Finite Group Databasebeta

← #366776

Subgroups — #366776

Order 512.

1857 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34810C211yesmaximal, characteristic
2C2×#1896C211yesmaximal, characteristic
3C2×#1871C211yesmaximal, characteristic
4#20837C211yesmaximal, characteristic, complemented
5C2×#1826C211yesmaximal, characteristic, complemented
6C22×#339C211yesmaximal, characteristic, complemented
7C42×C22⋊C4C211yesmaximal, characteristic, complemented
8#11370C212yesmaximal, complemented
9#11053C212yesmaximal, complemented
10#10923C212yesmaximal, complemented
11#10839C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C2×#525C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#423C2212yes—
17C22×#104C2211yescharacteristic
18C22×#104C2212yes—
19C22×#104C2211yescharacteristic
20C2×#407C2211yescharacteristic
21#2504C411yescharacteristic, complemented
22#2503C411yescharacteristic, complemented
23#2502C411yescharacteristic, complemented
24#2501C411yescharacteristic, complemented
25#2500C412yescomplemented
26#2499C412yescomplemented
27#2498—22no—
28#2412—22no—
29#2395—22no—
30C4×#123—22no—
31#2297C2212yes—
32#2285C2212yes—
33C2×C4×C22⋊C4—22no—
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C43C2212yes—
37#1896—24no—
38#1871—24no—
39#1851—22no—
40#1850—22no—
41#1826—24no—
42C2×#339—24no—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49C2×#339C2212yes—
50#1728C2212yes—
51C4×#95—22no—
52C4×#95C2212yes—
53C22×#41C2311yescharacteristic
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C22×M16—21no—
57C22×M16—22no—
58C22×M16—24no—
59C22×M16—22no—
60C22×M16—42no—
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2312yes—
64C23×C8—21no—
65#525—22no—
66#525—22no—
67#525—22no—
68#525—22no—
69C2×#123—21no—
70C2×#123—22no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×#120—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117—21no—
79C2×#117—21no—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117C2×C412yes—
85C2×#117C2312yes—
86C2×#117C2311yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4D412yes—
92C22×C42—24no—
93C22×C42—21no—
94C22×C42—21no—
95C22×C42—24no—
96C22×C42—22no—
97C22×C42—42no—
98C22×C42D414yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103#423—24no—
104#423—42no—
105C2×C4⋊C8—24no—
106C2×#104—24no—
107C2×#104—24no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—24no—
111C2×#104—42no—
112C2×#104—42no—
113#407—22no—
114#407—22no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117#400—22no—
118#399—22no—
119#398—22no—
120#397—24no—
121#397—22no—
122#394—24no—
123#391C2×C412yescomplemented
124#391C2×C412yescomplemented
125#390C2×C412yescomplemented
126#390C2×C412yescomplemented
127#388—24no—
128#387—24no—
129#386—24no—
130#386—22no—
131#385—22no—
132#384C2×C412yescomplemented
133#384C2×C412yescomplemented
134#382C2×C412yescomplemented
135#382C2×C412yescomplemented
136#381—22no—
137#380—24no—
138C4×C4⋊C4—24no—
139C4×C22⋊C4—216no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4C2×C412yes—
143C4×C22⋊C4C2×C411yescharacteristic
144C4×C22⋊C4C2×C411yescharacteristic
145C4×C22⋊C4D414yes—
146C4×C22⋊C4D412yes—
147C4×C22⋊C4D412yes—
148C2×#95—21no—
149C2×#95—24no—
150C2×#95D412yes—
151C43—24no—
152C43—42no—
153#339—24no—
154#339—24no—
155#339—24no—
156#339—22no—
157#339—22no—
158#339—24no—
159#339—24no—
160#336—24no—
161C25C2×D411yescharacteristic
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C22×Q8C22×C411yescharacteristic
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D414yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4C2411yescharacteristic
200C2×M16—22no—
201C2×M16—28no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—44no—
215C2×M16—48no—
216C2×M16—44no—
217C2×M16—42no—
218C2×M16#4114yes—
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—44no—
223C22×C8—42no—
224#123—42no—
225#123—42no—
226#123—48no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#120—42no—
236#120—42no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241#117—24no—
242#117—24no—
243#117—22no—
244#117—22no—
245#117—21no—
246#117—21no—
247#117—22no—
248#117—22no—
249#117—21no—
250#117—21no—
251#117—42no—
252#117—42no—
253#117—42no—
254#117—42no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4C22⋊C412yes—
284C2×C22⋊C4C22⋊C414yes—
285C2×C22⋊C4C22⋊C412yes—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—28no—
299C2×C42—216no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—21no—
306C2×C42—44no—
307C2×C42—48no—
308C2×C42—48no—
309C2×C42—44no—
310C2×C42—48no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42C22×C412yes—
314C2×C42C22×C412yes—
315C2×C42C2×D414yes—
316C2×C42C2×D412yes—
317C2×C42C2×D414yes—
318C2×C42C2×D412yes—
