Finite Group Databasebeta

← #6519047

Subgroups — #6519047

Order 512.

1904 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#34194C211yesmaximal, characteristic
3#27602C212yesmaximal, complemented
4#27601C212yesmaximal, complemented
5#25029C214yesmaximal, complemented
6#25015C212yesmaximal, complemented
7#25014C212yesmaximal, complemented
8#24983C214yesmaximal, complemented
9#24972C212yesmaximal, complemented
10#24971C212yesmaximal, complemented
11#24010C212yesmaximal, complemented
12#24003C212yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21981C211yesmaximal, characteristic, complemented
15#21788C211yesmaximal, characteristic
16#20740C211yesmaximal, characteristic, complemented
17#20735C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#485C2214yescomplemented
22C2×#483C2214yescomplemented
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2538—22no—
26#2538—21no—
27#2538—21no—
28#2538C2212yes—
29#2538C2212yes—
30#2538C2214yes—
31C2×#396—22no—
32#2354C2211yescharacteristic
33#2354C2211yescharacteristic
34#2352C2212yes—
35#2352C2212yes—
36D4×C4⋊C4C2211yescharacteristic
37D4×C4⋊C4C2211yescharacteristic
38#2345—21no—
39#2342—22no—
40D4×C22⋊C4—21no—
41#2319—21no—
42#2318—22no—
43C2×#390—21no—
44C2×#389—21no—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47#2283C2211yescharacteristic
48#2280C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51#2275—21no—
52#2077C2214yescomplemented
53#2077C2214yescomplemented
54#2077C2214yescomplemented
55#2076C2214yescomplemented
56#2036C2214yescomplemented
57#2035C2214yescomplemented
58#2035C2214yescomplemented
59#2035C2214yescomplemented
60#1955C2214yes—
61#1954C2212yes—
62#1954C2212yes—
63#1948C2212yescomplemented
64#1948C2214yescomplemented
65#1947C2212yescomplemented
66#1873C2214yescomplemented
67#1872C2214yescomplemented
68#1854C2214yescomplemented
69#1853C2212yes—
70#1853C2212yes—
71#1853C2212yes—
72#1853C2214yescomplemented
73#1852C2212yes—
74#1847C2212yescomplemented
75#1844C2212yescomplemented
76#1836C2212yescomplemented
77#1834C2212yescomplemented
78#1825C2214yescomplemented
79#1824C2214yescomplemented
80#1823C2212yes—
81#1823C2211yescharacteristic
82#1823C2211yescharacteristic
83#1805C2212yescomplemented
84#1805C2212yescomplemented
85#1803C2212yescomplemented
86#1803C2212yescomplemented
87#1787C2212yes—
88#1787C2211yescharacteristic
89#1787C2211yescharacteristic
90#1779C2212yes—
91#1779C2212yes—
92#1763C2214yescomplemented
93C4×#103C2212yescomplemented
94C4×#103C2212yescomplemented
95C23×Q8C2311yescharacteristic
96C23×D4D411yescharacteristic
97C24×C4C2312yes—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—48no—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4D412yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#485—48no—
138#483—48no—
139#432—24no—
140#431—24no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
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#396—24no—
163#396—24no—
164#396—24no—
165#396—24no—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—24no—
170#390—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—24no—
175#389—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#383—21no—
193#382—21no—
194#382C2311yescharacteristic
195#382C2311yescharacteristic
196#381—22no—
197#381—21no—
198#380—22no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380C2312yes—
204#380C2314yes—
205#380C2314yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2311yescharacteristic
210#380C2311yescharacteristic
211#380C2312yes—
212#379—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224#376—21no—
225C2×#95—22no—
226C2×#95—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237C25C2×D412yes—
238C22×Q8—24no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—42no—
243C22×Q8C2412yes—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—42no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—44no—
292C4×D4—44no—
293#117—44no—
294#117—44no—
295#117—82no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—28no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—84no—
349C2×C22⋊C4—84no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—44no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4114yes—
