Finite Group Databasebeta

← #6644495

Subgroups — #6644495

Order 512.

1908 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2536C211yesmaximal, characteristic
3#33008C212yesmaximal, complemented
4#32008C214yesmaximal, complemented
5#31390C214yesmaximal, complemented
6#30569C214yesmaximal, complemented
7#30411C214yesmaximal, complemented
8#29913C212yesmaximal, complemented
9#27600C211yesmaximal, characteristic, complemented
10#24551C212yesmaximal, complemented
11#24380C212yesmaximal, complemented
12#24003C211yesmaximal, characteristic, complemented
13#24003C211yesmaximal, characteristic, complemented
14#23921C211yesmaximal, characteristic, complemented
15#21738C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#483C2212yes—
22C2×#483C2212yescomplemented
23C2×#467C2214yescomplemented
24C2×#451C2212yes—
25C2×#451C2212yescomplemented
26C22×C22⋊C8C2212yes—
27#2855C2212yes—
28#2848C2212yes—
29#2627—21no—
30#2626—21no—
31#2579—22no—
32#2564—22no—
33#2536—22no—
34#2536—22no—
35#2536C2214yes—
36#2536C2214yes—
37#2487C2212yes—
38#2417C2212yes—
39#2417C2212yes—
40#2409C2212yes—
41C2×#398—21no—
42C2×#394—21no—
43#2315—21no—
44#2308—21no—
45C2×#388—21no—
46C2×#388—21no—
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51#2101C2214yescomplemented
52#2097C2214yescomplemented
53#2085C2214yescomplemented
54#2077C2214yescomplemented
55#2075C2214yescomplemented
56#2074C2214yescomplemented
57#2036C2214yescomplemented
58#2035C2214yescomplemented
59#2035C2214yescomplemented
60#2035C2214yescomplemented
61#2001C2214yescomplemented
62#2001C2214yescomplemented
63#1892C2214yescomplemented
64#1892C2214yescomplemented
65#1878C2214yescomplemented
66#1878C2214yescomplemented
67#1872C2212yes—
68#1872C2212yes—
69#1872C2212yescomplemented
70#1872C2212yescomplemented
71#1869C2214yes—
72#1869C2214yes—
73#1865C2214yescomplemented
74#1865C2214yescomplemented
75#1853C2212yes—
76#1853C2212yescomplemented
77#1852C2212yes—
78#1852C2212yescomplemented
79#1823C2212yes—
80#1823C2212yescomplemented
81#1817C2212yes—
82#1807C2214yescomplemented
83#1787C2212yes—
84#1787C2212yescomplemented
85#1779C2211yescharacteristic
86#1779C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4D411yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C23×C8C2314yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8—44no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2314yes—
112C2×C4×D4—44no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#483—44no—
139#483—44no—
140#467—48no—
141#451—44no—
142#451—44no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8—44no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160#400—22no—
161#399—22no—
162#398—22no—
163#398—22no—
164#398—24no—
165#398—24no—
166#396—22no—
167#396—22no—
168#395—22no—
169#394—24no—
170#394—24no—
171#393—22no—
172#391C2312yes—
173#391C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#388—24no—
178#388—24no—
179#388—24no—
180#387—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#382—22no—
193#382C2314yes—
194#381—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#379—22no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25C2×D411yescharacteristic
224C25C2×D411yescharacteristic
225C22×Q8—24no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—24no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—42no—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4—42no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—44no—
267C2×Q16—44no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8—44no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C22⋊Q8—24no—
274C22⋊Q8—28no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—42no—
279C22⋊Q8—44no—
280C22⋊Q8—42no—
281C22⋊Q8—42no—
282C22⋊Q8—42no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—48no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—48no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—42no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—42no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—44no—
363C2×C42C2×D412yes—
364C2×C42C2412yes—
365#107—44no—
366#106—44no—
367#106—44no—
