Finite Group Databasebeta

← #764164

Subgroups — #764164

Order 512.

2858 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3C2×#2580C211yesmaximal, characteristic
4C2×#2331C211yesmaximal, characteristic, complemented
5#15855C214yesmaximal, complemented
6#15765C214yesmaximal, complemented
7#15279C214yesmaximal, complemented
8#15276C214yesmaximal, complemented
9#15141C212yesmaximal
10#15137C212yesmaximal, complemented
11#14150C211yesmaximal, characteristic, complemented
12#14141C212yesmaximal
13#14005C212yesmaximal, complemented
14#13850C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2580—28no—
22#2567—28no—
23#2542—28no—
24C2×#399C2212yes—
25C2×#399C2214yes—
26C2×#398C2214yes—
27C2×#396C2214yes—
28C2×#395C2214yes—
29C2×#393C2212yes—
30#2331—28no—
31C2×#391C2214yes—
32C2×#391C2212yes—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#385C2214yes—
46C2×#384C2212yes—
47C2×#384C2212yes—
48C2×#382C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53#1446C2212yes—
54#1445C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8D418yes—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—48no—
89C2×C22⋊Q8—48no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127#399—216no—
128#399—28no—
129#398—48no—
130#396—48no—
131#395—216no—
132#393—28no—
133#391—28no—
134#391—216no—
135#390—28no—
136#390—28no—
137#390—24no—
138#390Q818yescomplemented
139#389—28no—
140#389—28no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—48no—
145#388—48no—
146#387—216no—
147#385—216no—
148#384—28no—
149#384—28no—
150#382D418yes—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154#376—24no—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2412yes—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—42no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2×D418yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4C2×Q814yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—232no—
217C22⋊Q8—432no—
218C22⋊Q8—432no—
219C4×Q8—416no—
220C4×D4—416no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
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—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
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⋊C4—44no—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C4⋊C4#4114yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C2×Q8116yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×Q814yes—
310C2×C42#4114yes—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—42no—
329#95—44no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334#95C2×D418yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24C22×Q814yes—
346C24C2511yescharacteristic
347C2×Q8—28no—
348C2×Q8—216no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—88no—
356C2×Q8—88no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4C22×D418yes—
437C22×C4C22×D418yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q812yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4H(5,𝔽2)14yes—
454C22×C42−1+414yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C42—28no—
482C42—48no—
483C42—48no—
484C42—44no—
485C42—48no—
486C42—88no—
487C42#121116yes—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23C23×D411yescharacteristic
512C23C23×D411yescharacteristic
513C23C23×Q811yescharacteristic
514C23C22×#4111yescharacteristic
515C23C22×#4111yescharacteristic
516C23C2×H(5,𝔽2)12yes—
517C23C2×H(5,𝔽2)12yes—
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+412yes—
521C23#58512yes—
522Q8—416no—
523Q8—816no—
524D4—416no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
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—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—816no—
605C2×C4C2×#12114yes—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4#53314yes—
609C2×C4#54114yes—
610C2×C4#54714yes—
611C2×C4#54814yes—
612C2×C4D4×Q814yes—
613C2×C4D4×Q814yes—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—84no—
624C22—84no—
625C22C22×#12111yescharacteristic
626C22C2×#53311yescharacteristic
627C22C2×#54111yescharacteristic
628C22C2×#54711yescharacteristic
629C22C2×#54811yescharacteristic
630C22C2×D4×Q811yescharacteristic
631C22C2×D4×Q811yescharacteristic
632C22#348411yescharacteristic
633C22#348411yescharacteristic
634C22#348611yescharacteristic
635C22#348711yescharacteristic
636C22#348912yes—
637C22#349512yes—
638C22#349812yes—
639C22#350112yes—
640C22#350312yes—
641C22#350812yes—
642C22#351312yes—
643C22#352212yes—
644C22#354412yes—
645C22#355012yes—
646C22#355412yes—
647C22#355912yes—
648C4—24no—
649C4—24no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—48no—
655C4—44no—
656C4—44no—
657C4—48no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4#242918yes—
662C2—84no—
663C2C2×#242911yescharacteristic
664C2#3552811yescharacteristic
665C2#3558012yes—
666C2#3566111yescharacteristic
667C2#3733712yes—
668C2#4493411yescharacteristic
669C2#4493711yescharacteristic
670C2#4494112yes—
671C2#4505112yes—
672C2#4527111yescharacteristic
673C2#4535711yescharacteristic
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
1. C2×#2567C2semidirect
4. C2×#2331C2semidirect
5. #15855C2semidirect
6. #15765C2semidirect
7. #15279C2semidirect
8. #15276C2semidirect
10. #15137C2semidirect
11. #14150C2semidirect
13. #14005C2semidirect
14. #13850C2semidirect
138. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25422
C2×#25802
C2×#23312
#158552
#157652
#152792
#152762
#151412
#151372
#141502
#141412
#140052
#138502
#137852