Finite Group Databasebeta

← #719156

Subgroups — #719156

Order 512.

2918 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C212yesmaximal
2C2×#2399C211yesmaximal, characteristic, complemented
3C2×#2349C211yesmaximal, characteristic
4#16303C214yesmaximal, complemented
5#15622C214yesmaximal, complemented
6#15545C214yesmaximal, complemented
7#15221C212yesmaximal, complemented
8#15158C212yesmaximal, complemented
9#14736C214yesmaximal, complemented
10#14276C211yesmaximal, characteristic, complemented
11#14018C212yesmaximal
12#13954C211yesmaximal, characteristic, complemented
13#13911C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic
15#13785C211yesmaximal, characteristic
16C22×C4×Q8C2212yes—
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2580—216no—
20#2399—28no—
21C2×#399C2214yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#395C2212yes—
25C2×#395C2212yes—
26C2×#395C2214yes—
27C2×#393C2214yes—
28#2349—28no—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2212yes—
43C2×#380C2212yes—
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×#376C2211yescharacteristic
50C22×#95C2212yes—
51#1445C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1437C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4×Q8—28no—
83C2×C4×Q8—28no—
84C2×C4×Q8—44no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—28no—
88C2×C4×D4—24no—
89C2×C4×D4—44no—
90C2×C4×D4Q818yescomplemented
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121#399—216no—
122#398—28no—
123#398—28no—
124#395—28no—
125#395—28no—
126#395—48no—
127#393—216no—
128#390—28no—
129#390D418yes—
130#390Q818yescomplemented
131#389—28no—
132#389—28no—
133#389—28no—
134#389—44no—
135#385—216no—
136#385—216no—
137#385—216no—
138#385—48no—
139#384—24no—
140#382—28no—
141#382—24no—
142#380—28no—
143#380—48no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4D418yes—
148#376—24no—
149C2×#95—28no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C25C2412yes—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4C2×D414yes—
188C22×D4C2×Q814yes—
189C22×D4C2×Q814yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C2×Q818yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4×Q8—432no—
217C4×D4—232no—
218C4×D4—432no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
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—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×Q814yes—
270C2×C4⋊C4C2×Q814yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×D4116yes—
287C2×C22⋊C4C2×Q818yes—
288C2×C22⋊C4C2×Q8116yes—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42C2×D418yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×Q814yes—
304C2×C42C2×Q814yes—
305C2×C42#4114yes—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—22no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24C22×D414yes—
340C24C22×Q814yes—
341C24C22×Q814yes—
342C24C2511yescharacteristic
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4C4⋊Q8116yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
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—22no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—88no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C22⋊C4—216no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—88no—
481C42—216no—
482C42—28no—
483C42—28no—
484C42—48no—
485C42—84no—
486C42C22⋊Q8116yes—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23C2×C4⋊Q818yes—
510C23C23×D411yescharacteristic
511C23C23×D411yescharacteristic
512C23C23×Q811yescharacteristic
513C23C23×Q811yescharacteristic
514C23C22×#4111yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)12yes—
517C23C2×2−1+411yescharacteristic
518C23#58512yes—
519C23#58512yes—
520C23#58512yes—
521Q8—432no—
522D4—232no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4—832no—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×C22⋊Q814yes—
607C2×C4C2×C4⋊Q814yes—
608C2×C4#54414yes—
609C2×C4D4214yes—
610C2×C4#54714yes—
611C2×C4D4×Q814yes—
612C2×C4#55414yes—
613C22—216no—
614C22—28no—
615C22—44no—
616C22—48no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—48no—
622C22—44no—
623C22—48no—
624C22C22×C22⋊Q811yescharacteristic
625C22C22×C4⋊Q811yescharacteristic
626C22#344912yes—
627C22C2×#54411yescharacteristic
628C22C2×D4211yescharacteristic
629C22C2×#54711yescharacteristic
630C22C2×D4×Q811yescharacteristic
631C22C2×#55411yescharacteristic
632C22#348512yes—
633C22#348812yes—
634C22#348812yes—
635C22#350112yes—
636C22#350112yes—
637C22#350412yes—
638C22#350412yes—
639C22#350811yescharacteristic
640C22#351011yescharacteristic
641C22#351011yescharacteristic
642C22#351011yescharacteristic
643C22#354012yes—
644C22#354212yes—
645C22#355412yes—
646C22#355712yes—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—84no—
659C4—84no—
660C4#244018yes—
661C2—48no—
662C2C2×#244011yescharacteristic
663C2#3533411yescharacteristic
664C2#3551511yescharacteristic
665C2#3595012yes—
666C2#3765812yes—
667C2#3959012yes—
668C2#4249312yes—
669C2#4494711yescharacteristic
670C2#4513711yescharacteristic
671C2#4542111yescharacteristic
672C2#4542511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2399C2semidirect
4. #16303C2semidirect
5. #15622C2semidirect
6. #15545C2semidirect
7. #15221C2semidirect
8. #15158C2semidirect
9. #14736C2semidirect
10. #14276C2semidirect
12. #13954C2semidirect
13. #13911C2semidirect
90. C2×C4×D4Q8semidirect
130. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23992
C2×#23492
#163032
#156222
#155452
#152212
#151582
#147362
#142762
#140182
#139542
#139112
#139062
#137852