Finite Group Databasebeta

← #6355333

Subgroups — #6355333

Order 512.

1903 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60941C211yesmaximal, characteristic, complemented
2C2×#3077C211yesmaximal, characteristic, complemented
3C2×#3036C211yesmaximal, characteristic
4C2×#2975C211yesmaximal, characteristic, complemented
5#19427C214yesmaximal, complemented
6#19426C214yesmaximal, complemented
7#19421C214yesmaximal, complemented
8#19420C214yesmaximal, complemented
9#19279C218yesmaximal, complemented
10#17814C211yesmaximal, characteristic, complemented
11#17708C211yesmaximal, characteristic, complemented
12#17704C211yesmaximal, characteristic, complemented
13C22×AGL(1,ℤ/8)—21no—
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#3246—22no—
18#3229C2212yescomplemented
19#3227C2212yescomplemented
20#3208C2212yescomplemented
21#3206C2212yescomplemented
22#3187—22no—
23#3167C2214yescomplemented
24#3103—22no—
25#3101—22no—
26#3077—22no—
27#3077—22no—
28#3077C2214yes—
29#3077C2214yescomplemented
30C2×#478—22no—
31C2×#476C2212yescomplemented
32C2×#474C2212yescomplemented
33#3036—22no—
34#3036—22no—
35#3036C2214yes—
36#3036C2214yes—
37C2×#463—22no—
38C2×#461C2212yes—
39C2×#459C2212yes—
40#2975—22no—
41#2975—22no—
42#2975C2214yes—
43#2975C2214yescomplemented
44C2×#436—22no—
45C2×#436C2214yescomplemented
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#423C2211yescharacteristic
51C2×#417—21no—
52C2×#418—21no—
53C2×#407—22no—
54#1699C22116yescomplemented
55#1697C22116yescomplemented
56#1691C22116yescomplemented
57#1685C22116yescomplemented
58#1571C2214yescomplemented
59#1571C2214yescomplemented
60#1567C2214yescomplemented
61#1550C2214yescomplemented
62#1549C2212yescomplemented
63#1548C2212yescomplemented
64#1547C2214yescomplemented
65#1546C2212yescomplemented
66#1545C2212yescomplemented
67#1542C2214yescomplemented
68#1540C2214yescomplemented
69C22×#41—21no—
70C23×D4D411yescharacteristic
71C24×C4—21no—
72C2×AGL(1,ℤ/8)—24no—
73C2×AGL(1,ℤ/8)—24no—
74C2×AGL(1,ℤ/8)—48no—
75C22×SD16—22no—
76C22×D8—22no—
77C22×M16—21no—
78C22×M16—21no—
79C22×M16—24no—
80#533—21no—
81#533—21no—
82#533—21no—
83#533—21no—
84#533—21no—
85#533—21no—
86#533C2312yes—
87#533C2312yes—
88#532—21no—
89#532—21no—
90#532—21no—
91#532—21no—
92#532—21no—
93#532—21no—
94#532C2312yes—
95#532C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#125—21no—
101C2×#124—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×#121—21no—
105C2×#121—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4D412yes—
117C2×C4×D4D412yes—
118C2×C4×D4D412yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C22⋊C4—21no—
125C22×C42C2311yescharacteristic
126#498C2314yescomplemented
127#492C2314yescomplemented
128#487C2314yescomplemented
129#486C2314yescomplemented
130#483—24no—
131#481—24no—
132#478—28no—
133#478—28no—
134#476—24no—
135#476—24no—
136#476C2318yes—
137#475—24no—
138#474—24no—
139#474C2318yes—
140#465—28no—
141#463—28no—
142#463—28no—
143#461—24no—
144#461—24no—
145#461C2318yes—
146#460—24no—
147#459—24no—
148#459C2318yes—
149#450—24no—
150#448—24no—
151C4×SD16—28no—
152#436—28no—
153#436—28no—
154#436—28no—
155#436C23116yes—
156#431—21no—
157#431—21no—
158#431D412yes—
159#431D412yes—
160#431C2312yes—
161#431C2312yes—
162#431C2312yes—
163#431C2312yes—
164#428—24no—
165#428—24no—
166C2×#107—22no—
167C2×#106—22no—
168#423—21no—
169#423—21no—
170#423—21no—
171#423—21no—
172#423C2312yes—
173#423C2312yes—
174#423C2312yes—
175#423C2312yes—
176C2×C4⋊C8—21no—
177C2×C4⋊C8—21no—
178C2×C4⋊C8D412yes—
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2312yes—
182#418—24no—
183#418—24no—
184#417—24no—
185#417—24no—
186C2×#103—22no—
187C2×#102—22no—
