Finite Group Databasebeta

← #6376544

Subgroups — #6376544

Order 512.

1892 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3171C211yesmaximal, characteristic
2#34227C211yesmaximal, characteristic, complemented
3#34226C211yesmaximal, characteristic, complemented
4#22272C212yesmaximal, complemented
5#22250C212yesmaximal, complemented
6#22226C212yesmaximal, complemented
7#22224C212yesmaximal, complemented
8#22200C214yesmaximal, complemented
9#22199C214yesmaximal, complemented
10#21408C211yesmaximal, characteristic, complemented
11#21406C211yesmaximal, characteristic, complemented
12#21136C212yesmaximal, complemented
13#21134C212yesmaximal, complemented
14#21133C212yesmaximal, complemented
15#21129C212yesmaximal, complemented
16#20819C211yesmaximal, characteristic, complemented
17#20367C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3171C4116yescomplemented
21C2×#501C2211yescharacteristic
22C2×#496C2211yescharacteristic
23C2×#490C2211yescharacteristic
24C2×#489C2211yescharacteristic
25C2×#469C2212yes—
26C2×#468C2212yes—
27C2×C4×SD16C2212yes—
28C2×C4×D8C2212yes—
29C2×C4×M16C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2374—22no—
32#2373—22no—
33#2367—22no—
34#2363—22no—
35#2357—21no—
36#2357—21no—
37#2353—21no—
38#2349—21no—
39#2347—21no—
40#2343—21no—
41#2341—21no—
42#2339—21no—
43#2324C2214yes—
44#2324C2212yes—
45#2323C2212yes—
46#2320C2211yescharacteristic
47#2319C2211yescharacteristic
48#2319C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×#127C2212yes—
51C4×#125C2212yes—
52C4×#124C2212yes—
53#2285C2211yescharacteristic
54#2283C2211yescharacteristic
55#2276C2211yescharacteristic
56#2275C2212yes—
57#2275C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#1993C2212yes—
60#1992C2212yes—
61#1961C2212yes—
62#1959C2212yes—
63#1927C2214yes—
64#1926C2214yes—
65#1925C2214yes—
66#1925C2214yes—
67#1921C2214yes—
68#1919C2214yes—
69#1919C2214yes—
70#1918C2214yes—
71#1893C2212yes—
72#1893C2214yes—
73#1892C2212yes—
74#1880C2214yes—
75#1879C2214yes—
76#1879C2214yes—
77#1877C2214yes—
78#1864C2214yes—
79#1864C2214yes—
80#1863C2214yes—
81#1862C2214yes—
82#1847C2212yes—
83#1836C2212yes—
84#1809C2214yes—
85C4×#107C2212yes—
86C4×#106C2212yes—
87#1763C2212yes—
88#1762C2212yes—
89#1762C2212yes—
90#1762C2212yes—
91#1746C2212yes—
92#1746C2212yes—
93#1745C2212yes—
94#1745C2212yes—
95#1729C2214yes—
96C22×#41—21no—
97C22×SD16C2312yes—
98C22×D8C2312yes—
99C22×M16C2312yes—
100C23×C8C2312yes—
101#532—24no—
102#532C2×C418yescomplemented
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105#529—24no—
106#529C2×C418yescomplemented
107C2×#126—21no—
108C2×#125C2311yescharacteristic
109C2×#124—21no—
110C2×#124C2311yescharacteristic
111C2×#123—21no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×#121—21no—
114C2×#121C2311yescharacteristic
115C2×#121C2312yes—
116C2×C4×Q8—22no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2314yes—
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117C2312yes—
130C22×C42C2311yescharacteristic
131#501C2×C418yescomplemented
132#496C2×C418yes—
133#490C2×C418yescomplemented
134#489C2×C418yescomplemented
135#469C2×C4116yescomplemented
136#468C2×C4116yescomplemented
137C4×SD16C2×C4116yescomplemented
138C4×D8C2×C4116yescomplemented
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C4×M16C2×C418yescomplemented
152C2×#97—24no—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2311yescharacteristic
156C2×#97C2311yescharacteristic
157C2×C4×C8C2312yes—
158#391—22no—
159#391C2312yes—
160#390—24no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2312yes—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#388—22no—
172#388—24no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—24no—
184#386—22no—
185#385—22no—
186#385—24no—
187#384C2312yes—
188#383—21no—
189#383—21no—
190#382—21no—
191#382—21no—
192#382C2312yes—
193#381—21no—
194#381—22no—
195#381—21no—
196#380—21no—
197#380—22no—
198#380—21no—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—24no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D411yescharacteristic
224C4×C22⋊C4D411yescharacteristic
225C4×C22⋊C4D412yes—
226#376—21no—
227#376—21no—
228#376C2311yescharacteristic
229#376C2311yescharacteristic
230C2×#95—22no—
231C2×#95—21no—
232C43C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—28no—
244C2×#41—84no—
245C22×Q8—21no—
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2411yescharacteristic
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—44no—
