Finite Group Databasebeta

← #555927

Subgroups — #555927

Order 512.

3870 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#2318C212yesmaximal, complemented
3#14683C211yesmaximal, characteristic, complemented
4#14682C211yesmaximal, characteristic, complemented
5#14475C214yesmaximal, complemented
6#14474C212yesmaximal, complemented
7#14473C212yesmaximal, complemented
8#14448C212yesmaximal, complemented
9#14417C212yesmaximal
10#14359C214yesmaximal, complemented
11#14346C212yesmaximal, complemented
12#13960C212yesmaximal, complemented
13#13958C212yesmaximal, complemented
14#13940C212yesmaximal, complemented
15#13939C212yesmaximal, complemented
16C2×#529—216no—
17C2×#529C4116yescomplemented
18C22×#126C2212yes—
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2212yes—
26C22×C4×D4C2211yescharacteristic
27C22×#117C2212yes—
28C23×C42C2211yescharacteristic
29#2318—216no—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#383C2212yes—
36C2×#382C2212yes—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C22×#95C2212yes—
42#1446C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1437C2214yes—
55#1436C2214yes—
56C4⋊#95C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2214yes—
59#1434C2212yes—
60#1434C2212yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2211yescharacteristic
64#1434C2211yescharacteristic
65#1434C2212yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C22⋊#95C2214yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C24×C4C2312yes—
86#529—2128no—
87C2×#126—28no—
88C2×#126—28no—
89C2×#126C2×C4116yescomplemented
90C2×#125—28no—
91C2×#125C2×C418yescomplemented
92C2×#124—28no—
93C2×#124C2×C418yes—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123C2×C4116yescomplemented
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8C2×C418yescomplemented
99C2×#121—28no—
100C2×#121C2×C418yes—
101C2×C4×Q8—28no—
102C2×C4×Q8C2×C418yescomplemented
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4C2×C418yes—
106C2×C4×D4C2×C4116yescomplemented
107C2×C4×D4D4116yes—
108C2×#117—28no—
109C2×#117—28no—
110C2×#117C2×C4116yescomplemented
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42—28no—
131C22×C42C2×C418yescomplemented
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138#388—216no—
139#387—216no—
140#386—216no—
141#386—48no—
142#385—216no—
143#383—28no—
144#382—28no—
145#381—28no—
146#380—28no—
147#380—44no—
148C4×C22⋊C4—28no—
149C2×#95—28no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8C22×C414yes—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4C22×C418yes—
191C22×D4C22×C414yes—
192C22×D4C2×D418yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C22×C414yes—
213C23×C4C22×C414yes—
214C23×C4C22×C414yes—
215C23×C4C22×C418yes—
216C23×C4C2×D418yes—
217C23×C4C2×D418yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#126—264no—
232#125—232no—
233#124—232no—
234#123—264no—
235C22⋊Q8—232no—
236#121—232no—
237C4×Q8—232no—
238C4×D4—264no—
239C4×D4C22⋊C4164yescomplemented
240#117—264no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C2×D418yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
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—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C2×D418yes—
295C2×C22⋊C4C2×D418yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—232no—
302C2×C42—28no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C22×C414yes—
306C2×C42C22×C418yes—
307C2×C42C22×C414yes—
308C2×C42C22×C414yes—
309C2×C42C22×C414yes—
310C2×C42C2×D418yes—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24C23×C414yes—
352C24C23×C412yes—
353C24C22×D414yes—
354C24C22×D414yes—
355C24H(5,𝔽2)12yes—
356C242−1+412yes—
357C24C2511yescharacteristic
358C2×Q8—28no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—416no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4C2×C22⋊C4116yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4C2×C22⋊C4132yes—
455C22×C4C23×C412yes—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—28no—
473C4⋊C4—216no—
474C4⋊C4—88no—
475C4⋊C4—88no—
476C4⋊C4C2×C22⋊C4116yes—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—416no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4C2×C22⋊C4132yes—
488C42—28no—
489C42—28no—
490C42—28no—
491C42—216no—
492C42—88no—
493C42C2×C22⋊C4116yes—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23C22×C22⋊C418yes—
533C23#52014yes—
534C23C24×C411yescharacteristic
535C23C23×D412yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23#58512yes—
539C23#58512yes—
540C23#58512yes—
541C23#58512yes—
542C23#58512yes—
543Q8—416no—
544D4—416no—
545D4—432no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—432no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—832no—
618C2×C4C22×C22⋊C414yes—
619C2×C4C22×C22⋊C414yes—
620C2×C4C22×C22⋊C414yes—
621C2×C4C22×C22⋊C418yes—
622C22—24no—
623C22—24no—
624C22—28no—
625C22—24no—
626C22—28no—
627C22—28no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—44no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22C23×C22⋊C411yescharacteristic
643C22C2×#52011yescharacteristic
644C22#342811yescharacteristic
645C22#342811yescharacteristic
646C22#342811yescharacteristic
647C22#342811yescharacteristic
648C22#342811yescharacteristic
649C22#344914yes—
650C22#348514yes—
651C22#348812yes—
652C22#348912yes—
653C22#354512yes—
654C22#355312yes—
655C22#356012yes—
656C22#356412yes—
657C22#356614yes—
658C22#356614yes—
659C4—24no—
660C4—28no—
661C4—24no—
662C4—24no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—84no—
669C4—84no—
670C4—84no—
671C4—84no—
672C2—44no—
673C2—48no—
674C2#3404411yescharacteristic
675C2#3417211yescharacteristic
676C2#3417511yescharacteristic
677C2#3439011yescharacteristic
678C2#3439511yescharacteristic
679C2#3440011yescharacteristic
680C2#3440111yescharacteristic
681C2#3712114yes—
682C2#3712214yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2318C2semidirect
3. #14683C2semidirect
4. #14682C2semidirect
5. #14475C2semidirect
6. #14474C2semidirect
7. #14473C2semidirect
8. #14448C2semidirect
10. #14359C2semidirect
11. #14346C2semidirect
12. #13960C2semidirect
13. #13958C2semidirect
14. #13940C2semidirect
15. #13939C2semidirect
17. C2×#529C4semidirect
89. C2×#126C2×C4semidirect
91. C2×#125C2×C4semidirect
96. C2×#123C2×C4semidirect
98. C2×C22⋊Q8C2×C4semidirect
102. C2×C4×Q8C2×C4semidirect
106. C2×C4×D4C2×C4semidirect
110. C2×#117C2×C4semidirect
131. C22×C42C2×C4semidirect
239. C4×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#23182
#146832
#146822
#144752
#144742
#144732
#144482
#144172
#143592
#143462
#139602
#139582
#139402
#139392