Finite Group Databasebeta

← #6508774

Subgroups — #6508774

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic, complemented
3C2×#2500C211yesmaximal, characteristic
4#34245C211yesmaximal, characteristic, complemented
5#28123C212yesmaximal, complemented
6#28120C212yesmaximal, complemented
7#27653C212yesmaximal, complemented
8#27647C212yesmaximal, complemented
9#25213C214yesmaximal, complemented
10#25203C214yesmaximal, complemented
11#24589C214yesmaximal, complemented
12#24541C214yesmaximal, complemented
13#21932C211yesmaximal, characteristic, complemented
14#21767C211yesmaximal, characteristic
15#20699C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3187—28no—
20#3095—28no—
21C2×#484C2212yes—
22C2×#481C2212yes—
23C2×#485C2214yescomplemented
24C2×#483C2212yes—
25C2×#483C2212yes—
26C2×#465C2214yes—
27C2×#448C2212yes—
28C2×#450C2212yes—
29C2×#428C2212yescomplemented
30C2×#419C2212yescomplemented
31C2×#407C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2500—21no—
34#2500—21no—
35#2500—21no—
36#2500—21no—
37#2500C2214yes—
38#2500C2214yes—
39C2×#394—22no—
40#2368—22no—
41#2362C2212yescomplemented
42#2362C2212yescomplemented
43#2344C2212yescomplemented
44#2342—22no—
45Q8×C22⋊C4C2212yescomplemented
46#2332—21no—
47#2332—21no—
48#2313—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#388—22no—
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2280C2211yescharacteristic
57#2274—21no—
58C2×C4×C22⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2084C2214yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2067C2214yescomplemented
64#2065C2214yescomplemented
65#2055C2214yescomplemented
66#2055C2214yescomplemented
67#1949C2214yescomplemented
68#1948C2214yescomplemented
69#1926C2214yescomplemented
70#1925C2214yescomplemented
71#1895C2214yes—
72#1880C2214yescomplemented
73#1880C2214yescomplemented
74#1875C2212yes—
75#1875C2212yes—
76#1851C2214yescomplemented
77#1850C2214yescomplemented
78#1825C2214yescomplemented
79#1824C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1805C2214yescomplemented
83#1803C2214yescomplemented
84#1797C2212yes—
85#1797C2212yes—
86#1786C2212yes—
87#1735C2212yescomplemented
88#1735C2212yescomplemented
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C23×C8C2312yes—
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—22no—
113C2×#117—24no—
114C2×#117—24no—
115C2×#117C2312yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4D412yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#485—216no—
135#484—28no—
136#483—28no—
137#483—28no—
138#481—28no—
139#465—216no—
140#450—28no—
141#448—28no—
142#428—28no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#419—28no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155#407—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161#394—24no—
162#394—24no—
163#394—24no—
164#394—24no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390C2314yes—
174#390C2314yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
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#384—21no—
194#384—21no—
195#384C2314yes—
196#382—21no—
197#382—21no—
198#382C2314yes—
199#381—21no—
200#381—22no—
201#380—22no—
202#380C2312yes—
203#380C2312yes—
204#379—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D411yescharacteristic
217C4×C22⋊C4D411yescharacteristic
218C4×C22⋊C4D411yescharacteristic
219C4×C22⋊C4D411yescharacteristic
220C4×C22⋊C4C2312yes—
221C4×C22⋊C4C2312yes—
222C2×#95—22no—
223C2×#95D412yes—
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—84no—
249C22×D4C2412yes—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—42no—
256C23×C4—42no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D414yes—
263C23×C4#4114yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×SD16—28no—
268C2×SD16—44no—
269C2×SD16—44no—
270C2×M16—24no—
271C2×M16—42no—
272C2×M16—42no—
273C22×C8—24no—
274C22×C8—42no—
275C22×C8—42no—
276C22×C8C2414yes—
277C22×C8C2414yes—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8C2×D418yes—
283C22⋊Q8C2×D418yes—
284#121—24no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121—44no—
289#121—44no—
290C4×Q8—44no—
291C4×Q8—44no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—48no—
297#117—44no—
298#117—44no—
299#117—42no—
300#117—42no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—84no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2412yes—
370#107—28no—
371#106—28no—
372#106—28no—
373C4⋊C8—44no—
374C4⋊C8—44no—
375#103—28no—
376#103—28no—
377#103—28no—
378#102—28no—
379#102—44no—
380#102—44no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383C22⋊C8—24no—
384C22⋊C8—24no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—24no—
402C24—22no—
403C24—42no—
404C24—41no—
405C24—41no—
406C24—44no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—44no—
411C24#12114yes—
412C24C22×D411yescharacteristic
413C24C22×D411yescharacteristic
414C24C22×D411yescharacteristic
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—48no—
420C2×Q8—81no—
421C2×Q8—81no—
422C2×Q8—84no—
423C2×Q8C22×D414yes—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—816no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4#12114yes—
477C22×C4#12114yes—
478C22×C4#12114yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2511yescharacteristic
493SD16—88no—
494M16—48no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—84no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4C22×D414yes—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C42—24no—
543C42—24no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—44no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—81no—
553C42—81no—
554C42—82no—
555C42—82no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—28no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—88no—
579C23—88no—
580C23—84no—
581C23—84no—
582C23C2×#12112yes—
583C23C2×#12112yes—
584C23C2×#12112yes—
585C23#57714yes—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—44no—
589Q8—84no—
590D4—44no—
591D4—44no—
592D4—84no—
593D4—88no—
594D4—88no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4C2×#12112yes—
647C2×C4C2×#12112yes—
648C2×C4C2×#12112yes—
649C2×C4#57514yes—
650C2×C4C23×D411yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4#58512yes—
653C2×C4#58512yes—
654C8—44no—
655C8—44no—
656C22—24no—
657C22—24no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—88no—
671C22—82no—
672C22—84no—
673C22—84no—
674C22—82no—
675C22—82no—
676C22—168no—
677C22#304314yes—
678C22C22×#12111yescharacteristic
679C22#348511yescharacteristic
680C22#348511yescharacteristic
681C22C2×#57512yes—
682C22C2×#57712yes—
683C4—24no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4#348512yes—
698C4#349112yes—
699C2—24no—
700C2—42no—
701C2—42no—
702C2—162no—
703C2#3544711yescharacteristic
704C2#6041511yescharacteristic
705C2C2×#304311yescharacteristic
706C2#6173112yes—
707C2#6173612yes—
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. #34245C2semidirect
5. #28123C2semidirect
6. #28120C2semidirect
7. #27653C2semidirect
8. #27647C2semidirect
9. #25213C2semidirect
10. #25203C2semidirect
11. #24589C2semidirect
12. #24541C2semidirect
13. #21932C2semidirect
15. #20699C2semidirect
23. C2×#485C22semidirect
29. C2×#428C22semidirect
30. C2×#419C22semidirect
41. #2362C22semidirect
42. #2362C22semidirect
43. #2344C22semidirect
45. Q8×C22⋊C4C22semidirect
59. #2085C22semidirect
60. #2084C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2055C22semidirect
66. #2055C22semidirect
67. #1949C22semidirect
68. #1948C22semidirect
69. #1926C22semidirect
70. #1925C22semidirect
72. #1880C22semidirect
73. #1880C22semidirect
76. #1851C22semidirect
77. #1850C22semidirect
78. #1825C22semidirect
82. #1805C22semidirect
83. #1803C22semidirect
87. #1735C22semidirect
88. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
C2×#30952
C2×#25002
#342452
#281232
#281202
#276532
#276472
#252132
#252032
#245892
#245412
#219322
#217672
#206992