Finite Group Databasebeta

← #6519416

Subgroups — #6519416

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2540C211yesmaximal, characteristic
2#34194C211yesmaximal, characteristic, complemented
3#26245C214yesmaximal, complemented
4#26242C214yesmaximal, complemented
5#26240C214yesmaximal, complemented
6#26239C214yesmaximal, complemented
7#24010C212yesmaximal, complemented
8#24009C212yesmaximal, complemented
9#24005C212yesmaximal, complemented
10#24003C212yesmaximal, complemented
11#21808C211yesmaximal, characteristic, complemented
12#21806C211yesmaximal, characteristic, complemented
13#21788C211yesmaximal, characteristic, complemented
14#21363C211yesmaximal, characteristic, complemented
15#21359C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2212yes—
19C22×C4×D4C2211yescharacteristic
20C2×#481C2214yescomplemented
21C2×#485C2214yescomplemented
22C2×#482C2214yescomplemented
23C2×#483C2214yescomplemented
24C2×#432C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2540—22no—
27#2540—21no—
28#2540—21no—
29#2540C2214yes—
30#2540C2212yes—
31#2540C2212yes—
32C2×#399—21no—
33C2×#392—21no—
34#2354C2212yescomplemented
35#2352C2212yescomplemented
36#2352C2212yescomplemented
37D4×C4⋊C4C2212yescomplemented
38#2345—21no—
39#2342—22no—
40D4×C22⋊C4—21no—
41#2319—21no—
42#2318—22no—
43C2×#390—22no—
44#2285C2211yescharacteristic
45#2283C2211yescharacteristic
46#2280C2212yes—
47C2×#380C2212yes—
48#2275—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2036C2214yescomplemented
51#2035C2214yescomplemented
52#2035C2214yescomplemented
53#2035C2214yescomplemented
54#2034C2214yescomplemented
55#2034C2214yescomplemented
56#2034C2214yescomplemented
57#2033C2214yescomplemented
58#1983C2214yescomplemented
59#1983C2212yescomplemented
60#1983C2212yescomplemented
61#1959C2214yescomplemented
62#1958C2214yescomplemented
63#1955C2212yescomplemented
64#1955C2212yescomplemented
65#1954C2214yescomplemented
66#1949C2214yescomplemented
67#1948C2214yescomplemented
68#1875C2214yescomplemented
69#1874C2214yescomplemented
70#1873C2214yescomplemented
71#1872C2214yescomplemented
72#1853C2212yes—
73#1853C2212yes—
74#1853C2212yes—
75#1852C2212yes—
76#1787C2211yescharacteristic
77#1787C2211yescharacteristic
78#1787C2212yes—
79#1786C2211yescharacteristic
80#1786C2211yescharacteristic
81#1786C2212yes—
82#1779C2212yes—
83#1779C2212yes—
84#1776C2214yescomplemented
85C4×#106C2212yescomplemented
86C4×#106C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4—22no—
89C23×D4D411yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#124—21no—
93C2×#124—21no—
94C2×#124—24no—
95C2×#124—24no—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—48no—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×#121—22no—
104C2×#121—22no—
105C2×#121—48no—
106C2×#121C2314yes—
107C2×#121C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4—24no—
110C2×C4×D4—24no—
111C2×C4×D4D412yes—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—21no—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4—22no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4D412yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133#485—48no—
134#483—48no—
135#482—48no—
136#481—48no—
137#432—24no—
138#431—24no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×#97C2312yes—
161C2×C4×C8C2312yes—
162#399—22no—
163#399—24no—
164#399—22no—
165#392—22no—
166#392—24no—
167#392—22no—
168#390—22no—
169#390—24no—
170#390—24no—
171#390—24no—
172#390—24no—
173#389—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#384C2312yes—
185#383—21no—
186#382—21no—
187#382C2312yes—
188#381—22no—
189#381—21no—
190#380—22no—
191#380—21no—
192#380—22no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2311yescharacteristic
198#380C2311yescharacteristic
199#380C2312yes—
200#379—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215#376—21no—
216C2×#95—22no—
217C2×#95—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222C25—21no—
223C25C2×D412yes—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—44no—
228C22×Q8C2412yes—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—44no—
236C22×D4—48no—
237C22×D4—44no—
238C22×D4—48no—
239C22×D4—84no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2×D412yes—
242C22×D4C2×D411yescharacteristic
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2414yes—
263#124—24no—
264#124—24no—
265#124—24no—
266#124—24no—
267#124—44no—
268#124—44no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C22⋊Q8—48no—
275#121—28no—
276#121—28no—
277#121—44no—
278#121—44no—
279#121—48no—
280#121—48no—
281C4×D4—24no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—48no—
