Finite Group Databasebeta

← #373779

Subgroups — #373779

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic, complemented
3C2×#1806C211yesmaximal, characteristic
4C4×#417C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic, complemented
6#11113C212yesmaximal
7#11078C212yesmaximal
8#10899C212yesmaximal, complemented
9#10891C212yesmaximal, complemented
10#10857C211yesmaximal, characteristic, complemented
11#10851C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#428C2212yes—
14C2×#423C2211yescharacteristic
15C22×#104C2212yes—
16C2×#417C2211yescharacteristic
17C2×#417C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2355—21no—
20#2355—21no—
21#2354—21no—
22#2354—21no—
23#2352—22no—
24#2352—22no—
25#2314—22no—
26#2311C2212yes—
27C2×#389—22no—
28#2285—21no—
29#2285—21no—
30#2285C2211yescharacteristic
31C2×#384C2212yes—
32C2×#384C2211yescharacteristic
33#2274C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1949—22no—
37#1948—21no—
38#1947—21no—
39#1806C418yescomplemented
40#1763—21no—
41#1762—21no—
42C4×#103—21no—
43C4×#102—21no—
44C2×#343C2212yes—
45C2×#343C2212yes—
46#1736C2212yes—
47#1736C2211yescharacteristic
48#1736C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1444—22no—
53C4⋊#95C2212yes—
54C22×#41C2311yescharacteristic
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C4×#41—22no—
59C4×#41—22no—
60C4×#41C2×C414yes—
61C4×#41Q814yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—21no—
71C2×#117C2312yes—
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#428—24no—
85#428—42no—
86C2×#107—22no—
87C2×#106—22no—
88#423—22no—
89#423C2×C414yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×#104—42no—
93C2×#104—42no—
94C2×#104C2×C418yes—
95#417—22no—
96#417—22no—
97#417—22no—
98#417—22no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C4×M16—22no—
104C4×M16—22no—
105C2×#97—21no—
106C2×C4×C8—21no—
107#391—22no—
108#390—22no—
109#389—48no—
110#388—24no—
111#388—24no—
112#387—24no—
113#387—24no—
114#386—24no—
115#384—24no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384—42no—
121#384D411yescharacteristic
122#384D411yescharacteristic
123#382—22no—
124#382—22no—
125#382D412yes—
126#380—22no—
127#380—22no—
128#380—21no—
129#380—21no—
130#380—22no—
131#380—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2×C414yes—
146C4×C22⋊C4—24no—
147C2×#95—21no—
148C2×#95—24no—
149C2×#95—21no—
150#343—24no—
151#343—42no—
152#340—22no—
153#340—22no—
154#340—21no—
155#340—21no—
156#340—21no—
157#340—21no—
158#339—24no—
159#339—42no—
160#339C2×C418yes—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41C22×C412yes—
172C2×#41C2×Q812yes—
173C22×Q8C2×D411yescharacteristic
174C22×D4—21no—
175C22×D4C2×D411yescharacteristic
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D414yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—22no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16C22×C414yes—
202C2×M16#4112yes—
203C2×M16#4112yes—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8C4⋊C414yescomplemented
210C4×Q8C4⋊C414yescomplemented
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4C4⋊C414yescomplemented
218C4×D4C4⋊C414yescomplemented
219#117—21no—
220#117—21no—
221#117—24no—
222#117—24no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—84no—
228#117—84no—
229#117C22×C412yes—
230#117C2×Q812yes—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
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—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4—84no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4#4112yes—
270C2×C4⋊C4#4112yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—44no—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42C22×C414yes—
294C2×C42C22×C412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×Q812yes—
300C2×C42#4112yes—
301C2×C42#4112yes—
302#107—44no—
303#106—44no—
304C4⋊C8—24no—
305C4⋊C8—24no—
306#104—48no—
307#103—24no—
308#103—24no—
309#102—24no—
310#102—24no—
311#97—42no—
312C4×C8—42no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—48no—
321#95—41no—
322#95—41no—
323C24—21no—
324C24C22×D411yescharacteristic
