Finite Group Databasebeta

← #368355

Subgroups — #368355

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 707 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×#1860C212yesmaximal, complemented
6C4×#382C211yesmaximal, characteristic, complemented
7#11113C212yesmaximal, complemented
8#11079C212yesmaximal, 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×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1983—22no—
37#1982—21no—
38#1981—21no—
39#1962C418yescomplemented
40#1949—22no—
41#1948—21no—
42#1947—21no—
43#1860—28no—
44C2×#343C2212yes—
45C2×#343C2212yes—
46#1736C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C4⋊#95C2211yescharacteristic
50C4⋊#95C2211yescharacteristic
51#1434—22no—
52C4×#95—22no—
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⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42D412yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#428—24no—
90#428—42no—
91C2×#107—22no—
92C2×#106—22no—
93#423—22no—
94#423—22no—
95#423—22no—
96#423C2×C414yescomplemented
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×#104—24no—
102C2×#104—24no—
103C2×#104—42no—
104C2×#104—42no—
105C2×#104C2×C418yescomplemented
106C2×#104C2×C418yescomplemented
107#417—24no—
108#417—24no—
109#417—42no—
110#417—42no—
111C2×#103—22no—
112C2×#103—22no—
113C2×#102—22no—
114C2×#102—22no—
115#391—22no—
116#390—22no—
117#390—22no—
118#389—22no—
119#388—24no—
120#388—24no—
121#387—24no—
122#386—24no—
123#386—24no—
124#385—24no—
125#384D411yescharacteristic
126#384D411yescharacteristic
127#382—22no—
128#382—22no—
129#382—22no—
130#382D412yes—
131#382D414yescomplemented
132#380—21no—
133#380—21no—
134#380—22no—
135#380—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
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×C22⋊C4—24no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95D412yes—
154C43—22no—
155C43C2×C414yescomplemented
156#343—24no—
157#343—42no—
158#340—22no—
159#340—22no—
160#339—24no—
161#339—42no—
162C2×#41—22no—
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×#41C22×C414yes—
173C2×#41C2×Q814yes—
174C22×Q8C2×D412yes—
175C22×D4—22no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C2×D412yes—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—24no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2411yescharacteristic
190C2×M16—22no—
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×M16C22×C414yes—
202C22×C8—22no—
203C22×C8—22no—
204C4⋊Q8—22no—
205C4⋊Q8C4⋊C414yes—
206#127—22no—
207#127C4⋊C414yes—
208#124—24no—
209#124C4⋊C418yes—
210#121—24no—
211#121—24no—
212#121—24no—
213#121—24no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—48no—
223C4×D4—48no—
224C4×D4—84no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—44no—
230#117—44no—
231C2×C4⋊C4—21no—
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—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
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—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4#4112yes—
265C2×C4⋊C4#4112yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C22×C412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×Q812yes—
303C2×C42#4112yes—
304C2×C42#4112yes—
305C2×C42#4112yes—
306#107—44no—
307#106—44no—
308C4⋊C8—24no—
309C4⋊C8—24no—
310C4⋊C8—42no—
311C4⋊C8—42no—
312#104—216no—
313#104—48no—
314#103—44no—
315#103—44no—
316#102—44no—
317#102—44no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—41no—
329#95—41no—
330C24—22no—
331C24C22×D411yescharacteristic
332#41—24no—
333#41—24no—
334#41—24no—
335#41—24no—
336#41—24no—
337#41—24no—
338#41—24no—
339#41—24no—
340#41—42no—
341#41—42no—
342#41—42no—
343#41—42no—
344#41—42no—
345#41—42no—
346#41—42no—
347#41—42no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8C2×C4⋊C414yes—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4C2×C4⋊C414yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—48no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4C4×D412yes—
422C22×C4C4×D412yes—
423C22×C4#12011yescharacteristic
424C22×C4#12011yescharacteristic
425C22×C4#12011yescharacteristic
426C22×C4#12011yescharacteristic
