Finite Group Databasebeta

← #988881

Subgroups — #988881

Order 512.

3250 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2680C212yesmaximal
2C2×#2486C212yesmaximal
3C2×#2440C212yesmaximal, complemented
4#17114C212yesmaximal, complemented
5#16522C212yesmaximal, complemented
6#16262C214yesmaximal, complemented
7#15949C212yesmaximal
8#15298C214yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15132C212yesmaximal, complemented
11#15128C211yesmaximal, characteristic, complemented
12#13994C212yesmaximal, complemented
13#13910C211yesmaximal, characteristic, complemented
14#13880C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#123C2214yes—
18C22×C22⋊Q8C2214yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21#2680—216no—
22#2486—216no—
23#2440—216no—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#397C2214yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#394C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#384C2212yes—
42C2×#384C2211yescharacteristic
43C2×#381C2214yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1446C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2212yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2211yescharacteristic
62#1442C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2211yescharacteristic
66#1440C2214yes—
67#1440C2212yes—
68#1440C2214yes—
69#1439C2212yes—
70#1438C2211yescharacteristic
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77C4×#95C2211yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2314yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—42no—
84C2×#123—216no—
85C2×#123—216no—
86C2×#123—48no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—48no—
90C2×C4×D4—28no—
91C2×C4×D4—42no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#398—216no—
126#398—216no—
127#398—216no—
128#397—216no—
129#397—216no—
130#395—28no—
131#395—216no—
132#394D418yes—
133#389—28no—
134#389—28no—
135#389—28no—
136#389—28no—
137#388—216no—
138#385—216no—
139#385—216no—
140#385—48no—
141#384—28no—
142#384—28no—
143#384—42no—
144#381—216no—
145#380—28no—
146#380—28no—
147#380D4116yes—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—42no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C25C2412yes—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—44no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4C2×D414yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C2×D418yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—416no—
217#123—464no—
218C22⋊Q8—464no—
219C4×D4—416no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—432no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D414yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C2×D418yes—
291C2×C22⋊C4C2×D4116yes—
292C2×C22⋊C4C2×D418yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42#4114yes—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—41no—
325#95—41no—
326#95—41no—
327#95—44no—
328#95—44no—
329#95C2×D4116yes—
330#95C2×D414yes—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24C22×D414yes—
344C24C22×D414yes—
345C24H(5,𝔽2)14yes—
346C24C2511yescharacteristic
347C2×Q8—416no—
348C2×Q8—48no—
349C2×Q8—88no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×D4—216no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—44no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C22×D414yes—
441C22×C4C22×D418yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D418yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+414yes—
453C22×C42−1+412yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C22×C42−1+414yes—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—416no—
462C4⋊C4—44no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C4⋊C4—88no—
473C4⋊C4—84no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—216no—
477C22⋊C4—232no—
478C22⋊C4—216no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C42—28no—
483C42—44no—
484C42—44no—
485C42—48no—
486C42—48no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23D4218yes—
522C23C23×D412yes—
523C23C23×D411yescharacteristic
524C23C22×#4111yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+412yes—
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+412yes—
532C23#58511yescharacteristic
533Q8—816no—
534D4—416no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—432no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—832no—
617C2×C4#53618yes—
618C2×C4#53914yes—
619C2×C4D4214yes—
620C2×C4#54714yes—
621C2×C4#54718yes—
622C2×C4#56314yes—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22C2×#53612yes—
639C22C2×#53911yescharacteristic
640C22C2×D4211yescharacteristic
641C22C2×D4212yes—
642C22C2×#54712yes—
643C22C2×#54711yescharacteristic
644C22#348612yes—
645C22#348812yes—
646C22#349212yes—
647C22#349611yescharacteristic
648C22#349912yes—
649C22#349912yes—
650C22#349911yescharacteristic
651C22#350711yescharacteristic
652C22C2×#56311yescharacteristic
653C22#353711yescharacteristic
654C22#353811yescharacteristic
655C22#354412yes—
656C22#354412yes—
657C22#354412yes—
658C22#355011yescharacteristic
659C22#356312yes—
660C22#356311yescharacteristic
661C4—24no—
662C4—42no—
663C4—42no—
664C4—48no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—48no—
670C4—44no—
671C4—81no—
672C4—84no—
673C4—84no—
674C4#268018yescomplemented
675C2—48no—
676C2#3933712yes—
677C2#3937611yescharacteristic
678C2#4250611yescharacteristic
679C2#4296412yes—
680C2#4492112yes—
681C2#4532111yescharacteristic
682C2#4614711yescharacteristic
683C2C2×#268011yescharacteristic
684C2#5303111yescharacteristic
685C2#5899711yescharacteristic
686C2#5931312yes—
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
3. C2×#2440C2semidirect
4. #17114C2semidirect
5. #16522C2semidirect
6. #16262C2semidirect
8. #15298C2semidirect
9. #15137C2semidirect
10. #15132C2semidirect
11. #15128C2semidirect
12. #13994C2semidirect
13. #13910C2semidirect
14. #13880C2semidirect
674. C4#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#26802
C2×#24862
C2×#24402
#171142
#165222
#162622
#159492
#152982
#151372
#151322
#151282
#139942
#139102
#138802
C4×#3942