—42no—
78C23×D4—44no—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85#563—2128no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—216no—
88C2×#126—216no—
89C2×#126—216no—
90C2×#126—216no—
91C2×#125—216no—
92C2×#125—28no—
93C2×#124—28no—
94C2×#124—24no—
95C2×#124—44no—
96C2×#124D418yes—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—44no—
101C2×#123—44no—
102C2×#123D4116yes—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—216no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—42no—
108C2×C22⋊Q8—42no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—28no—
118C22×C22⋊C4—44no—
119C22×C22⋊C4—44no—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#395—44no—
140#394—216no—
141#394—216no—
142#394—216no—
143#393—44no—
144#393—48no—
145#390—24no—
146#390—24no—
147#386—28no—
148#386—28no—
149#385—216no—
150#379—28no—
151#379—216no—
152#379—216no—
153C4×C22⋊C4—24no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25—24no—
172C25—24no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—28no—
177C25—42no—
178C25—42no—
179C25—44no—
180C25—44no—
181C25—44no—
182C25C2412yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C22×Q8—28no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—44no—
193C22×Q8—42no—
194C22×Q8C2×D414yes—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—42no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—416no—
211C22×D4—432no—
212C22×D4—416no—
213C22×D4—432no—
214C22×D4—416no—
215C22×D4C2×D414yes—
216C22×D4C2×D418yes—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C2×D418yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C4⋊Q8—264no—
245#126—2128no—
246#125—264no—
247#124—232no—
248#124—416no—
249#123—264no—
250#123—432no—
251C22⋊Q8—2128no—
252C22⋊Q8—416no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D4116yes—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—432no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×D4116yes—
322C2×C22⋊C4C2×D4116yes—
323C2×C22⋊C4C2×D418yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—22no—
328C2×C42—42no—
329C2×C42C2×D414yes—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—42no—
345#95—44no—
346#95—42no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—216no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—416no—
377C24—416no—
378C24—416no—
379C24—416no—
380C24—48no—
381C24—48no—
382C24—48no—
383C24—48no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24—84no—
388C24—88no—
389C24—84no—
390C24—88no—
391C24—88no—
392C24—88no—
393C24—88no—
394C24C22×D414yes—
395C24C22×D414yes—
396C24C22×D414yes—
397C24H(5,𝔽2)14yes—
398C24H(5,𝔽2)12yes—
399C242−1+414yes—
400C24C2511yescharacteristic
401C2×Q8—216no—
402C2×Q8—28no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—84no—
406C2×Q8—84no—
407C2×Q8—88no—
408C2×Q8—84no—
409C2×Q8—88no—
410C2×Q8—84no—
411C2×D4—216no—
412C2×D4—28no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—4128no—
426C2×D4—48no—
427C2×D4—44no—
428C2×D4#120116yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—28no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—216no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
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—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—88no—
497C22×C4C22×D418yes—
498C22×C4C22×D414yes—
499C22×C4C22×D414yes—
500C22×C4C22×D418yes—
501C22×C4C22×D414yes—
502C22×C4C22×D412yes—
503C22×C4H(5,𝔽2)12yes—
504C4⋊C4—216no—
505C4⋊C4—232no—
506C4⋊C4—232no—
507C4⋊C4—216no—
508C4⋊C4—232no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—44no—
512C22⋊C4—232no—
513C22⋊C4—232no—
514C22⋊C4—216no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C42—216no—
529C42—28no—
530C42—44no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—24no—
548C23—28no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—28no—
555C23—28no—
556C23—432no—
557C23—432no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—44no—
589C23—42no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—816no—
594C23—816no—
595C23—832no—
596C23—816no—
597C23—832no—
598C23—832no—
599C23—816no—
600C23—832no—
601C23—816no—
602C23C2×#12018yes—
603C23#56314yescomplemented
604C23C23×D411yescharacteristic
605C23C23×D412yes—
606C23C2×H(5,𝔽2)11yescharacteristic
607C23C2×H(5,𝔽2)12yes—
608C23C2×H(5,𝔽2)11yescharacteristic
609C23C2×2−1+412yes—
610C23#58514yes—
611C23#58512yes—
612Q8—88no—
613Q8—816no—
614D4—416no—
615D4—416no—
616D4—432no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—24no—
634C2×C4—28no—
635C2×C4—416no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
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—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
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—44no—
674C2×C4—44no—
675C2×C4—816no—
676C2×C4—816no—
677C2×C4C2×#12014yes—
678C22—28no—
679C22—24no—
680C22—28no—
681C22—216no—
682C22—44no—
683C22—42no—
684C22—42no—
685C22—44no—
686C22—48no—
687C22—48no—
688C22—48no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—48no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—48no—
699C22—44no—
700C22—44no—
701C22—48no—
702C22—864no—
703C22—82no—
704C22—82no—
705C22—84no—
706C22—82no—
707C22—82no—
708C22—84no—
709C22C22×#12011yescharacteristic
710C22#348411yescharacteristic
711C22#348412yes—
712C22#348412yes—
713C22#348411yescharacteristic
714C22#348912yes—
715C22#348914yes—
716C22#350012yes—
717C22#350014yes—
718C22C2×#56311yescharacteristic
719C22#353312yes—
720C22#353514yes—
721C22#354511yescharacteristic
722C22#355112yes—
723C22#356014yes—
724C22#356112yes—
725C4—24no—
726C4—42no—
727C4—48no—
728C4—44no—
729C4—48no—
730C4—44no—
731C4—82no—
732C4—84no—
733C4—84no—
734C4—82no—
735C2—48no—
736C2—44no—
737C2—84no—
738C2#3534611yescharacteristic
739C2#3607812yes—
740C2#4545612yes—
741C2#4613611yescharacteristic
742C2#4795614yes—
743C2#5086712yes—
744C2#5089712yes—
745C2#5197811yescharacteristic
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. C22×#563C2semidirect
4. #16633C2semidirect
5. #16251C2semidirect
6. #15986C2semidirect
7. #15908C2semidirect
8. #15761C2semidirect
9. #15265C2semidirect
10. #14916C2semidirect
11. #14717C2semidirect
12. #14505C2semidirect
13. #14473C2semidirect
17. C22×C4⋊Q8C22semidirect
18. C22×#126C22semidirect
19. C22×#125C22semidirect
25. C22×C22⋊Q8C22semidirect
29. C2×#395C22semidirect
34. C2×#393C22semidirect
44. #1446C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
64. #1437C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1434C22semidirect
68. C22⋊#95C22semidirect
69. C22⋊#95C22semidirect
70. C22⋊#95C22semidirect
603. C23#563semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
C2×#24352
C2×#24052
#166332
#162512
#159862
#159082
#157612
#152652
#149162
#147172
#145052
#144732