188C2×#102—22no—
189C2×#102—22no—
190#407—28no—
191#407—28no—
192C2×C22⋊C8—21no—
193C2×C22⋊C8—24no—
194C2×C22⋊C8D412yes—
195C2×C4×C8—21no—
196C2×C4×C8C2312yes—
197#335C23116yescomplemented
198#333C2318yescomplemented
199#332C2318yescomplemented
200#332C2318yescomplemented
201#331C2318yescomplemented
202C25—21no—
203C2×#41—22no—
204C2×#41—22no—
205C2×#41—41no—
206C2×#41—41no—
207C2×#41—44no—
208C2×#41—44no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—41no—
221C22×D4C2×D412yes—
222C22×D4C2×D412yes—
223C22×D4C2×D412yes—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—41no—
232C23×C4—84no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4#4111yescharacteristic
236AGL(1,ℤ/8)—416no—
237AGL(1,ℤ/8)—416no—
238C2×SD16—28no—
239C2×SD16—28no—
240C2×SD16—48no—
241C2×D8—28no—
242C2×D8—48no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—216no—
248C2×M16—44no—
249C2×M16—44no—
250C2×M16—88no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8C2×D414yes—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C2412yes—
265C4⋊Q8C2412yes—
266C4⋊Q8C2412yes—
267#127—21no—
268#127—21no—
269#127—21no—
270#127C2412yes—
271#125—22no—
272#125—22no—
273#125—41no—
274#125—41no—
275#124—22no—
276#124—22no—
277#124—41no—
278#124—41no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—41no—
284C22⋊Q8—41no—
285C22⋊Q8—41no—
286C22⋊Q8—41no—
287#121—22no—
288#121—22no—
289#121—22no—
290#121—22no—
291#121—41no—
292#121—41no—
293#121—41no—
294#121—41no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—41no—
300C4×Q8—41no—
301C4×Q8—41no—
302C4×Q8—41no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—41no—
320C4×D4—41no—
321C4×D4—41no—
322C4×D4—41no—
323C4×D4C2×D414yes—
324C4×D4C2×D414yes—
325C4×D4C2×D414yes—
326C4×D4C2414yes—
327#117—22no—
328#117—22no—
329#117—41no—
330#117—41no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—41no—
341C2×C4⋊C4—41no—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—41no—
352C2×C22⋊C4—41no—
353C2×C22⋊C4—41no—
354C2×C42—41no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42#4111yescharacteristic
362C2×C42#4111yescharacteristic
363C2×C42#4111yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—28no—
368#106—28no—
369#106—28no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8—22no—
375C4⋊C8—22no—
376C4⋊C8—22no—
377C4⋊C8C2×D414yes—
378C4⋊C8C2×D414yes—
379C4⋊C8C2414yes—
380C4⋊C8C2414yes—
381C4⋊C8C2414yes—
382#103—28no—
383#103—28no—
384#102—28no—
385#102—28no—
386#102—28no—
387#102—28no—
388C22⋊C8—22no—
389C22⋊C8—22no—
390C22⋊C8—216no—
391C22⋊C8C2×D414yes—
392C22⋊C8C2×D414yes—
393C4×C8—22no—
394C4×C8—22no—
395C4×C8C2×D414yes—
396C4×C8C2414yes—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—44no—
401C24—41no—
402C24—82no—
403C24—82no—
404C24#12111yescharacteristic
405#41—44no—
406#41—44no—
407#41—44no—
408#41—44no—
409#41—84no—
410#41—84no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4—84no—
446C2×D4C22×D414yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—81no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—82no—
468C22×C4—84no—
469C22×C4#12111yescharacteristic
470C22×C4#12111yescharacteristic
471C22×C4#12111yescharacteristic
472C22×C4#12111yescharacteristic
473C22×C4#12111yescharacteristic
474C22×C4#12111yescharacteristic
475C22×C4#12111yescharacteristic
476C22×C4#12111yescharacteristic
477C22×C4#12111yescharacteristic
478C22×C4AGL(1,ℤ/8)14yes—
479C22×C4#13714yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4111yescharacteristic
487C22×C4C2×#4111yescharacteristic
488C22×C4C2×#4111yescharacteristic
489SD16—416no—
490SD16—88no—
491D8—416no—
492M16—44no—
493M16—44no—
494M16—44no—
495M16—44no—