261C2×SD16C22×C418yes—
262C2×D8—44no—
263C2×D8C22×C418yes—
264C2×M16—44no—
265C2×M16C22×C418yes—
266C22×C8—24no—
267C22×C8—44no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—22no—
272C4⋊Q8C22×C414yes—
273#127—22no—
274#127C22×C414yes—
275#126—28no—
276#125—22no—
277#125C22×C414yes—
278#124—28no—
279#124—22no—
280#124C22×C414yes—
281#123—28no—
282C22⋊Q8—22no—
283C22⋊Q8C22×C414yes—
284#121—28no—
285#121—22no—
286#121—24no—
287#121C22×C414yes—
288#121C22×C418yes—
289C4×Q8—22no—
290C4×Q8—44no—
291C4×Q8—84no—
292C4×Q8C22×C414yes—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—22no—
296C4×D4—24no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—84no—
301C4×D4—84no—
302C4×D4C22×C414yes—
303C4×D4C22×C418yes—
304#117—28no—
305#117—44no—
306#117—82no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42C22×C414yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—44no—
364#107C22×C418yes—
365#106—44no—
366#106C22×C418yes—
367#103—44no—
368#103C22×C418yes—
369#102—44no—
370#102—44no—
371#102—44no—
372#102C22×C418yes—
373#102C22×C418yes—
374#102C22×C418yes—
375#97—24no—
376#97—24no—
377#97C22×C414yes—
378#97C22×C414yes—
379C4×C8C22×C418yes—
380#95—22no—
381#95—21no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95C2×D411yescharacteristic
396#95C2×D411yescharacteristic
397#95C2×D411yescharacteristic
398#95C2×D412yes—
399#95C2×D411yescharacteristic
400#95C2×D412yes—
401C24—21no—
402C24—22no—
403C24C22×D411yescharacteristic
404#41—816no—
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—24no—
408C2×Q8—44no—
409C2×Q8C23×C412yes—
410C2×D4—24no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—84no—
425C2×D4C23×C412yes—
426C2×D4C23×C414yes—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4C4×D414yes—
449C22×C4C4×D414yes—
450C22×C4C23×C412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462SD16—48no—
463D8—48no—
464M16—48no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—48no—
471C2×C8C23×C414yes—
472C2×C8C23×C414yes—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4C23×C412yes—
494C4⋊C4C23×C412yes—
495C4⋊C4C23×C414yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—28no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—24no—
514C42—24no—
515C42—24no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—81no—
522C42—81no—
523C42—82no—
524C42—82no—
525C42—82no—
526C42—84no—
527C42C4×D414yes—
528C42C4×D414yes—
529C42C23×C412yes—
530C42C23×C412yes—
531C42C2×#4112yes—
532C42C2×#4112yes—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—42no—
540C23—41no—
541C23—44no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23C2×C4×D412yes—
548C23C23×D411yescharacteristic
549Q8—24no—
550Q8—48no—
551Q8—42no—
552D4—24no—
553D4—24no—
554D4—24no—
555D4—48no—
556D4—48no—
557D4—42no—
558D4—44no—
559D4—84no—
560D4—88no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4—164no—
605C2×C4—168no—
606C2×C4C2×C4×D412yes—
607C2×C4C2×C4×D412yes—
608C2×C4C2×C4×D412yes—
609C2×C4C2×C4×D412yes—
610C2×C4C2×C4×D412yes—
611C2×C4C24×C411yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4#58514yes—
614C8—24no—
615C8—24no—
616C8—44no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—42no—
627C22—44no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22C22×C4×D411yescharacteristic
633C22#344912yes—
634C22#358512yes—
635C22#358511yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4#342812yes—
654C4#345512yes—
655C2—42no—
656C2—84no—
657C2—82no—
658C2#3407011yescharacteristic
659C2#6016811yescharacteristic
660C2#6017011yescharacteristic
661C2#6101612yes—
662C2#6101912yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3171C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34227C2semidirect
3. #34226C2semidirect
4. #22272C2semidirect
5. #22250C2semidirect
6. #22226C2semidirect
7. #22224C2semidirect
8. #22200C2semidirect
9. #22199C2semidirect
10. #21408C2semidirect
11. #21406C2semidirect
12. #21136C2semidirect
13. #21134C2semidirect
14. #21133C2semidirect
15. #21129C2semidirect
16. #20819C2semidirect
17. #20367C2semidirect
20. #3171C4semidirect
102. #532C2×C4semidirect
106. #529C2×C4semidirect
131. #501C2×C4semidirect
133. #490C2×C4semidirect
134. #489C2×C4semidirect
135. #469C2×C4semidirect
136. #468C2×C4semidirect
137. C4×SD16C2×C4semidirect
138. C4×D8C2×C4semidirect
151. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31712
#342272
#342262
#222722
#222502
#222262
#222242
#222002
#221992
#214082
#214062
#211362
#211342
#211332
#211292
#208192
#203672