Finite Group Databasebeta

← #770340

Subgroups — #770340

Order 512.

3398 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#395C211yesmaximal, characteristic, complemented
2C2×#2325C211yesmaximal, characteristic
3#15861C214yesmaximal, complemented
4#15772C214yesmaximal, complemented
5#15476C214yesmaximal, complemented
6#15475C214yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#14873C212yesmaximal, complemented
9#14152C211yesmaximal, characteristic
10#14137C211yesmaximal, characteristic
11#13966C211yesmaximal, characteristic, complemented
12#13960C212yesmaximal, complemented
13#13881C211yesmaximal, characteristic, complemented
14#13872C212yesmaximal, complemented
15#13778C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#395—28no—
23C2×#395C2214yes—
24C2×#395C2214yes—
25C2×#393C2214yes—
26C2×#393C2214yes—
27#2325—28no—
28C2×#390C2212yes—
29C2×#388C2212yes—
30C2×#387C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2214yes—
48#1446C2214yes—
49#1445C2212yes—
50#1442C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—44no—
79C23×Q8—44no—
80C23×Q8—44no—
81C23×Q8C2314yes—
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#121—24no—
93C2×#121—24no—
94C2×#121Q818yescomplemented
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#395—216no—
128#395—216no—
129#393—48no—
130#393—48no—
131#390—44no—
132#388—28no—
133#387—28no—
134#386—28no—
135#386—28no—
136#386—44no—
137#386—44no—
138#386—44no—
139#386—44no—
140#385—28no—
141#382—44no—
142#380—24no—
143#380—28no—
144#380—48no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95D418yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—28no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—416no—
185C22×Q8—416no—
186C22×Q8—416no—
187C22×Q8—464no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4C2×Q818yes—
199C22×D4C2×Q814yescomplemented
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×Q814yes—
232C23×C4#4114yes—
233C23×C4#4118yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247#121—232no—
248C4×D4—432no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4C2×Q814yescomplemented
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
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⋊C4C2×Q818yescomplemented
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C2×D418yes—
341#95C2×D418yes—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24C22×D412yes—
357C24C22×Q814yes—
358C24C22×Q812yes—
359C24C2×#4112yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)14yes—
362C24C2511yescharacteristic
363C2×Q8—416no—
364C2×Q8—416no—
365C2×Q8—416no—
366C2×Q8—416no—
367C2×Q8—416no—
368C2×Q8—416no—
369C2×Q8—4128no—
370C2×Q8—416no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—216no—
374C2×D4—28no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—84no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
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—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
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—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—416no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C22⋊Q8116yes—
473C22×C4#124116yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×Q814yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q812yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C42−1+414yes—
488C4⋊C4—28no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—88no—
497C22⋊C4—216no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—48no—
509C42—48no—
510C42—84no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23C2×C22⋊Q814yes—
547C23C2×#12414yes—
548C23#53514yes—
549C23#55418yes—
550C23#55518yes—
551C23C23×D411yescharacteristic
552C23C23×Q811yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×2−1+412yes—
559C23#58512yes—
560Q8—464no—
561D4—432no—
562D4—88no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
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—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—816no—
638C2×C4—88no—
639C2×C4C2×C22⋊Q814yes—
640C2×C4C2×C22⋊Q814yes—
641C2×C4C2×#12418yes—
642C22—24no—
643C22—24no—
644C22—216no—
645C22—24no—
646C22—216no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—44no—
651C22—48no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—82no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22—82no—
661C22#256318yes—
662C22C22×C22⋊Q811yescharacteristic
663C22C22×#12411yescharacteristic
664C22C2×#53511yescharacteristic
665C22C2×#55412yes—
666C22C2×#55512yes—
667C22#348411yescharacteristic
668C22#348411yescharacteristic
669C22#348511yescharacteristic
670C22#348611yescharacteristic
671C22#350112yes—
672C22#350312yes—
673C22#350512yes—
674C22#350514yes—
675C22#351114yes—
676C22#351112yes—
677C22#353214yes—
678C22#353514yes—
679C4—24no—
680C4—24no—
681C4—28no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—84no—
690C4—84no—
691C4—82no—
692C2—28no—
693C2—84no—
694C2C2×#256311yescharacteristic
695C2#3534911yescharacteristic
696C2#3537711yescharacteristic
697C2#3557411yescharacteristic
698C2#3559211yescharacteristic
699C2#3670311yescharacteristic
700C2#3711811yescharacteristic
701C2#4355114yes—
702C2#4357514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#395C2semidirect
3. #15861C2semidirect
4. #15772C2semidirect
5. #15476C2semidirect
6. #15475C2semidirect
7. #15091C2semidirect
8. #14873C2semidirect
11. #13966C2semidirect
12. #13960C2semidirect
13. #13881C2semidirect
14. #13872C2semidirect
15. #13778C2semidirect
94. C2×#121Q8semidirect
199. C22×D4C2×Q8semidirect
276. C2×C4⋊C4C2×Q8semidirect
307. C2×C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#3952
C2×#23252
#158612
#157722
#154762
#154752
#150912
#148732
#141522
#141372
#139662
#139602
#138812
#138722
#137782