Finite Group Databasebeta

← #6489121

Subgroups — #6489121

Order 512.

2036 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#32909C214yesmaximal, complemented
4#30042C212yesmaximal, complemented
5#30041C212yesmaximal, complemented
6#29147C212yesmaximal, complemented
7#28447C212yesmaximal, complemented
8#28164C214yesmaximal, complemented
9#27978C212yesmaximal, complemented
10#25726C212yesmaximal, complemented
11#23934C212yesmaximal, complemented
12#23931C212yesmaximal
13#22273C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2212yes—
19C2×#497C2214yes—
20C2×#464C2218yes—
21C2×#451C2214yes—
22C2×#448C2214yes—
23#2862C2211yescharacteristic
24#2862C2211yescharacteristic
25#2861—21no—
26#2861—21no—
27#2858C2211yescharacteristic
28#2858C2211yescharacteristic
29#2850—21no—
30#2850—21no—
31#2850C2212yes—
32#2850C2212yes—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×#124—21no—
42C4×#124—21no—
43C4×#124C2212yes—
44C4×#124C2212yes—
45#2297—21no—
46#2297—21no—
47C2×#391C2212yes—
48C2×#389—21no—
49C2×#389—21no—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4—21no—
53C2×C43C2211yescharacteristic
54#2072C2214yes—
55#2071C2214yes—
56#2024C2214yes—
57#2024C2214yes—
58#2016C2214yes—
59#2016C2214yes—
60#2014C2214yes—
61#2013C2214yes—
62#1960C2214yes—
63#1959C2214yes—
64#1880C2218yes—
65#1878C2218yes—
66#1878C2218yes—
67#1877C2218yes—
68#1874C2214yes—
69#1872C2214yes—
70#1839C2212yes—
71#1836C2214yes—
72#1831C2212yes—
73#1817C2212yes—
74#1805C2218yes—
75#1797C2212yes—
76#1767C2212yes—
77#1749C2212yes—
78#1745C2212yes—
79C22×#41—21no—
80C23×Q8C2312yes—
81C22×Q16C2318yes—
82C22×SD16C2318yes—
83#533—44no—
84#532—44no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#125—21no—
95C2×#124—21no—
96C2×#124—21no—
97C2×#124—21no—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—48no—
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×#117—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C42—22no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#497—216no—
129#464—416no—
130#451—48no—
131#448—48no—
132C2×C4⋊C8C2314yes—
133C2×#103C2318yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2314yes—
139#391—22no—
140#391—22no—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391C2314yes—
148#391C2314yes—
149#391C2312yes—
150#391C2312yes—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#384—21no—
180#384—21no—
181#384C2312yes—
182#384C2312yes—
183#382—21no—
184#382—21no—
185#382C2312yes—
186#382C2312yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
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×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C2×#95—22no—
200C2×#95—22no—
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215C2×#41—44no—
216C2×#41—44no—
217C2×#41—44no—
218C22×Q8—28no—
219C22×Q8—22no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—28no—
223C22×Q8—44no—
224C22×Q8C2412yes—
225C22×Q8C2414yes—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—216no—
241C2×Q16—48no—
242C2×Q16—48no—
243C2×SD16—216no—
244C2×SD16—48no—
245C2×SD16—48no—
246C22×C8C2418yes—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—24no—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253C4⋊Q8—42no—
254C4⋊Q8#4118yescomplemented
255#127—42no—
256#125—44no—
257#124—28no—
258#124—28no—
259#124—44no—
260#124—44no—
261#124—44no—
262C22⋊Q8—28no—
263C22⋊Q8—44no—
264#121—28no—
265#121—44no—
266C4×Q8—216no—
267C4×Q8—48no—
268C4×Q8—44no—
269C4×Q8—48no—
270C4×Q8—48no—
271C4×Q8—44no—
272C4×D4—44no—
273C4×D4—44no—
274#117—44no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—48no—
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×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C4⋊C8—28no—
352#103—48no—
353#103—48no—
354#103—48no—
355#102—48no—
356C22⋊C8—28no—
357#97—28no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
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—22no—
371C24C22×D411yescharacteristic
372#41—816no—
373C2×Q8—28no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—432no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8C2×#4118yes—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4#13714yes—
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×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4H(5,𝔽2)14yes—
450C22×C4C2511yescharacteristic
451Q16—816no—
452SD16—816no—
453C2×C8—28no—
454C2×C8—28no—
455C2×C8—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊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—
487C42—22no—
488C42—22no—
489C42—22no—
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—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—48no—
513C42—44no—
514C42—48no—
515C42#13714yes—
516C42C2×#4114yes—
517C42H(5,𝔽2)14yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23#53512yes—
530C23C2×#13712yes—
531C23#57512yes—
532C23C23×D411yescharacteristic
533Q8—216no—
534Q8—416no—
535Q8—48no—
536Q8—48no—
537Q8—84no—
538Q8—84no—
539D4—48no—
540D4—84no—
541D4—84no—
542D4—88no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
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—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4#52512yes—
597C2×C4#52512yes—
598C2×C4#53512yes—
599C2×C4#53512yes—
600C2×C4#53512yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4#57512yes—
607C2×C4C22×#4112yes—
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C8—48no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22#321514yes—
625C22C2×#52511yescharacteristic
626C22C2×#53511yescharacteristic
627C22C2×#53511yescharacteristic
628C22C22×#13711yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57511yescharacteristic
631C22C2×#57511yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—48no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4#351914yes—
651C2—24no—
652C2—162no—
653C2#3523711yescharacteristic
654C2#6056011yescharacteristic
655C2#6056011yescharacteristic
656C2C2×#321511yescharacteristic
657C2#6079011yescharacteristic
658C2#6080011yescharacteristic
659C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #32909C2semidirect
4. #30042C2semidirect
5. #30041C2semidirect
6. #29147C2semidirect
7. #28447C2semidirect
8. #28164C2semidirect
9. #27978C2semidirect
10. #25726C2semidirect
11. #23934C2semidirect
13. #22273C2semidirect
14. #20778C2semidirect
254. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#366392
C2×C4×C4⋊Q82
#329092
#300422
#300412
#291472
#284472
#281642
#279782
#257262
#239342
#239312
#222732
#207782