Finite Group Databasebeta

← #6500825

Subgroups — #6500825

Order 512.

2184 conjugacy classes of subgroups.

The subgroup lattice has 771 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2416C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic, complemented
3#27616C212yesmaximal, complemented
4#27601C211yesmaximal, characteristic, complemented
5#27600C211yesmaximal, characteristic, complemented
6#27588C212yesmaximal, complemented
7#26219C214yesmaximal, complemented
8#26215C214yesmaximal, complemented
9#26147C214yesmaximal, complemented
10#26143C214yesmaximal, complemented
11#24005C211yesmaximal, characteristic, complemented
12#24002C211yesmaximal, characteristic, complemented
13#21767C212yesmaximal, complemented
14#21742C211yesmaximal, characteristic, complemented
15#20844C212yesmaximal, complemented
16C22×#121—21no—
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#481C2212yescomplemented
22C2×#480C2212yescomplemented
23C2×#428C2214yescomplemented
24C2×#407C2212yes—
25#2416—22no—
26#2416—22no—
27#2416C2214yes—
28#2416C2214yes—
29C2×#396—21no—
30C2×#394—21no—
31#2353—22no—
32#2353—22no—
33#2352C2212yescomplemented
34#2352C2212yescomplemented
35#2352C2212yescomplemented
36#2352C2212yescomplemented
37#2313—21no—
38#2312—21no—
39#2308—21no—
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#386—21no—
43#2283C2212yes—
44#2280C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49#2101C2214yescomplemented
50#2092C2214yescomplemented
51#2085C2214yescomplemented
52#2077C2214yescomplemented
53#2075C2214yescomplemented
54#2073C2214yescomplemented
55#2034C2214yescomplemented
56#2033C2214yescomplemented
57#1926C2214yescomplemented
58#1925C2214yescomplemented
59#1921C2214yescomplemented
60#1920C2214yescomplemented
61#1895C2214yescomplemented
62#1895C2214yescomplemented
63#1875C2214yescomplemented
64#1875C2214yescomplemented
65#1874C2212yescomplemented
66#1874C2212yescomplemented
67#1870C2214yescomplemented
68#1870C2214yescomplemented
69#1853C2212yes—
70#1853C2212yes—
71#1853C2212yescomplemented
72#1853C2214yescomplemented
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yescomplemented
76#1852C2214yescomplemented
77#1824C2212yescomplemented
78#1823C2212yes—
79#1823C2212yes—
80#1823C2212yescomplemented
81#1817C2212yes—
82#1786C2212yes—
83#1786C2212yes—
84#1779C2211yescharacteristic
85#1779C2211yescharacteristic
86#1736C2214yescomplemented
87C23×D4—21no—
88C23×D4—21no—
89C23×D4D411yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C22×M16C2314yes—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121—22no—
100C2×#121—28no—
101C2×#121—44no—
102C2×#121—48no—
103C2×#121C2314yes—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×C4×D4—28no—
107C2×C4×D4D412yes—
108C2×C4×D4D412yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—44no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
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⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42C2311yescharacteristic
136#481—44no—
137#480—44no—
138#428—48no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147#407—28no—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155#396—24no—
156#396—24no—
157#394—24no—
158#394—24no—
159#390—22no—
160#390C2314yes—
161#389—22no—
162#389C2314yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#386—24no—
176#386—24no—
177#383—22no—
178#382C2312yes—
179#382C2312yes—
180#381—22no—
181#381—22no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
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—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C25C2×D411yescharacteristic
223C25C2×D411yescharacteristic
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4—82no—
240C22×D4C2×D412yes—
241C22×D4C2×D412yes—
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—42no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—82no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4#4118yes—
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2412yes—
271C2×M16—28no—
272C2×M16—44no—
273C2×M16—44no—
274C22×C8C2414yes—
275C22×C8C2414yes—
276#121—28no—
277#121—28no—
278#121—24no—
279#121—24no—
280#121—42no—
281#121—42no—
282#121—44no—
283#121—44no—
284#121—416no—
285#121—416no—
286#121—48no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293#117—44no—
294#117—44no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—21no—
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—44no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—48no—
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—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
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—24no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361#107—28no—
362#107—44no—
363#106—28no—
364#106—44no—
365#102—44no—
366#102—44no—
367#102—44no—
368#102—44no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—28no—
372C22⋊C8—42no—
373C22⋊C8—42no—
374C22⋊C8—42no—
375C22⋊C8—42no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—41no—
