Finite Group Databasebeta

← #6508759

Subgroups — #6508759

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4#34212C211yesmaximal, characteristic
5#28124C212yesmaximal, complemented
6#28121C212yesmaximal, complemented
7#27653C212yesmaximal, complemented
8#27647C212yesmaximal, complemented
9#25226C214yesmaximal, complemented
10#25220C214yesmaximal, complemented
11#24590C214yesmaximal, complemented
12#24540C214yesmaximal, complemented
13#21934C211yesmaximal, characteristic
14#21765C211yesmaximal, characteristic
15#20702C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3195—28no—
20#3095—28no—
21C2×#484C2212yes—
22C2×#485C2214yes—
23C2×#483C2212yes—
24C2×#483C2212yes—
25C2×#483C2212yes—
26C2×#470C2214yes—
27C2×#451C2212yes—
28C2×#451C2212yes—
29C2×#428C2212yes—
30C2×#419C2212yes—
31C2×#407C2211yescharacteristic
32C2×#407C2211yescharacteristic
33#2499—22no—
34#2499—22no—
35#2499C2214yes—
36#2499C2214yes—
37C2×#397—21no—
38C2×#397—21no—
39#2368—22no—
40#2360C2212yes—
41#2360C2212yes—
42#2345—22no—
43#2345C2212yes—
44Q8×C22⋊C4C2212yes—
45#2329—21no—
46#2329—21no—
47#2315—21no—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#387—21no—
53C2×#387—21no—
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#380C2211yescharacteristic
57#2273—21no—
58C2×C4×C22⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2084C2214yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2056C2214yescomplemented
66#2056C2214yescomplemented
67#1949C2214yescomplemented
68#1948C2214yescomplemented
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1894C2214yes—
72#1878C2214yescomplemented
73#1878C2214yescomplemented
74#1872C2212yes—
75#1872C2212yes—
76#1851C2214yes—
77#1851C2214yes—
78#1825C2214yes—
79#1824C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1805C2214yescomplemented
83#1803C2214yescomplemented
84#1797C2212yes—
85#1797C2212yes—
86#1787C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×M16C2312yes—
94C22×M16C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8D418yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×#117—21no—
110C2×#117—21no—
111C2×#117—24no—
112C2×#117—24no—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C22×C4⋊C4—21no—
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⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#485—216no—
135#484—28no—
136#483—28no—
137#483—28no—
138#483—28no—
139#470—216no—
140#451—28no—
141#451—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×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155#407—24no—
156#407—24no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161#397—24no—
162#397—24no—
163#397—24no—
164#397—24no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#391C2314yes—
172#391C2314yes—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—22no—
192#384C2314yes—
193#382—22no—
194#382C2314yes—
195#380—21no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214C4×C22⋊C4C2312yes—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95—21no—
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—
232C25C2×D411yescharacteristic
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8C2×D414yes—
243C22×Q8C2412yes—
244C22×Q8C2412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D414yes—
262C23×C4C2×D411yescharacteristic
263C23×C4#4114yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—42no—
273C2×M16—42no—
274C2×M16—42no—
275C2×M16—42no—
276C22×C8C2414yes—
277C22×C8C2414yes—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8C2×D418yes—
289C22⋊Q8C2×D418yes—
290C4×Q8—44no—
291C4×Q8—44no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—44no—
297#117—44no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
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—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—42no—
359C2×C42—42no—
360C2×C42—44no—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2412yes—
364C2×C42C2412yes—
365#107—28no—
366#106—28no—
367#106—28no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#103—44no—
375#103—44no—
376C22⋊C8—24no—
377C22⋊C8—24no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
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—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—22no—
401C24—22no—
402C24—41no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24#12114yes—
407C24C22×D411yescharacteristic
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—48no—
421C2×Q8—42no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×Q8—84no—
425C2×Q8—84no—
426C2×Q8—84no—
427C2×Q8C22×D414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4#12114yes—
486C22×C4#12114yes—
487C22×C4#12114yes—
488C22×C4#13714yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2511yescharacteristic
503Q16—88no—
504M16—48no—
505M16—84no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4C22×D414yes—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C42—24no—
551C42—24no—
552C42—42no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42—82no—
557C42—82no—
558C42—82no—
559C42—82no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—28no—
577C23—44no—
578C23—44no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23—82no—
583C23C2×#12112yes—
584C23C2×#12112yes—
585C23C2×#12112yes—
586C23C2×#13712yes—
587C23#57512yes—
588C23C23×D411yescharacteristic
589Q8—28no—
590Q8—44no—
591Q8—44no—
592Q8—44no—
593Q8—84no—
594Q8—84no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
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—44no—
624C2×C4—44no—
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—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—162no—
656C2×C4—162no—
657C2×C4C2×#12112yes—
658C2×C4C2×#12112yes—
659C2×C4C2×#12112yes—
660C2×C4C2×#13714yes—
661C2×C4C23×D411yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4#58512yes—
664C2×C4#58512yes—
665C8—44no—
666C8—44no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—88no—
688C22#303714yes—
689C22C22×#12111yescharacteristic
690C22#348511yescharacteristic
691C22#348511yescharacteristic
692C22C22×#13711yescharacteristic
693C22C2×#57511yescharacteristic
694C22#358611yescharacteristic
695C22#358611yescharacteristic
696C4—24no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#348512yes—
712C4#349112yes—
713C2—24no—
714C2—42no—
715C2—42no—
716C2#3544711yescharacteristic
717C2C2×#303711yescharacteristic
718C2#6042711yescharacteristic
719C2#6170511yescharacteristic
720C2#6171211yescharacteristic
721C2#6171611yescharacteristic
722C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #28124C2semidirect
6. #28121C2semidirect
7. #27653C2semidirect
8. #27647C2semidirect
9. #25226C2semidirect
10. #25220C2semidirect
11. #24590C2semidirect
12. #24540C2semidirect
59. #2085C22semidirect
60. #2084C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
65. #2056C22semidirect
66. #2056C22semidirect
67. #1949C22semidirect
68. #1948C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
72. #1878C22semidirect
73. #1878C22semidirect
82. #1805C22semidirect
83. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#30952
C2×#24992
#342122
#281242
#281212
#276532
#276472
#252262
#252202
#245902
#245402
#219342
#217652
#207022