Finite Group Databasebeta

← #368354

Subgroups — #368354

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21258C211yesmaximal, characteristic
3C2×#1962C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic, complemented
5C2×#1859C212yesmaximal, complemented
6C4×#384C211yesmaximal, characteristic, complemented
7#11113C212yesmaximal, complemented
8#11078C212yesmaximal, complemented
9#10847C211yesmaximal, characteristic, complemented
10#10839C211yesmaximal, characteristic, complemented
11#10781C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C2×#520C2212yes—
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C22×#104C2212yes—
18C22×#104C2212yes—
19C2×#417C2212yes—
20C2×#417C2212yes—
21#2367—22no—
22#2366—21no—
23#2365—21no—
24#2354—21no—
25#2352—22no—
26D4×C4⋊C4—21no—
27#2325—22no—
28#2311C2212yes—
29C4×#121—22no—
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4—22no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C43C2211yescharacteristic
37#1983—22no—
38#1982—21no—
39#1981—21no—
40#1962C418yescomplemented
41#1949—22no—
42#1948—21no—
43#1947—21no—
44#1859—28no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47#1736C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C4⋊#95—22no—
51C4⋊#95C2211yescharacteristic
52C4⋊#95C2211yescharacteristic
53C22×#41C2312yes—
54C22×M16C2312yes—
55C22×M16C2312yes—
56#532—22no—
57#532—22no—
58#532C2×C414yes—
59#532Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×#127D411yescharacteristic
62C2×#124D412yes—
63C2×#121—21no—
64C2×#121—21no—
65#520—24no—
66#520—24no—
67#520—42no—
68#520—42no—
69C2×C4×Q8—22no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×#117—22no—
76C2×#117C2312yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#428—24no—
92#428—42no—
93C2×#107—22no—
94C2×#106—22no—
95#423—22no—
96#423—22no—
97#423—22no—
98#423C2×C414yescomplemented
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×#104—24no—
104C2×#104—24no—
105C2×#104—42no—
106C2×#104—42no—
107C2×#104C2×C418yescomplemented
108C2×#104C2×C418yescomplemented
109#417—24no—
110#417—24no—
111#417—42no—
112#417—42no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#102—22no—
116C2×#102—22no—
117#391—22no—
118#390—22no—
119#390—22no—
120#389—22no—
121#388—24no—
122#388—24no—
123#387—24no—
124#386—24no—
125#386—24no—
126#385—24no—
127#384—22no—
128#384—22no—
129#384—22no—
130#384D411yescharacteristic
131#384D411yescharacteristic
132#384D414yescomplemented
133#382D412yes—
134#380—21no—
135#380—21no—
136#380—22no—
137#380—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—48no—
150C4×C22⋊C4—24no—
151C2×#95—21no—
152C2×#95—22no—
153C2×#95—22no—
154C2×#95D412yes—
155C43—22no—
156C43C2×C414yescomplemented
157#343—24no—
158#343—42no—
159#340—22no—
160#340—22no—
161#339—24no—
162#339—42no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41C22×C414yes—
174C2×#41C2×Q814yes—
175C22×Q8C2×D412yes—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C2×D412yes—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16C22×C414yes—
203C22×C8—22no—
204C22×C8—22no—
205C4⋊Q8—22no—
206C4⋊Q8C4⋊C414yescomplemented
207#127—22no—
208#127C4⋊C414yescomplemented
209#124—24no—
210#124C4⋊C418yescomplemented
211#121—24no—
212#121—24no—
213#121—24no—
214#121—24no—
215C4×Q8—44no—
216C4×Q8—44no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—44no—
222C4×D4—44no—
223C4×D4—48no—
224C4×D4—48no—
225C4×D4—84no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—44no—
231#117—44no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
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×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4#4112yes—
272C2×C4⋊C4#4112yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—41no—
300C2×C42—41no—
301C2×C42C22×C412yes—
302C2×C42C22×C412yes—
303C2×C42C22×C412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×Q812yes—
310C2×C42#4112yes—
311C2×C42#4112yes—
312C2×C42#4112yes—
313#107—44no—
314#106—44no—
315C4⋊C8—24no—
316C4⋊C8—24no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319#104—216no—
320#104—48no—
321#103—44no—
322#103—44no—
323#102—44no—
324#102—44no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—44no—
332#95—44no—
333#95—41no—
334#95—41no—
335C24—22no—
336C24C22×D411yescharacteristic
337#41—24no—
338#41—24no—
339#41—24no—
340#41—24no—
341#41—24no—
342#41—24no—
343#41—24no—
344#41—24no—
345#41—42no—
346#41—42no—
347#41—42no—
348#41—42no—
349#41—42no—
350#41—42no—
351#41—42no—
352#41—42no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8C2×C4⋊C414yes—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4C2×C4⋊C414yes—