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—22no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8C2×C4⋊C412yes—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4C2×C4⋊C412yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4C4×D412yes—
403C22×C4C4×D412yes—
404C22×C4#12011yescharacteristic
405C22×C4#12011yescharacteristic
406C22×C4#12011yescharacteristic
407C22×C4#12011yescharacteristic
408C22×C4C22⋊Q812yes—
409C22×C4C22⋊Q812yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×Q811yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424M16—24no—
425M16—24no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—42no—
431M16—42no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—42no—
445C4⋊C4—216no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—41no—
466C4⋊C4—42no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—816no—
475C4⋊C4C2×C4⋊C412yes—
476C4⋊C4C2×C4⋊C412yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—24no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—82no—
496C42—84no—
497C42—84no—
498C42—82no—
499C42C2×C4⋊C412yes—
500C42C2×C4⋊C412yes—
501C42C4×D412yes—
502C42C4×D412yes—
503C42C22⋊Q812yes—
504C42C22⋊Q812yes—
505C42#12312yes—
506C42#12312yes—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×C4×D411yescharacteristic
519C23C2×#12011yescharacteristic
520C23C2×C22⋊Q811yescharacteristic
521C23#54611yescharacteristic
522C23#54711yescharacteristic
523C23#54811yescharacteristic
524C23#55011yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529D4—22no—
530D4—22no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—41no—
536D4—41no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4#38512yes—
591C2×C4#38512yes—
592C2×C4#38512yes—
593C2×C4#38512yes—
594C2×C4#45312yes—
595C2×C4#45612yes—
596C2×C4C22×C4⋊C411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C4×#4111yescharacteristic
599C2×C4C4×#4111yescharacteristic
600C2×C4#51811yescharacteristic
601C2×C4#51911yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4C2×#12311yescharacteristic
604C2×C4#52511yescharacteristic
605C2×C4#52511yescharacteristic
606C2×C4#53311yescharacteristic
607C2×C4#53311yescharacteristic
608C2×C4#53711yescharacteristic
609C2×C4#53711yescharacteristic
610C2×C4#54411yescharacteristic
611C2×C4#54411yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54711yescharacteristic
614C2×C4#54811yescharacteristic
615C2×C4#55011yescharacteristic
616C2×C4#55211yescharacteristic
617C2×C4#55211yescharacteristic
618C2×C4#55311yescharacteristic
619C2×C4#55311yescharacteristic
620C2×C4#55511yescharacteristic
621C2×C4#55511yescharacteristic
622C2×C4#55611yescharacteristic
623C2×C4#55611yescharacteristic
624C8—44no—
625C8—42no—
626C8—42no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22C2×#38511yescharacteristic
647C22#236211yescharacteristic
648C22#236311yescharacteristic
649C22#240711yescharacteristic
650C22#240711yescharacteristic
651C22#243911yescharacteristic
652C22#244111yescharacteristic
653C22C2×#45311yescharacteristic
654C22C2×#45611yescharacteristic
655C22#301911yescharacteristic
656C22#302011yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—48no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#185914yes—
670C4#229611yescharacteristic
671C4#231611yescharacteristic
672C4#236211yescharacteristic
673C4#236311yescharacteristic
674C4#243911yescharacteristic
675C4#244111yescharacteristic
676C4#248511yescharacteristic
677C4#248511yescharacteristic
678C2—21no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—82no—
683C2#1412911yescharacteristic
684C2C2×#185911yescharacteristic
685C2#2086411yescharacteristic
686C2#2414411yescharacteristic
687C2#2414711yescharacteristic
688C2#2417011yescharacteristic
689C2#2417311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
2. #21170C2semidirect
5. #13910C2semidirect
8. #10899C2semidirect
9. #10891C2semidirect
10. #10857C2semidirect
11. #10851C2semidirect
39. #1806C4semidirect
209. C4×Q8C4⋊C4semidirect
210. C4×Q8C4⋊C4semidirect
217. C4×D4C4⋊C4semidirect
218. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341882
#211702
C2×#18062
C4×#4172
#139102
#111132
#110782
#108992
#108912
#108572
#108512