368#103—28no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—44no—
377C22⋊C8—42no—
378C22⋊C8—42no—
379C22⋊C8—42no—
380C22⋊C8—42no—
381#95—22no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—41no—
401C24—41no—
402C24—41no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24#13514yes—
409C24#13714yes—
410C24C22×D411yescharacteristic
411C24C22×D411yescharacteristic
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—416no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×Q8—88no—
429C2×Q8H(5,𝔽2)14yes—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—416no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—88no—
495C22×C4—88no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4H(5,𝔽2)14yes—
511C22×C4C2511yescharacteristic
512Q16—88no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—88no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—81no—
543C4⋊C4—81no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—48no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C42—24no—
565C42—22no—
566C42—22no—
567C42—48no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—22no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—22no—
581C23—22no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—21no—
589C23—21no—
590C23—28no—
591C23—28no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—82no—
597C23—82no—
598C23—82no—
599C23—82no—
600C23—82no—
601C23—82no—
602C23—82no—
603C23—82no—
604C23#53412yes—
605C23C2×#13512yes—
606C23C2×#13512yes—
607C23C2×#13512yes—
608C23C2×#13714yes—
609C23C2×#13712yes—
610C23C23×D411yescharacteristic
611Q8—28no—
612Q8—48no—
613Q8—44no—
614Q8—44no—
615Q8—44no—
616Q8—84no—
617D4—48no—
618D4—48no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—28no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—82no—
669C2×C4—84no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—162no—
679C2×C4—162no—
680C2×C4#53412yes—
681C2×C4C2×H(5,𝔽2)12yes—
682C2×C4C2×H(5,𝔽2)12yes—
683C2×C4#58512yes—
684C8—44no—
685C8—44no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—24no—
695C22—22no—
696C22—22no—
697C22—24no—
698C22—22no—
699C22—22no—
700C22—24no—
701C22—42no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—42no—
706C22—41no—
707C22—42no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—42no—
712C22—41no—
713C22—88no—
714C22—88no—
715C22#319014yes—
716C22C2×#53411yescharacteristic
717C22#348411yescharacteristic
718C22#348511yescharacteristic
719C22C22×#13511yescharacteristic
720C22C22×#13711yescharacteristic
721C22#358611yescharacteristic
722C22#358611yescharacteristic
723C4—22no—
724C4—42no—
725C4—42no—
726C4—41no—
727C4—42no—
728C4—41no—
729C4—44no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4#353414yes—
738C2—24no—
739C2—42no—
740C2—42no—
741C2—44no—
742C2#3711711yescharacteristic
743C2C2×#319011yescharacteristic
744C2#6075411yescharacteristic
745C2#6156411yescharacteristic
746C2#6157311yescharacteristic
747C2#6169611yescharacteristic
748C2#6170511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33008C2semidirect
4. #32008C2semidirect
5. #31390C2semidirect
6. #30569C2semidirect
7. #30411C2semidirect
8. #29913C2semidirect
9. #27600C2semidirect
10. #24551C2semidirect
11. #24380C2semidirect
12. #24003C2semidirect
13. #24003C2semidirect
14. #23921C2semidirect
22. C2×#483C22semidirect
23. C2×#467C22semidirect
25. C2×#451C22semidirect
51. #2101C22semidirect
52. #2097C22semidirect
53. #2085C22semidirect
54. #2077C22semidirect
55. #2075C22semidirect
56. #2074C22semidirect
57. #2036C22semidirect
58. #2035C22semidirect
59. #2035C22semidirect
60. #2035C22semidirect
61. #2001C22semidirect
62. #2001C22semidirect
63. #1892C22semidirect
64. #1892C22semidirect
65. #1878C22semidirect
66. #1878C22semidirect
69. #1872C22semidirect
70. #1872C22semidirect
73. #1865C22semidirect
74. #1865C22semidirect
76. #1853C22semidirect
78. #1852C22semidirect
80. #1823C22semidirect
82. #1807C22semidirect
84. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#25362
#330082
#320082
#313902
#305692
#304112
#299132
#276002
#245512
#243802
#240032
#239212
#217382