Finite Group Databasebeta

← #6479614

Subgroups — #6479614

Order 512.

2354 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#32953C212yesmaximal, complemented
4#32905C212yesmaximal, complemented
5#29909C212yesmaximal, complemented
6#29625C212yesmaximal
7#29498C214yesmaximal, complemented
8#29367C214yesmaximal, complemented
9#29130C212yesmaximal, complemented
10#28382C212yesmaximal, complemented
11#28201C212yesmaximal, complemented
12#28157C212yesmaximal, complemented
13#22049C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C22×C4×D4—21no—
17C22×C4×D4—21no—
18C22×C4×D4C2212yes—
19C2×#474C2214yescomplemented
20C2×#461C2214yescomplemented
21C2×#450C2214yes—
22#2864—21no—
23#2863—21no—
24#2863C2212yescomplemented
25#2862C2211yescharacteristic, complemented
26#2859C2211yescharacteristic, complemented
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2855—21no—
30#2855C2212yes—
31#2851—21no—
32C4×C4⋊Q8C2211yescharacteristic, complemented
33C4×C4⋊Q8C2211yescharacteristic, complemented
34C4×#127—21no—
35C4×#127—21no—
36C4×#127—21no—
37C4×#127—21no—
38C4×#127C2212yes—
39C4×#127C2212yes—
40C4×#127C2212yes—
41C4×#127C2212yes—
42C4×#125—21no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8C2212yes—
45#2295—21no—
46#2295C2212yescomplemented
47C2×#390—21no—
48C2×#390—21no—
49C2×#390C2212yes—
50C2×#384C2211yescharacteristic
51#2283—21no—
52C2×C43C2211yescharacteristic
53#2100C2214yescomplemented
54#2095C2214yescomplemented
55#2083C2214yescomplemented
56#2072C2214yescomplemented
57#2041C2214yes—
58#2041C2214yes—
59#2014C2214yes—
60#2014C2214yes—
61#1973C2218yescomplemented
62#1936C2218yescomplemented
63#1925C2218yescomplemented
64#1880C2218yescomplemented
65#1877C2218yescomplemented
66#1872C2214yes—
67#1862C2218yescomplemented
68#1852C2214yes—
69#1839C2212yescomplemented
70#1831C2212yescomplemented
71#1817C2212yes—
72#1802C2214yescomplemented
73#1802C2214yescomplemented
74#1797C2212yes—
75#1786C2214yes—
76#1767C2212yescomplemented
77#1749C2212yes—
78#1745C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C24×C4—22no—
83C22×SD16C2318yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127—28no—
87C2×#127—21no—
88C2×#127—21no—
89C2×#127C2312yes—
90C2×#127C2312yes—
91C2×#125—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—48no—
107C2×C4×D4—48no—
108C2×C4×D4—48no—
109C2×C4×D4C2314yes—
110C2×C4×D4C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#474—48no—
125#461—48no—
126#450—216no—
127C2×#107C2318yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2318yes—
130C2×#102C2318yes—
131C2×#102C2318yes—
132C2×#102C2318yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135#391—21no—
136#391—21no—
137#391—21no—
138#391—21no—
139#391C2312yes—
140#391C2312yes—
141#391C2312yes—
142#391C2312yes—
143#390—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#383—22no—
184#382—21no—
185#382—21no—
186#382—21no—
187#381—22no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380C2312yes—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199#376—22no—
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25—22no—
215C22×Q8—21no—
216C22×Q8C2412yes—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—24no—
225C22×D4—44no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4#4118yes—
232C22×D4C2414yes—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—48no—
237C23×C4—48no—
238C23×C4—48no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×SD16—216no—
247C2×SD16—48no—
248C2×SD16—48no—
249C22×C8C2418yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252#127—28no—
253#127—28no—
254#127—28no—
255#127—28no—
256#125—44no—
257C22⋊Q8—28no—
258C22⋊Q8—44no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—44no—
264C4×D4—44no—
265C4×D4—48no—
266C4×D4—48no—
267C4×D4—48no—
268C4×D4—48no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4—48no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—48no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—48no—
349C4⋊C8—28no—
350#103—48no—
351#102—216no—
352#102—48no—
353#102—48no—
354C22⋊C8—28no—
355#97—44no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24—28no—
371C24—24no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—44no—
376C24—44no—
377C24—44no—
378C24C22×D411yescharacteristic
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—416no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—432no—
401C2×D4—432no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—416no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4C2×#4118yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
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—24no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—832no—
443C22×C4AGL(1,ℤ/8)14yes—
444C22×C4#13714yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4114yes—
459C22×C4C2511yescharacteristic
460SD16—816no—
461C2×C8—28no—
462C2×C8—28no—
463C2×C8—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—44no—
522C42—44no—
523C42AGL(1,ℤ/8)14yes—
524C42H(5,𝔽2)14yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—28no—
530C23—416no—
531C23—416no—
532C23—416no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—88no—
540C23—88no—
541C23—88no—
542C23—88no—
543C23#52512yes—
544C23C2×AGL(1,ℤ/8)12yes—
545C23C2×#13712yes—
546C23C23×D411yescharacteristic
547Q8—48no—
548Q8—84no—
549D4—28no—
550D4—28no—
551D4—416no—
552D4—416no—
553D4—416no—
554D4—416no—
555D4—416no—
556D4—44no—
557D4—44no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—816no—
609C2×C4—88no—
610C2×C4—816no—
611C2×C4#52512yes—
612C2×C4#53512yes—
613C2×C4#53512yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57512yes—
622C2×C4C22×#4112yes—
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4#58512yes—
625C8—48no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—832no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22#311014yes—
641C22C2×#52511yescharacteristic
642C22C2×#53511yescharacteristic
643C22#344911yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57511yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—82no—
661C4—82no—
662C4—84no—
663C4—82no—
664C4—82no—
665C4#321714yes—
666C4#350514yes—
667C2—24no—
668C2—88no—
669C2#3510211yescharacteristic
670C2C2×#311011yescharacteristic
671C2#6055811yescharacteristic
672C2C2×#321711yescharacteristic
673C2#6080411yescharacteristic
674C2#6095111yescharacteristic
675C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36635C2semidirect
3. #32953C2semidirect
4. #32905C2semidirect
5. #29909C2semidirect
7. #29498C2semidirect
8. #29367C2semidirect
9. #29130C2semidirect
10. #28382C2semidirect
11. #28201C2semidirect
12. #28157C2semidirect
13. #22049C2semidirect
14. #20778C2semidirect
19. C2×#474C22semidirect
20. C2×#461C22semidirect
24. #2863C22semidirect
25. #2862C22semidirect
26. #2859C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×C4⋊Q8C22semidirect
46. #2295C22semidirect
53. #2100C22semidirect
54. #2095C22semidirect
55. #2083C22semidirect
56. #2072C22semidirect
61. #1973C22semidirect
62. #1936C22semidirect
63. #1925C22semidirect
64. #1880C22semidirect
65. #1877C22semidirect
67. #1862C22semidirect
69. #1839C22semidirect
70. #1831C22semidirect
72. #1802C22semidirect
73. #1802C22semidirect
76. #1767C22semidirect
78. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×#1272
#329532
#329052
#299092
#296252
#294982
#293672
#291302
#283822
#282012
#281572
#220492
#207782