291#117—44no—
292#117—44no—
293#117—82no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
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—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—48no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—84no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—82no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42#4114yes—
352C2×C42#4114yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—28no—
356#106—28no—
357#106—28no—
358#106—28no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#102—44no—
366#102—44no—
367#102—44no—
368#102—44no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—24no—
372C22⋊C8—44no—
373C22⋊C8—44no—
374C22⋊C8—44no—
375C22⋊C8—44no—
376#97—24no—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—22no—
392C24—24no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—22no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—44no—
401C24—44no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24C22×D412yes—
406C24C22×D412yes—
407C24C22×D411yescharacteristic
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—84no—
415C2×Q8—84no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—416no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—42no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4—832no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4#12314yes—
480C22×C4#12314yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2511yescharacteristic
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C4⋊C4—28no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4H(5,𝔽2)18yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—44no—
536C42—82no—
537C42—82no—
538C42—82no—
539C42—82no—
540C42—81no—
541C42—81no—
542C42#12314yes—
543C42#12314yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—88no—
576C23—84no—
577C23—84no—
578C23—88no—
579C23—84no—
580C23C2×#12312yes—
581C23#57514yes—
582C23#57712yes—
583C23#57712yes—
584C23C23×D411yescharacteristic
585Q8—42no—
586Q8—44no—
587Q8—42no—
588Q8—84no—
589D4—28no—
590D4—48no—
591D4—44no—
592D4—42no—
593D4—44no—
594D4—42no—
595D4—84no—
596D4—816no—
597D4—816no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—44no—
616C2×C4—416no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
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—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—164no—
642C2×C4—164no—
643C2×C4C2×#12312yes—
644C2×C4C2×#12312yes—
645C2×C4C2×#12312yes—
646C2×C4C2×#12312yes—
647C2×C4C2×#12312yes—
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)14yes—
651C8—42no—
652C8—42no—
653C8—44no—
654C22—22no—
655C22—24no—
656C22—24no—
657C22—22no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—816no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—168no—
679C22#309914yes—
680C22C22×#12311yescharacteristic
681C22#348412yes—
682C22C2×#57512yes—
683C22C2×#57711yescharacteristic
684C22C2×#57711yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—48no—
691C4—42no—
692C4—42no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#350812yes—
699C4#350812yes—
700C2—24no—
701C2—44no—
702C2—82no—
703C2—162no—
704C2#3555411yescharacteristic
705C2#6051111yescharacteristic
706C2C2×#309911yescharacteristic
707C2#6159412yes—
708C2#6160112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2540C2
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
2. #34194C2semidirect
3. #26245C2semidirect
4. #26242C2semidirect
5. #26240C2semidirect
6. #26239C2semidirect
7. #24010C2semidirect
8. #24009C2semidirect
9. #24005C2semidirect
10. #24003C2semidirect
11. #21808C2semidirect
12. #21806C2semidirect
13. #21788C2semidirect
14. #21363C2semidirect
15. #21359C2semidirect
20. C2×#481C22semidirect
21. C2×#485C22semidirect
22. C2×#482C22semidirect
23. C2×#483C22semidirect
34. #2354C22semidirect
35. #2352C22semidirect
36. #2352C22semidirect
37. D4×C4⋊C4C22semidirect
50. #2036C22semidirect
51. #2035C22semidirect
52. #2035C22semidirect
53. #2035C22semidirect
54. #2034C22semidirect
55. #2034C22semidirect
56. #2034C22semidirect
57. #2033C22semidirect
58. #1983C22semidirect
59. #1983C22semidirect
60. #1983C22semidirect
61. #1959C22semidirect
62. #1958C22semidirect
63. #1955C22semidirect
64. #1955C22semidirect
65. #1954C22semidirect
66. #1949C22semidirect
67. #1948C22semidirect
68. #1875C22semidirect
69. #1874C22semidirect
70. #1873C22semidirect
71. #1872C22semidirect
84. #1776C22semidirect
85. C4×#106C22semidirect
86. C4×#106C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25402
#341942
#262452
#262422
#262402
#262392
#240102
#240092
#240052
#240032
#218082
#218062
#217882
#213632
#213592