Finite Group Databasebeta

← #6519042

Subgroups — #6519042

Order 512.

1872 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×#2538C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27604C212yesmaximal, complemented
4#27603C211yesmaximal, characteristic, complemented
5#27601C211yesmaximal, characteristic, complemented
6#25029C214yesmaximal, complemented
7#25028C214yesmaximal, complemented
8#24984C214yesmaximal, complemented
9#24983C214yesmaximal, complemented
10#24010C212yesmaximal, complemented
11#24006C211yesmaximal, characteristic, complemented
12#24003C211yesmaximal, characteristic, complemented
13#21987C212yesmaximal, complemented
14#21790C211yesmaximal, characteristic
15#20741C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#484C2212yescomplemented
21C2×#485C2214yescomplemented
22C2×#483C2212yescomplemented
23C2×#432C2212yes—
24#2538—22no—
25#2538—22no—
26#2538C2214yes—
27#2538C2214yes—
28C2×#396—21no—
29C2×#396—21no—
30#2353—22no—
31#2353—22no—
32#2352C2212yes—
33#2352C2212yes—
34#2352C2212yes—
35#2352C2212yes—
36#2313—21no—
37#2312—21no—
38#2308—21no—
39C2×#390—21no—
40C2×#389—21no—
41C2×#384C2211yescharacteristic
42#2283C2212yes—
43#2280C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49#2077C2214yescomplemented
50#2077C2214yescomplemented
51#2076C2214yescomplemented
52#2075C2214yescomplemented
53#2036C2214yescomplemented
54#2036C2214yescomplemented
55#2035C2214yescomplemented
56#2035C2214yescomplemented
57#1955C2214yes—
58#1955C2214yes—
59#1948C2214yescomplemented
60#1947C2214yescomplemented
61#1873C2214yescomplemented
62#1872C2212yescomplemented
63#1872C2212yescomplemented
64#1854C2214yescomplemented
65#1853C2212yes—
66#1853C2212yes—
67#1853C2212yescomplemented
68#1852C2212yes—
69#1852C2212yes—
70#1852C2212yescomplemented
71#1847C2214yescomplemented
72#1836C2214yescomplemented
73#1825C2214yescomplemented
74#1824C2212yes—
75#1824C2212yes—
76#1824C2212yescomplemented
77#1823C2212yescomplemented
78#1805C2214yescomplemented
79#1805C2214yescomplemented
80#1787C2212yes—
81#1787C2212yes—
82#1779C2211yescharacteristic
83#1779C2211yescharacteristic
84#1779C2212yes—
85#1763C2214yescomplemented
86#1763C2214yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—44no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—28no—
101C2×C4×Q8C2314yes—
102C2×C4×D4—28no—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#485—48no—
133#484—44no—
134#483—44no—
135#432—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2314yes—
154#396—24no—
155#396—24no—
156#396—24no—
157#396—24no—
158#390—24no—
159#390—24no—
160#389—24no—
161#389—24no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#384—22no—
175#384C2314yes—
176#383—22no—
177#382C2312yes—
178#382C2312yes—
179#381—22no—
180#381—22no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380C2312yes—
189#380C2314yes—
190#380C2312yes—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#379—22no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207#376—22no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220C25C2×D411yescharacteristic
221C25C2×D411yescharacteristic
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—42no—
230C22×D4C2×D412yes—
231C22×D4C2×D412yes—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C22×C8C2414yes—
258C22×C8C2414yes—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—28no—
262C22⋊Q8—42no—
263C22⋊Q8—44no—
264C22⋊Q8—42no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—48no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277#117—44no—
278#117—44no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D412yes—
347C2×C42#4118yes—
348C2×C42C2412yes—
349#107—44no—
350#107—44no—
351#106—44no—
352#106—44no—
353C4⋊C8—28no—
354#103—28no—
355#103—28no—
356#103—44no—
357#103—44no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C22⋊C8—44no—
363C22⋊C8—42no—
364C22⋊C8—42no—
365C22⋊C8—44no—
366#97—28no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24#13714yes—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—416no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8H(5,𝔽2)14yes—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—416no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4#12314yes—
481C22×C4#12314yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C2×#4114yes—
489C22×C4C2511yescharacteristic
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—48no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—84no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—84no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C42—48no—
538C42—82no—
539C42—82no—
540C42—82no—
541C42—82no—
542C42#12318yes—
543C42C2×#4114yes—
544C42C2×#4114yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—28no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23C2×#12312yes—
583C23C2×#13712yes—
584C23C2×#13712yes—
585C23C2×#13712yes—
586C23#57512yes—
587C23C23×D411yescharacteristic
588Q8—28no—
589Q8—48no—
590Q8—44no—
591D4—48no—
592D4—48no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
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—82no—
641C2×C4—82no—
642C2×C4—84no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—162no—
649C2×C4—162no—
650C2×C4—162no—
651C2×C4—162no—
652C2×C4C2×#12314yes—
653C2×C4C2×#12312yes—
654C2×C4C2×#12314yes—
655C2×C4C22×#4112yes—
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4#58512yes—
658C8—44no—
659C8—44no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—88no—
688C22—88no—
689C22#309514yes—
690C22C22×#12311yescharacteristic
691C22#348411yescharacteristic
692C22#348511yescharacteristic
693C22C22×#13711yescharacteristic
694C22C2×#57511yescharacteristic
695C22#358511yescharacteristic
696C22#358611yescharacteristic
697C4—24no—
698C4—24no—
699C4—22no—
700C4—42no—
701C4—41no—
702C4—41no—
703C4—44no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#349414yes—
712C2—24no—
713C2—42no—
714C2—42no—
715C2—82no—
716C2#3554011yescharacteristic
717C2C2×#309511yescharacteristic
718C2#6052011yescharacteristic
719C2#6157411yescharacteristic
720C2#6157711yescharacteristic
721C2#6170911yescharacteristic
722C2#6171211yescharacteristic
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. #27604C2semidirect
4. #27603C2semidirect
5. #27601C2semidirect
6. #25029C2semidirect
7. #25028C2semidirect
8. #24984C2semidirect
9. #24983C2semidirect
10. #24010C2semidirect
11. #24006C2semidirect
12. #24003C2semidirect
13. #21987C2semidirect
15. #20741C2semidirect
20. C2×#484C22semidirect
21. C2×#485C22semidirect
22. C2×#483C22semidirect
49. #2077C22semidirect
50. #2077C22semidirect
51. #2076C22semidirect
52. #2075C22semidirect
53. #2036C22semidirect
54. #2036C22semidirect
55. #2035C22semidirect
56. #2035C22semidirect
59. #1948C22semidirect
60. #1947C22semidirect
61. #1873C22semidirect
62. #1872C22semidirect
63. #1872C22semidirect
64. #1854C22semidirect
67. #1853C22semidirect
70. #1852C22semidirect
71. #1847C22semidirect
72. #1836C22semidirect
73. #1825C22semidirect
76. #1824C22semidirect
77. #1823C22semidirect
78. #1805C22semidirect
79. #1805C22semidirect
85. #1763C22semidirect
86. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341852
#276042
#276032
#276012
#250292
#250282
#249842
#249832
#240102
#240062
#240032
#219872
#217902
#207412