Finite Group Databasebeta

← #196692

Subgroups — #196692

Order 512.

1740 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
1C22×#423C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic, complemented
4C2×#1560C212yesmaximal, complemented
5#9141C212yesmaximal, complemented
6#9135C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#9127C212yesmaximal, complemented
9#8041C212yesmaximal, complemented
10C23×M16—21no—
11C2×#532C2211yescharacteristic
12C2×#525C412yes—
13C23×C42C2211yescharacteristic
14C2×#423—21no—
15C2×#423—21no—
16C2×#423—28no—
17C2×#423C2212yes—
18C2×#423C2212yes—
19C2×#423C2212yes—
20C22×C4⋊C8—21no—
21C22×C4⋊C8C2212yes—
22C2×#407—21no—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854—21no—
28#2854—21no—
29#2850C412yes—
30#2850C412yes—
31#2847—21no—
32#2847—21no—
33C4×#121—21no—
34C4×#121—21no—
35C4×#121—21no—
36C4×#121—21no—
37#2297C412yes—
38#2285C2211yescharacteristic
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43#1728—21no—
44#1560—28no—
45C2×#333C2214yescomplemented
46C2×#331C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C22×#41C2×C412yes—
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C22×M16—28no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—44no—
58C22×M16—44no—
59C23×C8—22no—
60#532—24no—
61#532—24no—
62C2×C4⋊Q8C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#124C2×C412yes—
65#525—48no—
66C2×#123—22no—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8C2×C412yes—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121C2×C412yes—
76C2×#121C2×C412yes—
77C2×#120—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×#117C2×C412yes—
83C22×C42—24no—
84C22×C42—24no—
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#423—28no—
98#423—28no—
99#423—24no—
100#423—28no—
101#423—24no—
102#423—24no—
103#423D418yescomplemented
104C2×C4⋊C8—28no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—28no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8C2314yes—
111C2×C4⋊C8C2314yes—
112#407—44no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115#391C2×C412yes—
116#391C2×C412yes—
117#390—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390C2×C412yes—
122#390C2×C412yes—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#388—22no—
128#388—22no—
129#388—22no—
130#388—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#386—22no—
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#382C2×C412yes—
140#380—21no—
141#380—21no—
142#380—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150#336—22no—
151#336—22no—
152#333—28no—
153#331—28no—
154C25C22⋊C411yescharacteristic
155C2×#41—28no—
156C2×#41—44no—
157C2×#41—44no—
158C22×Q8C22×C412yes—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C412yes—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C2×M16—28no—
198C2×M16—28no—
199C2×M16—432no—
200C2×M16—44no—
201C2×M16—44no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16—44no—
205C2×M16—44no—
206C2×M16—44no—
207C22×C8—28no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—44no—
211C22×C8—44no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214#127—22no—
215#127—22no—
216#124—28no—
217#123—48no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220#121—44no—
221#121—44no—
222#121—42no—
223#121—44no—
224#121—42no—
225#121—42no—
226#121—42no—
227#121—42no—
228#121—42no—
229#121—42no—
230#121—42no—
231#121—44no—
232#120—48no—
233C4×D4—48no—
234C4×D4—48no—
235C4×D4—42no—
236C4×D4—44no—
237C4×D4—42no—
238#117—44no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D414yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4114yes—
294C2×C42C2411yescharacteristic
295C4⋊C8—28no—
296C4⋊C8—28no—
297C4⋊C8—28no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—24no—
301C4⋊C8—44no—
302C4⋊C8C2×D418yes—
303C22⋊C8—44no—
304C22⋊C8—44no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—41no—
318C24—41no—
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322#41—816no—
323C2×Q8—24no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×D4—24no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—42no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
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—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—416no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4#10014yes—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C4×D414yes—
427C22×C4#12012yes—
428C22×C4#12114yes—
429C22×C4#12312yes—
430C22×C4AGL(1,ℤ/8)12yes—
431C22×C4AGL(1,ℤ/8)12yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4#13712yes—
434C22×C4#13712yes—
435C22×C4#13712yes—
436C22×C4C23×C411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439M16—416no—
440M16—416no—
441M16—88no—
442C2×C8—28no—
443C2×C8—416no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C2×C8—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C42—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—44no—
489C42C4×D414yes—
490C42#12012yes—
491C42#12114yes—
492C42#12312yes—
493C42C22×D412yes—
494C42C2×#4112yes—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—28no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23#38612yes—
526C23C2×#10012yes—
527C23#41712yes—
528C23#41712yes—
529C23#41712yes—
530C23C22×C22⋊C411yescharacteristic
531Q8—84no—
532D4—84no—
533D4—88no—
534D4—88no—
535D4—88no—
536D4—81no—
537D4—81no—
538D4—81no—
539D4—81no—
540D4—82no—
541D4—82no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
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—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—88no—
602C2×C4#38612yes—
603C2×C4#38612yes—
604C2×C4#38612yes—
605C2×C4C2×#10012yes—
606C2×C4C2×#10012yes—
607C2×C4#41312yes—
608C2×C4#41712yes—
609C2×C4#41712yes—
610C2×C4#41712yes—
611C2×C4C2×C4×D412yes—
612C2×C4C2×#12011yescharacteristic
613C2×C4C2×#12112yes—
614C2×C4C2×#12311yescharacteristic
615C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
616C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
617C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
618C2×C4C2×#13711yescharacteristic
619C2×C4C2×#13711yescharacteristic
620C2×C4C2×#13711yescharacteristic
621C2×C4#57512yes—
622C8—48no—
623C8—84no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—416no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#156014yescomplemented
653C22C2×#38611yescharacteristic
654C22C22×#10011yescharacteristic
655C22C2×#41311yescharacteristic
656C22C2×#41711yescharacteristic
657C22C2×#41711yescharacteristic
658C22C2×#41711yescharacteristic
659C22#289611yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—41no—
668C4—41no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#190014yes—
673C4#300311yescharacteristic
674C4#300411yescharacteristic
675C4#305512yes—
676C4#305612yes—
677C4#309211yescharacteristic
678C4#309211yescharacteristic
679C2—24no—
680C2—42no—
681C2—42no—
682C2—162no—
683C2C2×#156011yescharacteristic
684C2#1773711yescharacteristic
685C2#1775011yescharacteristic
686C2#1775211yescharacteristic
687C2#2090911yescharacteristic
688C2#2097111yescharacteristic
689C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#423C2semidirect
3. #20626C2semidirect
4. C2×#1560C2semidirect
5. #9141C2semidirect
6. #9135C2semidirect
7. #9133C2semidirect
8. #9127C2semidirect
9. #8041C2semidirect
45. C2×#333C22semidirect
46. C2×#331C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
103. #423D4semidirect
652. C22#1560semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
#366242
#206262
C2×#15602
#91412
#91352
#91332
#91272
#80412