397C24—41no—
398C24—41no—
399C24—41no—
400C24—41no—
401C24—41no—
402C24—41no—
403C24—41no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24#12314yes—
412C24AGL(1,ℤ/8)14yes—
413C24C22×D412yes—
414C24C22×D411yescharacteristic
415C24C22×D411yescharacteristic
416C24C22×D411yescharacteristic
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—416no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—88no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—816no—
442C2×D4—816no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—416no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—48no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—48no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—88no—
496C22×C4—84no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4—81no—
505C22×C4—816no—
506C22×C4#12314yes—
507C22×C4#12318yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C2×#4114yes—
515C22×C4C2×#4114yes—
516C22×C4C2×#4114yes—
517C22×C4C2511yescharacteristic
518M16—416no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—48no—
536C4⋊C4—42no—
537C4⋊C4—48no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4H(5,𝔽2)14yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—48no—
558C22⋊C4—84no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—24no—
564C42—22no—
565C42—22no—
566C42—82no—
567C42—82no—
568C42—82no—
569C42—82no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—22no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—21no—
579C23—21no—
580C23—28no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—21no—
585C23—21no—
586C23—22no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—21no—
591C23—21no—
592C23—22no—
593C23—28no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—82no—
610C23—82no—
611C23—82no—
612C23—82no—
613C23—84no—
614C23—84no—
615C23—84no—
616C23—84no—
617C23—84no—
618C23—84no—
619C23—84no—
620C23C2×#12314yes—
621C23C2×#12312yes—
622C23C2×AGL(1,ℤ/8)12yes—
623C23C2×AGL(1,ℤ/8)12yes—
624C23C2×AGL(1,ℤ/8)12yes—
625C23#57512yes—
626C23C23×D411yescharacteristic
627D4—48no—
628D4—48no—
629D4—44no—
630D4—44no—
631D4—84no—
632D4—88no—
633D4—88no—
634D4—816no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—24no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—48no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—82no—
682C2×C4—82no—
683C2×C4—82no—
684C2×C4—82no—
685C2×C4—816no—
686C2×C4—84no—
687C2×C4—84no—
688C2×C4—162no—
689C2×C4—162no—
690C2×C4—162no—
691C2×C4—162no—
692C2×C4C2×#12312yes—
693C2×C4C2×#12314yes—
694C2×C4C22×#4112yes—
695C2×C4C2×H(5,𝔽2)12yes—
696C2×C4#58512yes—
697C8—44no—
698C8—44no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—24no—
708C22—24no—
709C22—22no—
710C22—22no—
711C22—24no—
712C22—22no—
713C22—22no—
714C22—416no—
715C22—42no—
716C22—41no—
717C22—42no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—42no—
724C22—41no—
725C22—42no—
726C22—41no—
727C22—88no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—168no—
736C22#309414yes—
737C22C22×#12311yescharacteristic
738C22#348411yescharacteristic
739C22#348511yescharacteristic
740C22C22×AGL(1,ℤ/8)11yescharacteristic
741C22C2×#57511yescharacteristic
742C22#358511yescharacteristic
743C22#358611yescharacteristic
744C4—22no—
745C4—24no—
746C4—41no—
747C4—41no—
748C4—42no—
749C4—42no—
750C4—44no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4—82no—
757C4#350514yes—
758C2—24no—
759C2—44no—
760C2—42no—
761C2—42no—
762C2—162no—
763C2#3535111yescharacteristic
764C2C2×#309411yescharacteristic
765C2#6052011yescharacteristic
766C2#6157111yescharacteristic
767C2#6158311yescharacteristic
768C2#6170611yescharacteristic
769C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2416C2
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. #34185C2semidirect
3. #27616C2semidirect
4. #27601C2semidirect
5. #27600C2semidirect
6. #27588C2semidirect
7. #26219C2semidirect
8. #26215C2semidirect
9. #26147C2semidirect
10. #26143C2semidirect
11. #24005C2semidirect
12. #24002C2semidirect
13. #21767C2semidirect
14. #21742C2semidirect
15. #20844C2semidirect
21. C2×#481C22semidirect
22. C2×#480C22semidirect
23. C2×#428C22semidirect
33. #2352C22semidirect
34. #2352C22semidirect
35. #2352C22semidirect
36. #2352C22semidirect
49. #2101C22semidirect
50. #2092C22semidirect
51. #2085C22semidirect
52. #2077C22semidirect
53. #2075C22semidirect
54. #2073C22semidirect
55. #2034C22semidirect
56. #2033C22semidirect
57. #1926C22semidirect
58. #1925C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1895C22semidirect
62. #1895C22semidirect
63. #1875C22semidirect
64. #1875C22semidirect
65. #1874C22semidirect
66. #1874C22semidirect
67. #1870C22semidirect
68. #1870C22semidirect
71. #1853C22semidirect
72. #1853C22semidirect
75. #1852C22semidirect
76. #1852C22semidirect
77. #1824C22semidirect
80. #1823C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24162
#341852
#276162
#276012
#276002
#275882
#262192
#262152
#261472
#261432
#240052
#240022
#217672
#217422
#208442