Finite Group Databasebeta

← #6467215

Subgroups — #6467215

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3#34221C211yesmaximal, characteristic, complemented
4C2×#2326C211yesmaximal, characteristic
5#28288C212yesmaximal, complemented
6#28258C212yesmaximal, complemented
7#28211C212yesmaximal, complemented
8#28168C212yesmaximal, complemented
9#28123C212yesmaximal
10#28121C212yesmaximal, complemented
11#27648C212yesmaximal, complemented
12#24217C212yesmaximal
13#23695C214yesmaximal, complemented
14#21512C214yesmaximal, complemented
15#20814C211yesmaximal, characteristic, complemented
16#20741C211yesmaximal, characteristic, complemented
17#20733C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#3110—28no—
23#3104—28no—
24C2×#484C2212yes—
25C2×#484C2212yes—
26C2×#477C2212yescomplemented
27C2×#474C2212yes—
28C2×#464C2214yes—
29C2×#462C2212yescomplemented
30C2×#461C2212yes—
31C2×#451C2212yes—
32C2×#450C2212yes—
33C2×#440C2214yes—
34C2×C4×Q16C2214yescomplemented
35C2×#432C2211yescharacteristic
36C2×#431C2211yescharacteristic
37#2371—22no—
38#2368—22no—
39#2356—22no—
40#2343—22no—
41#2331C2211yescharacteristic, complemented
42#2331C2211yescharacteristic, complemented
43#2327C2211yescharacteristic, complemented
44#2327C2211yescharacteristic, complemented
45#2326—22no—
46#2326—22no—
47#2326C2212yes—
48#2326C2212yes—
49#2326C2212yes—
50#2326C2212yes—
51#2320C2212yescomplemented
52C4×C22⋊Q8C2212yes—
53C4×C22⋊Q8C2212yes—
54C2×#391C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#388—22no—
57C2×#385—22no—
58C2×#384C2211yescharacteristic
59#2283C2211yescharacteristic
60#2276C2212yescomplemented
61C2×#376C2211yescharacteristic
62#2083C2214yescomplemented
63#2072C2214yescomplemented
64#2068C2214yescomplemented
65#2067C2214yes—
66#2056C2214yescomplemented
67#2055C2214yes—
68#2042C2214yes—
69#2015C2214yes—
70#1983C2214yes—
71#1947C2214yescomplemented
72#1928C2214yescomplemented
73#1925C2214yes—
74#1880C2214yes—
75#1878C2214yescomplemented
76#1843C2212yescomplemented
77#1832C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1805C2212yescomplemented
81#1805C2212yescomplemented
82#1804C2212yes—
83#1804C2212yes—
84#1803C2214yescomplemented
85#1797C2212yes—
86#1797C2212yes—
87#1787C2212yes—
88#1786C2212yes—
89#1768C2212yescomplemented
90#1768C2212yescomplemented
91#1763C2214yescomplemented
92C4×#102C2214yes—
93C23×Q8C2311yescharacteristic
94C23×D4C2311yescharacteristic
95C24×C4—21no—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98#533—24no—
99#533D418yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#124—28no—
104C2×#124—22no—
105C2×#124C2312yes—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—24no—
122C2×C4×D4—24no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#484—28no—
138#484—28no—
139#477—28no—
140#474—28no—
141#464—216no—
142#462—28no—
143#461—28no—
144#451—28no—
145#450—28no—
146#440—216no—
147C4×Q16—216no—
148#432—24no—
149#431—24no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2312yes—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#390—22no—
176#390C2312yes—
177#390C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—24no—
181#388—24no—
182#388—24no—
183#388—24no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#385—22no—
195#385—22no—
196#385—24no—
197#385—24no—
198#385—24no—
199#385—24no—
200#384—22no—
201#384C2312yes—
202#384C2312yes—
203#383—21no—
204#383—22no—
205#382—22no—
206#382C2311yescharacteristic
207#382C2311yescharacteristic
208#381—21no—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#376—21no—
222#376—22no—
223#376D412yes—
224#376D412yes—
225#376C2311yescharacteristic
226#376C2311yescharacteristic
227#376C2311yescharacteristic
228#376C2311yescharacteristic
229C2×#95—22no—
230C2×#95D412yes—
231C2×#95D412yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8C2412yes—
246C22×Q8C2412yes—
247C22×D4—24no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—42no—
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—84no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—44no—
268C2×SD16—28no—
269C2×SD16—28no—
270C2×SD16—44no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8C2×D414yes—
276C4⋊Q8C2×D414yes—
277#125—28no—