319C2×C42#4114yes—
320C4⋊C8—48no—
321#104—48no—
322#104—44no—
323#104—44no—
324C22⋊C8—42no—
325C22⋊C8—42no—
326#95—216no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95C22⋊C414yes—
333#95C22⋊C412yes—
334#95C22⋊C412yes—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—21no—
339C24—42no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24#12112yes—
343C24#12112yes—
344C24C22×D411yescharacteristic
345#41—41no—
346#41—41no—
347#41—42no—
348#41—41no—
349#41—41no—
350#41—42no—
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—44no—
370C2×D4—82no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
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—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—432no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—82no—
425C22×C4C2×C22⋊C414yes—
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C414yes—
429C22×C4C4×D412yes—
430C22×C4C4×D412yes—
431C22×C4C4×D412yes—
432C22×C4C4×D412yes—
433C22×C4#12112yes—
434C22×C4#12112yes—
435C22×C4#12412yes—
436C22×C4#12412yes—
437C22×C4#12712yes—
438C22×C4#12712yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449M16—24no—
450M16—24no—
451M16—48no—
452M16—42no—
453M16—42no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—42no—
459M16—42no—
460C2×C8—24no—
461C2×C8—48no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—42no—
467C2×C8—42no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C22⋊C4—216no—
483C22⋊C4—24no—
484C22⋊C4—28no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—48no—
495C22⋊C4—84no—
496C22⋊C4#10418yescomplemented
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—48no—
510C42—432no—
511C42—42no—
512C42—41no—
513C42—42no—
514C42—42no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42#12314yescomplemented
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—48no—
535C23—48no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23#39012yes—
541C23#39012yes—
542C23C2×#10414yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×#12111yescharacteristic
545C23C2×#12111yescharacteristic
546C23#54612yes—
547C23#54712yes—
548Q8—41no—
549Q8—42no—
550Q8—41no—
551D4—41no—
552D4—42no—
553D4—41no—
554D4—84no—
555D4—81no—
556D4—81no—
557D4—82no—
558D4—84no—
559C2×C4—216no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—432no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—82no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4#39012yes—
613C2×C4#39012yes—
614C2×C4C2×#10418yes—
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C4×#4112yes—
618C2×C4#52012yes—
619C2×C4C2×#12311yescharacteristic
620C2×C4#52511yescharacteristic
621C2×C4C2×#12411yescharacteristic
622C2×C4#52912yes—
623C2×C4C2×#12711yescharacteristic
624C2×C4#53411yescharacteristic
625C2×C4#53611yescharacteristic
626C2×C4#53912yes—
627C2×C4#54612yes—
628C2×C4#54612yes—
629C2×C4#54712yes—
630C2×C4#54712yes—
631C2×C4#55112yes—
632C2×C4#55212yes—
633C8—44no—
634C8—81no—
635C8—81no—
636C22—22no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—22no—
642C22—24no—
643C22—24no—
644C22—28no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—44no—
649C22—48no—
650C22—44no—
651C22—44no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—84no—
656C22#196214yes—
657C22C2×#39011yescharacteristic
658C22#233912yes—
659C22#241211yescharacteristic
660C22#241311yescharacteristic
661C22#242512yes—
662C22C22×#10412yes—
663C22#290112yes—
664C4—28no—
665C4—22no—
666C4—22no—
667C4—28no—
668C4—48no—
669C4—44no—
670C4—44no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4C4×#12311yescharacteristic
677C4#232211yescharacteristic
678C4#235912yes—
679C4#248812yes—
680C4#250111yescharacteristic
681C4#253911yescharacteristic
682C2—21no—
683C2—22no—
684C2—21no—
685C2—24no—
686C2—42no—
687C2—161no—
688C2#1399611yescharacteristic
689C2C2×#196211yescharacteristic
690C2#2121011yescharacteristic
691C2#2250514yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20837C2semidirect
5. C2×#1826C2semidirect
6. C22×#339C2semidirect
7. C42×C22⋊C4C2semidirect
8. #11370C2semidirect
9. #11053C2semidirect
10. #10923C2semidirect
11. #10839C2semidirect
21. #2504C4semidirect
22. #2503C4semidirect
23. #2502C4semidirect
24. #2501C4semidirect
25. #2500C4semidirect
26. #2499C4semidirect
123. #391C2×C4semidirect
124. #391C2×C4semidirect
125. #390C2×C4semidirect
126. #390C2×C4semidirect
132. #384C2×C4semidirect
133. #384C2×C4semidirect
134. #382C2×C4semidirect
135. #382C2×C4semidirect
496. C22⋊C4#104semidirect
519. C42#123semidirect

Maximal subgroups

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

subgroupindex
#348102
C2×#18962
C2×#18712
#208372
C2×#18262
C22×#3392
C42×C22⋊C42
#113702
#110532
#109232
#108392