Finite Group Databasebeta

← #763384

Subgroups — #763384

Order 512.

3504 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C212yesmaximal, complemented
2C2×#2311C212yesmaximal
3#15965C212yesmaximal, complemented
4#15964C212yesmaximal, complemented
5#15506C212yesmaximal, complemented
6#15429C212yesmaximal, complemented
7#15335C214yesmaximal, complemented
8#15331C214yesmaximal, complemented
9#15220C212yesmaximal, complemented
10#15135C211yesmaximal, characteristic, complemented
11#15125C211yesmaximal, characteristic, complemented
12#14130C212yesmaximal, complemented
13#13907C212yesmaximal, complemented
14#13879C212yesmaximal, complemented
15#13851C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2212yes—
20#2439—216no—
21C2×#399C2212yes—
22C2×#396C2214yes—
23C2×#393C2214yes—
24C2×#393C2212yes—
25#2311—216no—
26C2×#391C2214yes—
27C2×#391C2214yes—
28C2×#390C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#387C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1446C2212yes—
48#1446C2211yescharacteristic
49#1446C2211yescharacteristic
50#1445C2212yes—
51#1444C2212yes—
52#1444C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2211yescharacteristic
66#1438C2211yescharacteristic
67#1438C2212yes—
68#1438C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22×#41—28no—
78C22×#41—28no—
79C22×#41—42no—
80C22×#41—42no—
81C22×#41Q818yescomplemented
82C23×Q8C2311yescharacteristic
83C23×Q8C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—42no—
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—44no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2314yes—
124#399—44no—
125#396—216no—
126#393—216no—
127#393—44no—
128#391—216no—
129#391—216no—
130#390—216no—
131#389—28no—
132#389—28no—
133#389—44no—
134#387—216no—
135#386—216no—
136#386—216no—
137#385—48no—
138#384—42no—
139#382—28no—
140#380—28no—
141#380—28no—
142#380—28no—
143#380—48no—
144#380D4116yes—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—42no—
148C4×C22⋊C4—42no—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C25C2411yescharacteristic
175C25C2412yes—
176C2×#41—264no—
177C2×#41—416no—
178C2×#41—416no—
179C2×#41—432no—
180C2×#41—416no—
181C22×Q8—24no—
182C22×Q8—28no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—42no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8C2×Q814yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4C2×Q814yes—
202C22×D4C2×Q818yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C2×D418yes—
219C23×C4C2×Q814yes—
220C23×C4C2×Q818yes—
221C23×C4#4118yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C4⋊Q8—416no—
236#117—432no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—44no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—41no—
332#95—41no—
333#95—41no—
334#95—41no—
335#95—44no—
336#95—44no—
337#95—42no—
338#95—42no—
339#95C2×D4116yes—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24C22×D414yes—
363C24C22×Q812yes—
364C24C22×Q814yes—
365C24C2×#4114yes—
366C24C2511yescharacteristic
367#41—4128no—
368C2×Q8—216no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—416no—
375C2×Q8—88no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—216no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—44no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—216no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—416no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—416no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C22⋊Q8132yes—
473C22×C4C22×D418yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×Q812yes—
478C22×C4C22×Q814yes—
479C22×C4C22×Q812yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C22×C42−1+412yes—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—84no—
497C4⋊C4—88no—
498C4⋊C4—84no—
499C22⋊C4—416no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—416no—
512C42—44no—
513C42—48no—
514C42—44no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×C22⋊Q818yes—
573C23C23×D412yes—
574C23C23×Q811yescharacteristic
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×2−1+411yescharacteristic
581C23C2×2−1+411yescharacteristic
582C23#58512yes—
583C23#58512yes—
584Q8—416no—
585Q8—816no—
586D4—432no—
587D4—416no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—264no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—416no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—832no—
664C2×C4C2×C22⋊Q818yes—
665C2×C4C2×C22⋊Q818yes—
666C2×C4#54314yes—
667C2×C4D4214yes—
668C2×C4#54618yes—
669C2×C4#55114yes—
670C22—28no—
671C22—28no—
672C22—28no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—82no—
691C22—81no—
692C22—81no—
693C22—82no—
694C22—81no—
695C22—81no—
696C22C22×C22⋊Q812yes—
697C22C2×#54311yescharacteristic
698C22C2×D4211yescharacteristic
699C22C2×#54612yes—
700C22C2×#55111yescharacteristic
701C22#348812yes—
702C22#348912yes—
703C22#349412yes—
704C22#349512yes—
705C22#349612yes—
706C22#349712yes—
707C22#350311yescharacteristic
708C22#350311yescharacteristic
709C22#350412yes—
710C22#350612yes—
711C22#350712yes—
712C22#353611yescharacteristic
713C22#353611yescharacteristic
714C22#354012yes—
715C22#354911yescharacteristic
716C22#355011yescharacteristic
717C22#355111yescharacteristic
718C22#355111yescharacteristic
719C4—28no—
720C4—48no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—48no—
725C4—42no—
726C4—42no—
727C4—82no—
728C4—82no—
729C4—84no—
730C4—84no—
731C4#243718yes—
732C2—48no—
733C2—82no—
734C2C2×#243711yescharacteristic
735C2#3551912yes—
736C2#3557512yes—
737C2#3730611yescharacteristic
738C2#3742611yescharacteristic
739C2#4488712yes—
740C2#4510311yescharacteristic
741C2#4511511yescharacteristic
742C2#4524411yescharacteristic
743C2#4525111yescharacteristic
744C2#4532512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2439C2semidirect
3. #15965C2semidirect
4. #15964C2semidirect
5. #15506C2semidirect
6. #15429C2semidirect
7. #15335C2semidirect
8. #15331C2semidirect
9. #15220C2semidirect
10. #15135C2semidirect
11. #15125C2semidirect
12. #14130C2semidirect
13. #13907C2semidirect
14. #13879C2semidirect
81. C22×#41Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#23112
#159652
#159642
#155062
#154292
#153352
#153312
#152202
#151352
#151252
#141302
#139072
#138792
#138512