Finite Group Databasebeta

← #6508760

Subgroups — #6508760

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic, complemented
3C2×#2500C211yesmaximal, characteristic
4#34212C211yesmaximal, characteristic, complemented
5#28123C212yesmaximal, complemented
6#28120C212yesmaximal, complemented
7#27653C212yesmaximal, complemented
8#27648C212yesmaximal, complemented
9#25224C214yesmaximal, complemented
10#25222C214yesmaximal, complemented
11#24591C214yesmaximal, complemented
12#24541C214yesmaximal, complemented
13#21934C211yesmaximal, characteristic, complemented
14#21764C211yesmaximal, characteristic
15#20702C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3193—28no—
20#3095—28no—
21C2×#484C2212yes—
22C2×#484C2212yes—
23C2×#485C2214yescomplemented
24C2×#483C2212yes—
25C2×#480C2212yes—
26C2×#468C2214yes—
27C2×#448C2212yes—
28C2×#450C2212yes—
29C2×#428C2212yescomplemented
30C2×#419C2212yescomplemented
31C2×#407C2211yescharacteristic
32C2×#407C2211yescharacteristic
33#2500—22no—
34#2500—22no—
35#2500C2214yes—
36#2500C2214yes—
37C2×#394—21no—
38C2×#394—21no—
39#2368—22no—
40#2360C2212yescomplemented
41#2360C2212yescomplemented
42#2345—22no—
43#2345C2212yescomplemented
44Q8×C22⋊C4C2212yescomplemented
45#2329—21no—
46#2329—21no—
47#2315—21no—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#380C2211yescharacteristic
57#2273—21no—
58C2×C4×C22⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2083C2214yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2067C2214yescomplemented
64#2065C2214yescomplemented
65#2055C2214yescomplemented
66#2055C2214yescomplemented
67#1949C2214yescomplemented
68#1947C2214yescomplemented
69#1926C2214yescomplemented
70#1925C2214yescomplemented
71#1895C2214yes—
72#1880C2214yescomplemented
73#1880C2214yescomplemented
74#1874C2212yes—
75#1874C2212yes—
76#1851C2214yescomplemented
77#1851C2214yescomplemented
78#1825C2214yescomplemented
79#1824C2212yes—
80#1824C2212yes—
81#1823C2212yes—
82#1805C2214yescomplemented
83#1803C2214yescomplemented
84#1797C2212yes—
85#1797C2212yes—
86#1786C2212yes—
87#1736C2212yescomplemented
88#1736C2212yescomplemented
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C22×M16C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8D418yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—24no—
105C2×#121—24no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×#117—21no—
111C2×#117—21no—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#485—216no—
139#484—28no—
140#484—28no—
141#483—28no—
142#480—28no—
143#468—216no—
144#450—28no—
145#448—28no—
146#428—28no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152#419—28no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159#407—24no—
160#407—24no—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165#394—24no—
166#394—24no—
167#394—24no—
168#394—24no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#391C2312yes—
173#390—22no—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#388—24no—
182#388—24no—
183#388—24no—
184#388—24no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#384—22no—
196#384C2314yes—
197#382—22no—
198#382C2314yes—
199#380—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380C2312yes—
204#380C2312yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C4×C22⋊C4C2312yes—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95D411yescharacteristic
223C2×#95D411yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C25—21no—
233C25C2×D411yescharacteristic
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—42no—
237C22×Q8—42no—
238C22×Q8C2×D414yes—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4—42no—
247C22×D4—44no—
248C22×D4—82no—
249C22×D4—82no—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—42no—
259C23×C4—42no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D414yes—
267C23×C4C2×D411yescharacteristic
268C23×C4#4114yes—
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×SD16—28no—
273C2×SD16—44no—
274C2×SD16—44no—
275C2×M16—24no—
276C2×M16—24no—
277C2×M16—42no—
278C2×M16—42no—
279C2×M16—42no—
280C2×M16—42no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8C2×D418yes—
288C22⋊Q8C2×D418yes—
289#121—24no—
290#121—24no—
291#121—24no—
292#121—24no—
293#121—44no—
294#121—44no—
295C4×Q8—44no—
296C4×Q8—44no—
297#117—24no—
298#117—24no—
299#117—24no—
300#117—24no—
301#117—44no—
302#117—44no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—82no—
357C2×C22⋊C4—82no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—42no—
369C2×C42—42no—
370C2×C42—44no—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2412yes—