365C2×C42#4114yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—44no—
369#106—44no—
370#106—44no—
371#106—44no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374#103—24no—
375#103—28no—
376#103—24no—
377#103—42no—
378#103—44no—
379#103—42no—
380C22⋊C8—22no—
381C22⋊C8—22no—
382C22⋊C8—24no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385C22⋊C8—44no—
386C22⋊C8—44no—
387#97—24no—
388C4×C8—24no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—21no—
404#95—21no—
405C24—24no—
406C24—24no—
407C24—24no—
408C24—24no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24C22×D412yes—
413C24C22×D412yes—
414C24C22×D411yescharacteristic
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8H(5,𝔽2)14yes—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—416no—
434C2×D4—42no—
435C2×D4#13714yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
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—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
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—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4#12314yes—
486C22×C4#12314yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2511yescharacteristic
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C42—44no—
553C42—44no—
554C42—42no—
555C42—42no—
556C42—82no—
557C42—82no—
558C42—82no—
559C42#12314yes—
560C42#12314yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C42C2×#4112yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23—48no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23C2×#12312yes—
590C23C2×#13714yes—
591C23#57712yes—
592C23#57712yes—
593C23C23×D411yescharacteristic
594Q8—24no—
595Q8—24no—
596Q8—48no—
597Q8—42no—
598Q8—42no—
599D4—28no—
600D4—48no—
601D4—44no—
602C2×C4—28no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—48no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—88no—
656C2×C4—84no—
657C2×C4—164no—
658C2×C4—164no—
659C2×C4C2×#12312yes—
660C2×C4C2×#12312yes—
661C2×C4C2×#12312yes—
662C2×C4C2×#12312yes—
663C2×C4C2×#12312yes—
664C2×C4C2×#13712yes—
665C2×C4#57512yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)12yes—
669C2×C4#58512yes—
670C8—42no—
671C8—42no—
672C8—44no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—816no—
694C22#309914yes—
695C22C22×#12311yescharacteristic
696C22#348411yescharacteristic
697C22#348511yescharacteristic
698C22C22×#13711yescharacteristic
699C22C2×#57511yescharacteristic
700C22C2×#57711yescharacteristic
701C22C2×#57711yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—81no—
710C4—82no—
711C4—81no—
712C4—82no—
713C4—82no—
714C4—81no—
715C4—81no—
716C4—84no—
717C4#309514yes—
718C4#349412yes—
719C4#349412yes—
720C2—24no—
721C2—44no—
722C2—82no—
723C2#3554011yescharacteristic
724C2C2×#309511yescharacteristic
725C2C2×#309911yescharacteristic
726C2#6159311yescharacteristic
727C2#6159611yescharacteristic
728C2#6173511yescharacteristic
729C2#6173911yescharacteristic
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. #27602C2semidirect
4. #27601C2semidirect
5. #25029C2semidirect
6. #25015C2semidirect
7. #25014C2semidirect
8. #24983C2semidirect
9. #24972C2semidirect
10. #24971C2semidirect
11. #24010C2semidirect
12. #24003C2semidirect
13. #21987C2semidirect
14. #21981C2semidirect
16. #20740C2semidirect
17. #20735C2semidirect
21. C2×#485C22semidirect
22. C2×#483C22semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2077C22semidirect
55. #2076C22semidirect
56. #2036C22semidirect
57. #2035C22semidirect
58. #2035C22semidirect
59. #2035C22semidirect
63. #1948C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1873C22semidirect
67. #1872C22semidirect
68. #1854C22semidirect
72. #1853C22semidirect
74. #1847C22semidirect
75. #1844C22semidirect
76. #1836C22semidirect
77. #1834C22semidirect
78. #1825C22semidirect
79. #1824C22semidirect
83. #1805C22semidirect
84. #1805C22semidirect
85. #1803C22semidirect
86. #1803C22semidirect
92. #1763C22semidirect
93. C4×#103C22semidirect
94. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341942
#276022
#276012
#250292
#250152
#250142
#249832
#249722
#249712
#240102
#240032
#219872
#219812
#217882
#207402
#207352