427C22×C4#12114yes—
428C22×C4C22⋊Q812yes—
429C22×C4C22⋊Q812yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×Q811yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C42−1+411yescharacteristic
444M16—24no—
445M16—24no—
446M16—42no—
447M16—42no—
448M16—42no—
449M16—42no—
450M16—42no—
451M16—42no—
452C2×C8—24no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—48no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C22⋊C4—28no—
486C22⋊C4—28no—
487C22⋊C4—48no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—41no—
501C42—41no—
502C42—42no—
503C42—42no—
504C42—44no—
505C42—82no—
506C42—82no—
507C42—82no—
508C42C2×C4⋊C412yes—
509C42C2×C4⋊C412yes—
510C42C4×D412yes—
511C42C4×D412yes—
512C42C4×D414yescomplemented
513C42C22⋊Q812yes—
514C42C22⋊Q812yes—
515C42#12312yes—
516C42#12312yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—24no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C2×C4×D411yescharacteristic
529C23C2×#12011yescharacteristic
530C23C2×#12111yescharacteristic
531C23C2×C22⋊Q811yescharacteristic
532C23#52511yescharacteristic
533C23#53511yescharacteristic
534C23#53711yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—42no—
538Q8—42no—
539D4—24no—
540D4—24no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—48no—
546D4—42no—
547D4—42no—
548D4—84no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
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—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
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—42no—
597C2×C4—42no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—168no—
607C2×C4#38512yes—
608C2×C4#38512yes—
609C2×C4#38512yes—
610C2×C4#38512yes—
611C2×C4#45412yes—
612C2×C4#45512yes—
613C2×C4C22×C4⋊C411yescharacteristic
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C4×#4111yescharacteristic
617C2×C4#51811yescharacteristic
618C2×C4#52011yescharacteristic
619C2×C4C2×C22⋊Q811yescharacteristic
620C2×C4C2×#12311yescharacteristic
621C2×C4#52911yescharacteristic
622C2×C4#53211yescharacteristic
623C2×C4#54011yescharacteristic
624C2×C4#54211yescharacteristic
625C2×C4D4211yescharacteristic
626C2×C4D4211yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#54611yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54811yescharacteristic
632C2×C4D4×Q811yescharacteristic
633C2×C4#55011yescharacteristic
634C2×C4#55011yescharacteristic
635C2×C4#55411yescharacteristic
636C2×C4#55411yescharacteristic
637C2×C4#55411yescharacteristic
638C2×C4#55611yescharacteristic
639C2×C4#55911yescharacteristic
640C2×C4#55911yescharacteristic
641C8—44no—
642C8—82no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22C2×#38511yescharacteristic
663C22C4×#12111yescharacteristic
664C22#232411yescharacteristic
665C22#240711yescharacteristic
666C22#240811yescharacteristic
667C22#253811yescharacteristic
668C22#258011yescharacteristic
669C22C2×#45511yescharacteristic
670C22C2×#45411yescharacteristic
671C22#301811yescharacteristic
672C22#301811yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#186014yescomplemented
686C4D4×C4⋊C411yescharacteristic
687C4#234911yescharacteristic
688C4#236611yescharacteristic
689C4#236711yescharacteristic
690C4#244511yescharacteristic
691C4#244711yescharacteristic
692C4#245211yescharacteristic
693C4#245411yescharacteristic
694C2—21no—
695C2—21no—
696C2—21no—
697C2—21no—
698C2—84no—
699C2#1401811yescharacteristic
700C2C2×#186011yescharacteristic
701C2#2086311yescharacteristic
702C2#2417711yescharacteristic
703C2#2417811yescharacteristic
704C2#2422311yescharacteristic
705C2#2422411yescharacteristic
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×#1860C2semidirect
6. C4×#382C2semidirect
7. #11113C2semidirect
8. #11079C2semidirect
9. #10847C2semidirect
10. #10839C2semidirect
11. #10781C2semidirect
39. #1962C4semidirect
96. #423C2×C4semidirect
105. C2×#104C2×C4semidirect
106. C2×#104C2×C4semidirect
131. #382D4semidirect
155. C43C2×C4semidirect
512. C42C4×D4semidirect
685. C4#1860semidirect

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×#18602
C4×#3822
#111132
#110792
#108472
#108392
#107812