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—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—48no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4C4×D412yes—
427C22×C4C4×D412yes—
428C22×C4#12011yescharacteristic
429C22×C4#12011yescharacteristic
430C22×C4#12011yescharacteristic
431C22×C4#12011yescharacteristic
432C22×C4#12114yes—
433C22×C4C22⋊Q812yes—
434C22×C4C22⋊Q812yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×Q811yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C42−1+411yescharacteristic
449M16—24no—
450M16—24no—
451M16—42no—
452M16—42no—
453M16—42no—
454M16—42no—
455M16—42no—
456M16—42no—
457C2×C8—24no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—416no—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—48no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C42—216no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—48no—
506C42—48no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42—44no—
512C42—82no—
513C42—82no—
514C42—82no—
515C42C2×C4⋊C412yes—
516C42C2×C4⋊C412yes—
517C42C4×D412yes—
518C42C4×D412yes—
519C42C4×D414yescomplemented
520C42C22⋊Q812yes—
521C42C22⋊Q812yes—
522C42#12312yes—
523C42#12312yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C2×C4×D411yescharacteristic
536C23C2×#12011yescharacteristic
537C23C2×#12111yescharacteristic
538C23C2×C22⋊Q811yescharacteristic
539C23#52511yescharacteristic
540C23#53511yescharacteristic
541C23#53711yescharacteristic
542Q8—24no—
543Q8—24no—
544Q8—42no—
545Q8—42no—
546D4—24no—
547D4—24no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—44no—
552D4—48no—
553D4—42no—
554D4—42no—
555D4—84no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—42no—
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—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—82no—
613C2×C4—168no—
614C2×C4#38512yes—
615C2×C4#38512yes—
616C2×C4#38512yes—
617C2×C4#38512yes—
618C2×C4#45312yes—
619C2×C4#45612yes—
620C2×C4C22×C4⋊C411yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C4×#4111yescharacteristic
624C2×C4#51811yescharacteristic
625C2×C4#52011yescharacteristic
626C2×C4C2×C22⋊Q811yescharacteristic
627C2×C4C2×#12311yescharacteristic
628C2×C4#52911yescharacteristic
629C2×C4#53211yescharacteristic
630C2×C4#54011yescharacteristic
631C2×C4#54211yescharacteristic
632C2×C4D4211yescharacteristic
633C2×C4D4211yescharacteristic
634C2×C4#54611yescharacteristic
635C2×C4#54611yescharacteristic
636C2×C4#54611yescharacteristic
637C2×C4#54711yescharacteristic
638C2×C4#54811yescharacteristic
639C2×C4D4×Q811yescharacteristic
640C2×C4#55011yescharacteristic
641C2×C4#55011yescharacteristic
642C2×C4#55411yescharacteristic
643C2×C4#55411yescharacteristic
644C2×C4#55411yescharacteristic
645C2×C4#55611yescharacteristic
646C2×C4#55911yescharacteristic
647C2×C4#55911yescharacteristic
648C8—44no—
649C8—82no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22C2×#38511yescharacteristic
670C22C4×#12111yescharacteristic
671C22#232411yescharacteristic
672C22#240711yescharacteristic
673C22#240811yescharacteristic
674C22#253811yescharacteristic
675C22#258011yescharacteristic
676C22C2×#45311yescharacteristic
677C22C2×#45611yescharacteristic
678C22#301811yescharacteristic
679C22#301811yescharacteristic
680C4—28no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#185914yescomplemented
693C4D4×C4⋊C411yescharacteristic
694C4#234911yescharacteristic
695C4#236611yescharacteristic
696C4#236711yescharacteristic
697C4#244511yescharacteristic
698C4#244711yescharacteristic
699C4#245211yescharacteristic
700C4#245411yescharacteristic
701C2—21no—
702C2—21no—
703C2—21no—
704C2—21no—
705C2—84no—
706C2#1401811yescharacteristic
707C2C2×#185911yescharacteristic
708C2#2086311yescharacteristic
709C2#2417611yescharacteristic
710C2#2417911yescharacteristic
711C2#2422211yescharacteristic
712C2#2422511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21170C2semidirect
5. C2×#1859C2semidirect
6. C4×#384C2semidirect
7. #11113C2semidirect
8. #11078C2semidirect
9. #10847C2semidirect
10. #10839C2semidirect
11. #10781C2semidirect
40. #1962C4semidirect
98. #423C2×C4semidirect
107. C2×#104C2×C4semidirect
108. C2×#104C2×C4semidirect
132. #384D4semidirect
156. C43C2×C4semidirect
206. C4⋊Q8C4⋊C4semidirect
208. #127C4⋊C4semidirect
210. #124C4⋊C4semidirect
519. C42C4×D4semidirect
692. C4#1859semidirect

Maximal subgroups

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

subgroupindex
#342062
#212582
C2×#19622
#211702
C2×#18592
C4×#3842
#111132
#110782
#108472
#108392
#107812