Finite Group Databasebeta

← #988950

Subgroups — #988950

Order 512.

3110 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C212yesmaximal, complemented
2C2×#2445C212yesmaximal
3#17107C212yesmaximal, complemented
4#16878C212yesmaximal, complemented
5#16269C214yesmaximal, complemented
6#15949C212yesmaximal, complemented
7#15384C214yesmaximal, complemented
8#15221C212yesmaximal, complemented
9#15138C212yesmaximal, complemented
10#15127C211yesmaximal, characteristic
11#14535C212yesmaximal, complemented
12#14002C212yesmaximal, complemented
13#13977C212yesmaximal, complemented
14#13910C211yesmaximal, characteristic, complemented
15C4×#395C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123C2214yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2212yes—
20#2580—216no—
21#2445—216no—
22C2×#399C2214yes—
23C2×#397C2214yes—
24C2×#395C2211yescharacteristic
25C2×#394C2212yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#393C2214yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#385C2214yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#382C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#1446C2211yescharacteristic
50#1445C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2211yescharacteristic
54#1444C2212yes—
55#1444C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2211yescharacteristic
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1438C2212yes—
69#1437C2214yes—
70#1437C2214yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C24×C4C2314yes—
82C2×#127—28no—
83C2×#127—42no—
84C2×#123—216no—
85C2×#123—48no—
86C2×#123—48no—
87C2×C4×Q8—28no—
88C2×C4×Q8—42no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#399—216no—
124#397—48no—
125#395—24no—
126#394—28no—
127#394—216no—
128#394—216no—
129#394—44no—
130#393—216no—
131#390—28no—
132#390—28no—
133#390—28no—
134#390—28no—
135#390D4116yes—
136#389—28no—
137#389—28no—
138#389—28no—
139#386—216no—
140#385—216no—
141#385—216no—
142#385—48no—
143#385—48no—
144#384—42no—
145#384—44no—
146#382—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—42no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C25C2414yes—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—41no—
177C22×Q8—42no—
178C22×Q8—41no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—44no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4C2×D418yes—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#127—416no—
218#123—464no—
219C4×Q8—416no—
220C4×D4—432no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
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—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4#4114yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
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⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D4116yes—
293C2×C22⋊C4C2×D4116yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42C2×D418yes—
309C2×C42#4114yes—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—44no—
326#95—42no—
327#95—41no—
328#95—41no—
329#95—41no—
330#95—44no—
331#95—42no—
332#95—44no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24C22×D418yes—
348C24C2511yescharacteristic
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—84no—
370C2×D4—84no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4C22×D418yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C42−1+414yes—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C4⋊C4—84no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—28no—
490C42—216no—
491C42—48no—
492C42—48no—
493C42—44no—
494C42—44no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23C23×D412yes—
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23C2×2−1+411yescharacteristic
538C23C2×2−1+412yes—
539C23#58512yes—
540C23#58511yescharacteristic
541Q8—416no—
542D4—432no—
543D4—816no—
544D4—816no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—832no—
626C2×C4#53618yes—
627C2×C4#53914yes—
628C2×C4D4214yes—
629C2×C4#54718yes—
630C2×C4#55914yes—
631C2×C4#56314yes—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22C2×#53612yes—
648C22C2×#53911yescharacteristic
649C22C2×D4211yescharacteristic
650C22C2×#54712yes—
651C22D4×#4112yes—
652C22C2×#55911yescharacteristic
653C22#348412yes—
654C22#348512yes—
655C22#349812yes—
656C22#350511yescharacteristic
657C22#350612yes—
658C22#350611yescharacteristic
659C22#351111yescharacteristic
660C22#352212yes—
661C22C2×#56311yescharacteristic
662C22#353711yescharacteristic
663C22#354011yescharacteristic
664C22#354911yescharacteristic
665C22#355412yes—
666C22#355412yes—
667C22#355412yes—
668C22#355912yes—
669C22#355911yescharacteristic
670C4—24no—
671C4—42no—
672C4—48no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—81no—
680C4—84no—
681C4—82no—
682C4—84no—
683C4#268018yes—
684C2—88no—
685C2#3976711yescharacteristic
686C2#3987712yes—
687C2#4251711yescharacteristic
688C2#4265112yes—
689C2#4487011yescharacteristic
690C2#4535412yes—
691C2#4634511yescharacteristic
692C2C2×#268011yescharacteristic
693C2#5298911yescharacteristic
694C2#5900011yescharacteristic
695C2#5964112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2580C2semidirect
3. #17107C2semidirect
4. #16878C2semidirect
5. #16269C2semidirect
6. #15949C2semidirect
7. #15384C2semidirect
8. #15221C2semidirect
9. #15138C2semidirect
11. #14535C2semidirect
12. #14002C2semidirect
13. #13977C2semidirect
14. #13910C2semidirect
15. C4×#395C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#24452
#171072
#168782
#162692
#159492
#153842
#152212
#151382
#151272
#145352
#140022
#139772
#139102
C4×#3952