496M16—816no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—44no—
501C2×C8—44no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8C22×D418yes—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4C22×D414yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C42—42no—
536C42—42no—
537C42#12111yescharacteristic
538C42#12111yescharacteristic
539C42#12111yescharacteristic
540C42#12111yescharacteristic
541C42#12111yescharacteristic
542C42#12111yescharacteristic
543C42C22×D411yescharacteristic
544C42C22×D411yescharacteristic
545C42C22×D411yescharacteristic
546C42C22×D411yescharacteristic
547C42C22×D411yescharacteristic
548C42C22×D411yescharacteristic
549C42C2×#4111yescharacteristic
550C42C2×#4111yescharacteristic
551C42C2×#4111yescharacteristic
552C42C2511yescharacteristic
553C23—24no—
554C23—41no—
555C23—41no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23—816no—
569C23—82no—
570C23#46814yes—
571C23C2×#12111yescharacteristic
572C23C2×#12111yescharacteristic
573C23C2×#12111yescharacteristic
574Q8—44no—
575Q8—42no—
576Q8—42no—
577Q8—82no—
578Q8—82no—
579Q8—82no—
580Q8—82no—
581D4—24no—
582D4—24no—
583D4—44no—
584D4—44no—
585D4—44no—
586D4—44no—
587D4—42no—
588D4—42no—
589D4—82no—
590D4—82no—
591D4—82no—
592D4—84no—
593D4—84no—
594D4—82no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4#46814yes—
624C2×C4#46814yes—
625C2×C4#46814yes—
626C2×C4C2×#12111yescharacteristic
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×#12111yescharacteristic
630C2×C4C2×#12111yescharacteristic
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×#12111yescharacteristic
633C2×C4C2×#12111yescharacteristic
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4C2×#13712yes—
641C2×C4C23×D411yescharacteristic
642C2×C4C23×D411yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C8—28no—
645C8—44no—
646C8—44no—
647C8—44no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—88no—
659C22—88no—
660C22—88no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—164no—
665C22C2×#46812yes—
666C22C2×#46812yes—
667C22C2×#46812yes—
668C22C22×#12111yescharacteristic
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4C2×#46812yes—
681C4C2×#46812yes—
682C4C2×#46812yes—
683C4C22×AGL(1,ℤ/8)11yescharacteristic
684C4C22×#13711yescharacteristic
685C4#358512yes—
686C2—22no—
687C2—22no—
688C2—22no—
689C2—84no—
690C2—162no—
691C2#6042311yescharacteristic
692C2C22×#46811yescharacteristic
693C2#6046911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60941C2semidirect
2. C2×#3077C2semidirect
4. C2×#2975C2semidirect
5. #19427C2semidirect
6. #19426C2semidirect
7. #19421C2semidirect
8. #19420C2semidirect
9. #19279C2semidirect
10. #17814C2semidirect
11. #17708C2semidirect
12. #17704C2semidirect
18. #3229C22semidirect
19. #3227C22semidirect
20. #3208C22semidirect
21. #3206C22semidirect
23. #3167C22semidirect
29. #3077C22semidirect
31. C2×#476C22semidirect
32. C2×#474C22semidirect
43. #2975C22semidirect
45. C2×#436C22semidirect
54. #1699C22semidirect
55. #1697C22semidirect
56. #1691C22semidirect
57. #1685C22semidirect
58. #1571C22semidirect
59. #1571C22semidirect
60. #1567C22semidirect
61. #1550C22semidirect
62. #1549C22semidirect
63. #1548C22semidirect
64. #1547C22semidirect
65. #1546C22semidirect
66. #1545C22semidirect
67. #1542C22semidirect
68. #1540C22semidirect
126. #498C23semidirect
127. #492C23semidirect
128. #487C23semidirect
129. #486C23semidirect
197. #335C23semidirect
198. #333C23semidirect
199. #332C23semidirect
200. #332C23semidirect
201. #331C23semidirect

Maximal subgroups

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

subgroupindex
#609412
C2×#30772
C2×#30362
C2×#29752
#194272
#194262
#194212
#194202
#192792
#178142
#177082
#177042