374C2×C42C2412yes—
375#107—28no—
376#107—28no—
377#106—28no—
378C4⋊C8—44no—
379C4⋊C8—44no—
380#103—28no—
381#103—28no—
382#103—28no—
383#102—28no—
384#102—44no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—24no—
388C22⋊C8—24no—
389C22⋊C8—24no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—41no—
410C24—41no—
411C24—41no—
412C24—41no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24#12114yes—
421C24C22×D411yescharacteristic
422C24C22×D411yescharacteristic
423C24C22×D411yescharacteristic
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—48no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×Q8—84no—
432C2×Q8C22×D414yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—82no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4—82no—
448C2×D4—816no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
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—48no—
468C22×C4—48no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
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—81no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4#12114yes—
503C22×C4#12114yes—
504C22×C4#12114yes—
505C22×C4#13714yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2×#4112yes—
519C22×C4C2511yescharacteristic
520SD16—88no—
521M16—48no—
522M16—84no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—81no—
547C4⋊C4—81no—
548C4⋊C4—84no—
549C4⋊C4—84no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4C22×D414yes—
553C22⋊C4—28no—
554C22⋊C4—28no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—48no—
567C22⋊C4—84no—
568C22⋊C4—84no—
569C42—24no—
570C42—24no—
571C42—42no—
572C42—42no—
573C42—44no—
574C42—44no—
575C42—82no—
576C42—82no—
577C42—82no—
578C42—82no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—28no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—82no—
606C23—82no—
607C23—82no—
608C23—82no—
609C23—84no—
610C23—84no—
611C23—84no—
612C23—84no—
613C23—84no—
614C23—84no—
615C23—84no—
616C23C2×#12112yes—
617C23C2×#12112yes—
618C23C2×#12112yes—
619C23C2×#13712yes—
620C23#57512yes—
621C23C23×D411yescharacteristic
622Q8—28no—
623Q8—44no—
624Q8—84no—
625D4—44no—
626D4—44no—
627D4—84no—
628D4—88no—
629D4—88no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
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—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—82no—
680C2×C4—82no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—84no—
685C2×C4—84no—
686C2×C4—162no—
687C2×C4—162no—
688C2×C4C2×#12112yes—
689C2×C4C2×#12112yes—
690C2×C4C2×#12112yes—
691C2×C4C2×#13714yes—
692C2×C4C23×D411yescharacteristic
693C2×C4C22×#4111yescharacteristic
694C2×C4#58512yes—
695C2×C4#58512yes—
696C8—44no—
697C8—44no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—88no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—168no—
727C22#303714yes—
728C22C22×#12111yescharacteristic
729C22#348511yescharacteristic
730C22#348511yescharacteristic
731C22C22×#13711yescharacteristic
732C22C2×#57511yescharacteristic
733C22#358511yescharacteristic
734C22#358511yescharacteristic
735C4—24no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—44no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—44no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4#348512yes—
750C4#349112yes—
751C2—24no—
752C2—42no—
753C2—42no—
754C2—162no—
755C2#3544711yescharacteristic
756C2C2×#303711yescharacteristic
757C2#6042611yescharacteristic
758C2#6170811yescharacteristic
759C2#6171011yescharacteristic
760C2#6171811yescharacteristic
761C2#6172111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
4. #34212C2semidirect
5. #28123C2semidirect
6. #28120C2semidirect
7. #27653C2semidirect
8. #27648C2semidirect
9. #25224C2semidirect
10. #25222C2semidirect
11. #24591C2semidirect
12. #24541C2semidirect
13. #21934C2semidirect
15. #20702C2semidirect
23. C2×#485C22semidirect
29. C2×#428C22semidirect
30. C2×#419C22semidirect
40. #2360C22semidirect
41. #2360C22semidirect
43. #2345C22semidirect
44. Q8×C22⋊C4C22semidirect
59. #2085C22semidirect
60. #2083C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2055C22semidirect
66. #2055C22semidirect
67. #1949C22semidirect
68. #1947C22semidirect
69. #1926C22semidirect
70. #1925C22semidirect
72. #1880C22semidirect
73. #1880C22semidirect
76. #1851C22semidirect
77. #1851C22semidirect
78. #1825C22semidirect
82. #1805C22semidirect
83. #1803C22semidirect
87. #1736C22semidirect
88. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#30952
C2×#25002
#342122
#281232
#281202
#276532
#276482
#252242
#252222
#245912
#245412
#219342
#217642
#207022