Finite Group Databasebeta

← #6500819

Subgroups — #6500819

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2415C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27616C212yesmaximal, complemented
4#27602C212yesmaximal, complemented
5#27589C212yesmaximal, complemented
6#26226C214yesmaximal, complemented
7#26222C214yesmaximal, complemented
8#26154C214yesmaximal, complemented
9#26150C214yesmaximal, complemented
10#24010C212yesmaximal, complemented
11#21765C211yesmaximal, characteristic, complemented
12#21765C211yesmaximal, characteristic, complemented
13#21742C211yesmaximal, characteristic
14#20846C211yesmaximal, characteristic, complemented
15#20846C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#485C2214yescomplemented
21C2×#428C2212yescomplemented
22C2×#428C2212yescomplemented
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25#2415—22no—
26#2415—22no—
27#2415C2214yes—
28#2415C2214yes—
29C2×#397—21no—
30C2×#394—21no—
31#2353—22no—
32#2353—22no—
33#2352C2212yes—
34#2352C2212yes—
35#2352C2212yes—
36#2352C2212yes—
37#2313—21no—
38#2312—21no—
39#2308—21no—
40C2×#391C2211yescharacteristic
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44#2283C2212yes—
45#2280C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49#2273—21no—
50#2101C2214yescomplemented
51#2097C2214yescomplemented
52#2085C2214yescomplemented
53#2077C2214yescomplemented
54#2076C2214yescomplemented
55#2074C2214yescomplemented
56#2036C2214yescomplemented
57#2035C2214yescomplemented
58#1928C2214yescomplemented
59#1927C2214yescomplemented
60#1921C2214yescomplemented
61#1920C2214yescomplemented
62#1894C2214yescomplemented
63#1894C2214yescomplemented
64#1873C2214yescomplemented
65#1872C2212yescomplemented
66#1872C2212yescomplemented
67#1872C2212yescomplemented
68#1872C2212yescomplemented
69#1870C2214yes—
70#1870C2214yes—
71#1854C2214yescomplemented
72#1854C2214yescomplemented
73#1854C2214yescomplemented
74#1853C2212yes—
75#1853C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1825C2214yescomplemented
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1787C2212yes—
83#1787C2212yes—
84#1779C2212yes—
85#1736C2212yescomplemented
86#1736C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C22×M16C2312yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2314yes—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117—21no—
110C2×#117—44no—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#485—48no—
137#428—44no—
138#428—44no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147#407—24no—
148#407—24no—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155#397—24no—
156#397—24no—
157#394—24no—
158#394—24no—
159#391—22no—
160#391C2314yes—
161#389—24no—
162#389—24no—
163#389—22no—
164#389C2314yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—24no—
170#388—24no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#383—22no—
180#382C2312yes—
181#382C2312yes—
182#381—22no—
183#381—22no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380C2312yes—
190#380C2314yes—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#379—22no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206#376—22no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25C2×D411yescharacteristic
224C25C2×D411yescharacteristic
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C2412yes—
231C22×D4—24no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4C2×D412yes—
235C22×D4C2×D412yes—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4#4114yes—
258C23×C4#4114yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—42no—
266C2×M16—42no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8C2414yes—
270C22×C8C2414yes—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—28no—
276C22⋊Q8—44no—
277C22⋊Q8—48no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286#117—24no—
287#117—24no—
288#117—42no—
289#117—42no—
290#117—44no—
291#117—44no—
292#117—44no—
293#117—44no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
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—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
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—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
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—22no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D412yes—
362C2×C42C2412yes—
363#107—28no—
364#107—44no—
365#106—28no—
366#106—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—44no—
376C22⋊C8—44no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—41no—
399C24—41no—
400C24—41no—
401C24—41no—
402C24—41no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24#12314yes—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—84no—
419C2×Q8—84no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—416no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
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—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4#12314yes—
492C22×C4#12314yes—
493C22×C4#12314yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507M16—48no—
508M16—48no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C4⋊C4H(5,𝔽2)14yes—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—44no—
561C42—82no—
562C42—82no—
563C42—82no—
564C42—82no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—28no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—82no—
595C23—82no—
596C23—82no—
597C23—82no—
598C23—82no—
599C23—82no—
600C23—82no—
601C23—82no—
602C23C2×#12312yes—
603C23C2×#12312yes—
604C23C2×#12312yes—
605C23#57512yes—
606C23#57512yes—
607C23C23×D411yescharacteristic
608Q8—44no—
609Q8—44no—
610Q8—84no—
611D4—48no—
612D4—48no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—84no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—162no—
672C2×C4—162no—
673C2×C4—162no—
674C2×C4—162no—
675C2×C4C2×#12312yes—
676C2×C4C2×#12312yes—
677C2×C4C2×#12312yes—
678C2×C4C22×#4111yescharacteristic
679C2×C4C22×#4111yescharacteristic
680C2×C4C2×H(5,𝔽2)12yes—
681C2×C4#58512yes—
682C8—44no—
683C8—44no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—88no—
712C22—88no—
713C22#309314yes—
714C22C22×#12311yescharacteristic
715C22#348411yescharacteristic
716C22#348511yescharacteristic
717C22C2×#57511yescharacteristic
718C22C2×#57511yescharacteristic
719C22#358611yescharacteristic
720C22#358611yescharacteristic
721C4—22no—
722C4—22no—
723C4—22no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—44no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4#350512yes—
736C4#350512yes—
737C2—24no—
738C2—42no—
739C2—42no—
740C2—82no—
741C2#3535111yescharacteristic
742C2C2×#309311yescharacteristic
743C2#6051911yescharacteristic
744C2#6157511yescharacteristic
745C2#6158311yescharacteristic
746C2#6171211yescharacteristic
747C2#6171611yescharacteristic
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. #27616C2semidirect
4. #27602C2semidirect
5. #27589C2semidirect
6. #26226C2semidirect
7. #26222C2semidirect
8. #26154C2semidirect
9. #26150C2semidirect
10. #24010C2semidirect
11. #21765C2semidirect
12. #21765C2semidirect
14. #20846C2semidirect
15. #20846C2semidirect
20. C2×#485C22semidirect
21. C2×#428C22semidirect
22. C2×#428C22semidirect
50. #2101C22semidirect
51. #2097C22semidirect
52. #2085C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2074C22semidirect
56. #2036C22semidirect
57. #2035C22semidirect
58. #1928C22semidirect
59. #1927C22semidirect
60. #1921C22semidirect
61. #1920C22semidirect
62. #1894C22semidirect
63. #1894C22semidirect
64. #1873C22semidirect
65. #1872C22semidirect
66. #1872C22semidirect
67. #1872C22semidirect
68. #1872C22semidirect
71. #1854C22semidirect
72. #1854C22semidirect
73. #1854C22semidirect
78. #1825C22semidirect
85. #1736C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24152
#341852
#276162
#276022
#275892
#262262
#262222
#261542
#261502
#240102
#217652
#217422
#208462