Finite Group Databasebeta

← #894658

Subgroups — #894658

Order 512.

3834 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2449C214yesmaximal, complemented
3C2×C4×#127C211yesmaximal, characteristic
4#17071C212yesmaximal, complemented
5#16522C212yesmaximal, complemented
6#15507C214yesmaximal, complemented
7#15325C214yesmaximal, complemented
8#14812C214yesmaximal, complemented
9#14723C212yesmaximal, complemented
10#14699C212yesmaximal, complemented
11#14440C211yesmaximal, characteristic, complemented
12#14419C212yesmaximal
13#14015C212yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2214yes—
17C22×C4×D4C2214yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2683—28no—
22#2449—232no—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#395C2214yes—
29C2×#395C2212yes—
30C2×#393C2212yes—
31C4×#127—28no—
32C2×#390C2214yes—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2212yes—
46C2×#384C2211yescharacteristic
47C2×C43C2211yescharacteristic
48#1444C2214yes—
49#1444C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C23×Q8C2312yes—
71C23×D4C2314yes—
72C23×D4C2311yescharacteristic
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#127—24no—
79C2×#127—24no—
80C2×#127Q818yescomplemented
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—48no—
85C2×C22⋊Q8—44no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—216no—
89C2×C4×D4—216no—
90C2×C4×D4—48no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C22×C4⋊C4—216no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#399—216no—
128#398—216no—
129#396—216no—
130#396—216no—
131#396—48no—
132#395—28no—
133#395—216no—
134#393—44no—
135#390—216no—
136#390—24no—
137#390D418yes—
138#389—28no—
139#389D4116yescomplemented
140#387—216no—
141#386—28no—
142#386—28no—
143#386—216no—
144#386—216no—
145#385—216no—
146#385—216no—
147#385—216no—
148#385—44no—
149#384—24no—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C43—24no—
166C25C2412yes—
167C25C2412yes—
168C22×Q8—24no—
169C22×Q8—28no—
170C22×Q8—44no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—22no—
176C22×D4—28no—
177C22×D4—22no—
178C22×D4—28no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—44no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4—44no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4C2×D414yes—
190C22×D4C2×Q8116yes—
191C22×D4C2×Q814yes—
192C22×D4C2×Q814yes—
193C22×D4#4114yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225#127—232no—
226C22⋊Q8—464no—
227C22⋊Q8—432no—
228C4×D4—464no—
229C4×D4—416no—
230C4×D4—416no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—432no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D4116yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
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—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D4116yes—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—41no—
316C2×C42—41no—
317C2×C42—42no—
318C2×C42C2×D418yes—
319C2×C42C2×D414yes—
320C2×C42C2×Q814yes—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—42no—
334#95—44no—
335#95C2×D4116yes—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24C22×D414yes—
355C24C22×Q814yes—
356C24C22×Q814yes—
357C24C2×#4114yes—
358C24C2511yescharacteristic
359C2×Q8—416no—
360C2×Q8—48no—
361C2×Q8—84no—
362C2×Q8—88no—
363C2×Q8—84no—
364C2×D4—232no—
365C2×D4—28no—
366C2×D4—232no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4C22⋊Q8116yescomplemented
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C22×D418yes—
460C22×C4C22×D414yes—
461C22×C4C22×D418yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q814yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+414yes—
474C4⋊C4—232no—
475C4⋊C4—216no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—416no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C22⋊C4—232no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—88no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C42—28no—
500C42—28no—
501C42—216no—
502C42—44no—
503C42—48no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23C2×C22⋊Q818yes—
550C23C23×D411yescharacteristic
551C23C23×D412yes—
552C23C23×Q811yescharacteristic
553C23C22×#4111yescharacteristic
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+412yes—
558C23#58511yescharacteristic
559C23#58512yes—
560C23#58511yescharacteristic
561Q8—816no—
562D4—416no—
563D4—464no—
564D4—416no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—432no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4C2×C22⋊Q814yes—
635C2×C4D4218yes—
636C2×C4D4214yes—
637C2×C4D4×Q818yes—
638C22—28no—
639C22—216no—
640C22—48no—
641C22—48no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—48no—
652C22—44no—
653C22—48no—
654C22—84no—
655C22—84no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C22×C22⋊Q811yescharacteristic
661C22#344912yes—
662C22#345911yescharacteristic
663C22#345911yescharacteristic
664C22C2×D4212yes—
665C22C2×D4211yescharacteristic
666C22C2×D4×Q812yes—
667C22D4×#4112yes—
668C22#348411yescharacteristic
669C22#348511yescharacteristic
670C22#348912yes—
671C22#349712yes—
672C22#349812yes—
673C22#350111yescharacteristic
674C22#350111yescharacteristic
675C22#350112yes—
676C22#352012yes—
677C22#352911yescharacteristic
678C22#354311yescharacteristic
679C22#355011yescharacteristic
680C22#355912yes—
681C22#355912yes—
682C22#356012yes—
683C4—28no—
684C4—24no—
685C4—24no—
686C4—28no—
687C4—48no—
688C4—42no—
689C4—44no—
690C4—48no—
691C4—84no—
692C4—84no—
693C4—82no—
694C2—48no—
695C2—84no—
696C2#3498511yescharacteristic
697C2D4×C22⋊Q812yes—
698C2#3725411yescharacteristic
699C2#3767912yes—
700C2#3944512yes—
701C2#3961112yes—
702C2#4622811yescharacteristic
703C2#4628211yescharacteristic
704C2#5210311yescharacteristic
705C2#5912611yescharacteristic
706C2#5981711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2683C2semidirect
2. C2×#2449C2semidirect
4. #17071C2semidirect
5. #16522C2semidirect
6. #15507C2semidirect
7. #15325C2semidirect
8. #14812C2semidirect
9. #14723C2semidirect
10. #14699C2semidirect
11. #14440C2semidirect
13. #14015C2semidirect
80. C2×#127Q8semidirect
139. #389D4semidirect
384. C2×D4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#24492
C2×C4×#1272
#170712
#165222
#155072
#153252
#148122
#147232
#146992
#144402
#144192
#140152