278#124—216no—
279#124—24no—
280#124—24no—
281#124—24no—
282#124#4118yes—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8C2×D418yes—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—28no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8C2×D418yes—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—44no—
302C4×D4—44no—
303#117—28no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—84no—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—216no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—84no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—42no—
351C2×C42—44no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D414yes—
357C2×C42#4114yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—28no—
365#107—28no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—28no—
369C4⋊C8—44no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#103—44no—
375#102—28no—
376#102—28no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380#97—24no—
381C4×C8—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—24no—
407C24—22no—
408C24—44no—
409C24—42no—
410C24—42no—
411C24—44no—
412C24C22×D411yescharacteristic
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—82no—
428C2×Q8C22×D414yes—
429C2×Q8C2×#4114yes—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C2×#4114yes—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—816no—
470C22×C4#12114yes—
471C22×C4#12114yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484Q16—416no—
485SD16—416no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22×D414yes—
515C22⋊C4—216no—
516C22⋊C4—28no—
517C22⋊C4—216no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—28no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—42no—
529C42—44no—
530C42—44no—
531C42—44no—
532C42—44no—
533C42—82no—
534C42—82no—
535C42—82no—
536C42#12114yes—
537C42#12114yes—
538C42C22×D412yes—
539C42C22×D412yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C42C2×#4114yes—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—24no—
547C23—48no—
548C23—416no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—84no—
554C23—88no—
555C23—84no—
556C23C2×#12112yes—
557C23#57814yes—
558C23C23×D411yescharacteristic
559Q8—28no—
560Q8—24no—
561Q8—24no—
562Q8—48no—
563Q8—42no—
564Q8—44no—
565Q8—42no—
566Q8—84no—
567D4—24no—
568D4—24no—
569D4—48no—
570D4—42no—
571D4—42no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4#52514yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C22×#4112yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4#58512yes—
629C8—44no—
630C8—44no—
631C22—22no—
632C22—24no—
633C22—22no—
634C22—24no—
635C22—44no—
636C22—48no—
637C22—816no—
638C22—82no—
639C22—82no—
640C22—84no—
641C22#304414yes—
642C22C22×#12111yescharacteristic
643C22C2×#52511yescharacteristic
644C22#344911yescharacteristic
645C22C2×#57812yes—
646C22#358512yes—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—24no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4D4×#4112yes—
664C4#347512yes—
665C2—24no—
666C2—84no—
667C2#3493511yescharacteristic
668C2C2×#304411yescharacteristic
669C2#6042611yescharacteristic
670C2#6058912yes—
671C2#6098112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3104C2semidirect
3. #34221C2semidirect
5. #28288C2semidirect
6. #28258C2semidirect
7. #28211C2semidirect
8. #28168C2semidirect
10. #28121C2semidirect
11. #27648C2semidirect
13. #23695C2semidirect
14. #21512C2semidirect
15. #20814C2semidirect
16. #20741C2semidirect
26. C2×#477C22semidirect
29. C2×#462C22semidirect
34. C2×C4×Q16C22semidirect
41. #2331C22semidirect
42. #2331C22semidirect
43. #2327C22semidirect
44. #2327C22semidirect
51. #2320C22semidirect
60. #2276C22semidirect
62. #2083C22semidirect
63. #2072C22semidirect
64. #2068C22semidirect
66. #2056C22semidirect
71. #1947C22semidirect
72. #1928C22semidirect
75. #1878C22semidirect
76. #1843C22semidirect
77. #1832C22semidirect
80. #1805C22semidirect
81. #1805C22semidirect
84. #1803C22semidirect
89. #1768C22semidirect
90. #1768C22semidirect
91. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#31042
#342212
C2×#23262
#282882
#282582
#282112
#281682
#281232
#281212
#276482
#242172
#236952
#215122
#208142
#207412
#207332