Finite Group Databasebeta

← #722644

Subgroups — #722644

Order 512.

3302 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic, complemented
2C2×#2626C211yesmaximal, characteristic
3C22×#399C211yesmaximal, characteristic
4C2×#2368C211yesmaximal, characteristic, complemented
5#16463C214yesmaximal, complemented
6#15765C214yesmaximal, complemented
7#15499C214yesmaximal, complemented
8#14870C214yesmaximal, complemented
9#14786C212yesmaximal, complemented
10#14726C212yesmaximal
11#14130C212yesmaximal, complemented
12#13991C212yesmaximal
13#13924C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21#2626—28no—
22#2502—28no—
23C2×#399—28no—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#399C2214yes—
27C2×#399C2212yes—
28C2×#398C2214yescomplemented
29C2×#398C2214yescomplemented
30C2×#396C2214yes—
31C2×#396C2214yes—
32C2×#393C2212yes—
33#2368—28no—
34C2×#391C2212yescomplemented
35C2×#391C2212yescomplemented
36C2×#389C2214yescomplemented
37C2×#388C2214yes—
38C2×#387C2212yescomplemented
39C2×#387C2212yescomplemented
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#384C2212yescomplemented
45C2×#380C2212yes—
46C2×#380C2211yescharacteristic
47C2×#379C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2211yescharacteristic
50C2×C4×C4⋊C4C2212yescomplemented
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C22×#95C2211yescharacteristic
55#1445C2214yescomplemented
56#1445C2214yescomplemented
57#1444C2212yescomplemented
58#1444C2214yescomplemented
59#1444C2214yescomplemented
60#1442C2214yescomplemented
61#1442C2212yescomplemented
62#1441C2214yescomplemented
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72C4⋊#95C2214yescomplemented
73C4⋊#95C2214yescomplemented
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C26C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8D418yes—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—48no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—48no—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129#399—28no—
130#399—28no—
131#399—28no—
132#399—216no—
133#398—216no—
134#398—48no—
135#396—216no—
136#396—48no—
137#393—28no—
138#391—28no—
139#391—28no—
140#389—48no—
141#388—48no—
142#387—28no—
143#387—28no—
144#385—28no—
145#385—28no—
146#385—28no—
147#385—28no—
148#384—28no—
149#380—24no—
150#380—28no—
151#379—28no—
152#379—24no—
153#379Q818yescomplemented
154C4×C4⋊C4—28no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—24no—
157C4×C22⋊C4—24no—
158C2×#95D418yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C25—22no—
182C25—22no—
183C25—22no—
184C25—22no—
185C25—22no—
186C25—22no—
187C25—44no—
188C25—44no—
189C25C2×Q814yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—44no—
195C22×Q8C2×D414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4#4114yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C22⋊Q8—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
265C2×C4⋊C4C2×D418yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—432no—
298C2×C22⋊C4—416no—
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⋊C4C2×D418yes—
306C2×C22⋊C4C2×Q814yes—
307C2×C22⋊C4C2×Q814yes—
308C2×C22⋊C4C2×Q8116yes—
309C2×C22⋊C4#4114yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318#95—24no—
319#95—24no—
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—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C2×D414yes—
338#95C2×D414yes—
339#95C2×D418yes—
340C24—24no—
341C24—24no—
342C24—28no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—28no—
354C24—28no—
355C24—416no—
356C24—48no—
357C24—48no—
358C24—416no—
359C24—44no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—84no—
364C24—84no—
365C24—84no—
366C24—84no—
367C24—88no—
368C24—88no—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C22×Q814yes—
372C24C22×Q812yes—
373C24C2×#4112yes—
374C24C2×#4112yes—
375C242−1+412yes—
376C242−1+412yes—
377C24C2511yescharacteristic
378C2×Q8—28no—
379C2×Q8—48no—
380C2×Q8—84no—
381C2×Q8—84no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—216no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
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—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—84no—
449C22×C4—88no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4#121116yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×Q814yes—
462C22×C4C22×Q814yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)18yes—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—48no—
495C42—48no—
496C42—84no—
497C42—84no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—432no—
514C23—42no—
515C23—44no—
516C23—416no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—816no—
534C23—816no—
535C23—816no—
536C23—88no—
537C23—816no—
538C23—816no—
539C23—88no—
540C23C2×#12114yes—
541C23#53314yes—
542C23#54114yes—
543C23#54714yes—
544C23#54814yes—
545C23D4×Q814yes—
546C23D4×Q814yes—
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C23×Q811yescharacteristic
550C23C22×#4111yescharacteristic
551C23C22×#4111yescharacteristic
552C23C2×H(5,𝔽2)14yes—
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23#58514yes—
556Q8—88no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—432no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—816no—
624C2×C4—88no—
625C2×C4C2×#12114yes—
626C2×C4C2×#12114yes—
627C2×C4#53518yes—
628C22—28no—
629C22—24no—
630C22—28no—
631C22—28no—
632C22—28no—
633C22—24no—
634C22—28no—
635C22—28no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—832no—
642C22—84no—
643C22—82no—
644C22—84no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22#242918yes—
649C22C22×#12111yescharacteristic
650C22C2×#53311yescharacteristic
651C22C2×#53512yes—
652C22C2×#54111yescharacteristic
653C22C2×#54711yescharacteristic
654C22C2×#54811yescharacteristic
655C22C2×D4×Q811yescharacteristic
656C22C2×D4×Q811yescharacteristic
657C22#348512yes—
658C22#348512yes—
659C22#349112yes—
660C22#349812yes—
661C22#350114yes—
662C22#350812yes—
663C22#351314yes—
664C22#353514yes—
665C22#354114yes—
666C4—24no—
667C4—24no—
668C4—28no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4—84no—
679C2—28no—
680C2—84no—
681C2C2×#242911yescharacteristic
682C2#3527211yescharacteristic
683C2#3544711yescharacteristic
684C2#3583912yes—
685C2#3803912yes—
686C2#3959412yes—
687C2#4138812yes—
688C2#4349412yes—
689C2#4410312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2502C2semidirect
4. C2×#2368C2semidirect
5. #16463C2semidirect
6. #15765C2semidirect
7. #15499C2semidirect
8. #14870C2semidirect
9. #14786C2semidirect
11. #14130C2semidirect
13. #13924C2semidirect
14. #13908C2semidirect
28. C2×#398C22semidirect
29. C2×#398C22semidirect
34. C2×#391C22semidirect
35. C2×#391C22semidirect
36. C2×#389C22semidirect
38. C2×#387C22semidirect
39. C2×#387C22semidirect
44. C2×#384C22semidirect
50. C2×C4×C4⋊C4C22semidirect
55. #1445C22semidirect
56. #1445C22semidirect
57. #1444C22semidirect
58. #1444C22semidirect
59. #1444C22semidirect
60. #1442C22semidirect
61. #1442C22semidirect
62. #1441C22semidirect
71. #1436C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
153. #379Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#26262
C22×#3992
C2×#23682
#164632
#157652
#154992
#148702
#147862
#147262
#141302
#139912
#139242
#139082
#137752