Finite Group Databasebeta

← #6489012

Subgroups — #6489012

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#32957C212yesmaximal, complemented
4#32903C212yesmaximal
5#30130C214yesmaximal, complemented
6#29132C212yesmaximal, complemented
7#28386C212yesmaximal, complemented
8#28284C212yesmaximal, complemented
9#28252C212yesmaximal, complemented
10#27677C212yesmaximal, complemented
11#24509C214yesmaximal, complemented
12#24209C212yesmaximal, complemented
13#22152C214yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529—21no—
17C22×C4×Q8C2212yes—
18C2×#477C2214yes—
19C2×#460C2214yes—
20C2×#461C2214yes—
21C2×#448C2214yes—
22C2×#432C2212yes—
23#2861—21no—
24#2860—21no—
25#2860C2212yes—
26#2859C2211yescharacteristic
27#2858C2211yescharacteristic
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2854—21no—
31#2854C2212yes—
32#2850—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#127C2211yescharacteristic
43C4×#124—21no—
44C4×#121—21no—
45C4×#121C2212yes—
46C2×#391C2212yes—
47#2293—21no—
48#2293C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4—21no—
54C2×C43C2211yescharacteristic
55#2099C2214yes—
56#2093C2214yes—
57#2068C2218yes—
58#2055C2218yes—
59#2043C2214yes—
60#2013C2214yes—
61#1981C2214yes—
62#1973C2218yes—
63#1947C2214yes—
64#1935C2218yes—
65#1880C2218yes—
66#1874C2214yes—
67#1863C2218yes—
68#1852C2214yes—
69#1842C2212yes—
70#1831C2212yes—
71#1803C2214yes—
72#1803C2214yes—
73#1802C2214yes—
74#1797C2212yes—
75#1797C2212yes—
76#1787C2214yes—
77#1767C2212yes—
78#1767C2212yes—
79C23×Q8C2312yes—
80C22×SD16C2318yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127C2311yescharacteristic
88#529—48no—
89C2×#126—22no—
90C2×#125—21no—
91C2×#124—21no—
92C2×#124—21no—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121C2312yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—48no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C42—22no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#477—48no—
124#461—216no—
125#460—48no—
126#448—48no—
127#432—44no—
128C2×#107C2318yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2318yes—
133C2×#103C2318yes—
134C2×#103C2318yes—
135C2×#102C2318yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2314yes—
138#391—22no—
139#391—22no—
140#391—21no—
141#391—21no—
142#391C2314yes—
143#391C2314yes—
144#390—21no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#386—22no—
179#386—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#382—21no—
187#380—21no—
188#380C2312yes—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C2×#95—22no—
202C2×#95—22no—
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43D411yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210C43C2311yescharacteristic
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C22×Q8—28no—
216C22×Q8—22no—
217C22×Q8—21no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—44no—
221C22×Q8C2414yes—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C2×SD16—216no—
237C2×SD16—48no—
238C2×SD16—48no—
239C22×C8C2418yes—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242C4⋊Q8—28no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8#4118yes—
246#127—24no—
247#126—48no—
248#125—44no—
249#124—44no—
250#124—44no—
251#123—48no—
252C22⋊Q8—44no—
253#121—24no—
254#121—24no—
255#121—44no—
256#121—44no—
257C4×Q8—28no—
258C4×Q8—28no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×Q8—44no—
264C4×D4—28no—
265C4×D4—44no—
266C4×D4—48no—
267C4×D4—44no—
268#117—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—42no—
321C2×C42—48no—
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×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—48no—
351C4⋊C8—28no—
352C4⋊C8—28no—
353C4⋊C8—44no—
354#103—216no—
355#103—48no—
356#103—48no—
357#102—48no—
358C22⋊C8—28no—
359#97—44no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24—22no—
371C24C22×D411yescharacteristic
372C2×Q8—28no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—416no—
379C2×Q8—416no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8C2×#4118yes—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4AGL(1,ℤ/8)14yes—
432C22×C4#13714yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+412yes—
448C22×C4C2511yescharacteristic
449SD16—816no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
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—21no—
511C42—21no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—41no—
517C42—48no—
518C42—48no—
519C42—44no—
520C42#13714yes—
521C42C2×#4114yes—
522C42H(5,𝔽2)14yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#53712yes—
535C23C2×AGL(1,ℤ/8)12yes—
536C23C2×#13712yes—
537C23C23×D411yescharacteristic
538Q8—28no—
539Q8—28no—
540Q8—416no—
541Q8—44no—
542Q8—44no—
543Q8—84no—
544D4—44no—
545D4—44no—
546D4—84no—
547D4—88no—
548D4—84no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—416no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—88no—
600C2×C4—88no—
601C2×C4#52512yes—
602C2×C4#52512yes—
603C2×C4#53512yes—
604C2×C4#53512yes—
605C2×C4#53712yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×#13712yes—
609C2×C4C2×#13712yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4#57512yes—
614C2×C4C22×#4112yes—
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×H(5,𝔽2)12yes—
617C2×C4C2×2−1+411yescharacteristic
618C8—48no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22#325414yes—
633C22C2×#52511yescharacteristic
634C22C2×#53511yescharacteristic
635C22C2×#53711yescharacteristic
636C22C22×AGL(1,ℤ/8)11yescharacteristic
637C22C22×#13711yescharacteristic
638C22C22×#13711yescharacteristic
639C22C2×#57511yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—41no—
648C4—48no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4—82no—
656C4#311214yes—
657C4#321814yes—
658C4#351912yes—
659C4#352012yes—
660C2—24no—
661C2—162no—
662C2#3523611yescharacteristic
663C2C2×#311211yescharacteristic
664C2#6056011yescharacteristic
665C2C2×#321811yescharacteristic
666C2#6080311yescharacteristic
667C2C2×#325411yescharacteristic
668C2#6091811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. C2×C4×C4⋊Q8C2semidirect
3. #32957C2semidirect
5. #30130C2semidirect
6. #29132C2semidirect
7. #28386C2semidirect
8. #28284C2semidirect
9. #28252C2semidirect
10. #27677C2semidirect
11. #24509C2semidirect
12. #24209C2semidirect
13. #22152C2semidirect
14. #20810C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#329572
#329032
#301302
#291322
#283862
#282842
#282522
#276772
#245092
#242092
#221522
#208102