Finite Group Databasebeta

← #6456410

Subgroups — #6456410

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3184C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic, complemented
3#35237C211yesmaximal, characteristic, complemented
4#34232C211yesmaximal, characteristic, complemented
5#25452C212yesmaximal, complemented
6#25364C212yesmaximal, complemented
7#25321C212yesmaximal, complemented
8#25272C212yesmaximal, complemented
9#25227C214yesmaximal, complemented
10#25097C214yesmaximal, complemented
11#23688C214yesmaximal, complemented
12#21513C214yesmaximal, complemented
13#21390C211yesmaximal, characteristic, complemented
14#21354C211yesmaximal, characteristic, complemented
15C4×#418C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3184—28no—
20C2×#498C2211yescharacteristic, complemented
21C2×#493C2211yescharacteristic, complemented
22C2×#486C2211yescharacteristic, complemented
23C2×#486C2211yescharacteristic, complemented
24#3100—28no—
25C2×#480C2212yes—
26C2×#474C2212yes—
27C2×#469C2214yes—
28C2×#459C2214yes—
29C2×#459C2212yes—
30C2×#447C2212yes—
31C2×#442C2214yescomplemented
32C2×C4×D8C2214yes—
33C2×#431C2211yescharacteristic
34C2×#418C2212yes—
35C2×C4×M16C2211yescharacteristic
36#2847—21no—
37#2847—21no—
38#2631—22no—
39#2563C2212yes—
40#2541C2212yes—
41#2535—22no—
42#2504C2212yes—
43#2433C2212yes—
44#2404—22no—
45#2366C2212yescomplemented
46#2365C2212yescomplemented
47#2363C2214yes—
48#2353—22no—
49#2341—21no—
50#2339—21no—
51#2333—22no—
52#2328C2214yes—
53#2319C2212yescomplemented
54C4×#124—22no—
55#2283C2211yescharacteristic
56#2275C2212yescomplemented
57C4×#117C2211yescharacteristic
58#2085C2214yescomplemented
59#2073C2214yescomplemented
60#2067C2214yescomplemented
61#2065C2214yes—
62#2054C2214yes—
63#2054C2214yescomplemented
64#2041C2214yes—
65#2011C2214yes—
66#1987C2212yescomplemented
67#1982C2214yes—
68#1982C2212yescomplemented
69#1949C2214yescomplemented
70#1939C2212yescomplemented
71#1939C2212yescomplemented
72#1926C2214yes—
73#1925C2214yescomplemented
74#1879C2214yes—
75#1879C2214yescomplemented
76#1875C2214yes—
77#1851C2214yes—
78#1804C2214yescomplemented
79#1768C2214yescomplemented
80#1762C2212yes—
81#1762C2212yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84C4×#102C2212yescomplemented
85C4×#102C2212yescomplemented
86#1736C2212yes—
87C22×#41—21no—
88C22×#41—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×D8C2314yes—
92C22×D8C2314yes—
93C22×M16C2312yes—
94#532—24no—
95#532—24no—
96#532—24no—
97#532D418yescomplemented
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#127C2314yes—
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×#124—21no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—22no—
112C2×#120—22no—
113C2×C4×Q8—22no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—24no—
120C2×C4×D4—24no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2312yes—
126C2×#117C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C42C2311yescharacteristic
130#498—24no—
131#493—24no—
132#486—24no—
133#486—24no—
134#480—28no—
135#474—28no—
136#469—216no—
137#459—28no—
138#459—216no—
139#447—28no—
140#442—216no—
141C4×D8—216no—
142#431—24no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148#418—28no—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×C22⋊C8C2312yes—
160C4×M16—24no—
161C2×#97C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2312yes—
165#397—24no—
166#393—24no—
167#392—24no—
168#392—24no—
169#391—22no—
170#390—22no—
171#390—24no—
172#390C2314yes—
173#390C2314yes—
174#390C2314yes—
175#389C2314yes—
176#388—24no—
177#388—24no—
178#387—24no—
179#386—24no—
180#386—22no—
181#386—22no—
182#386—24no—
183#386—24no—
184#386—24no—
185#386—24no—
186#386—24no—
187#383D412yes—
188#382D412yes—
189#382C2314yes—
190#381—22no—
191#381D412yes—
192#380—22no—
193#380—22no—
194#380—22no—
195#380D411yescharacteristic
196#380D411yescharacteristic
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—24no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4C2312yes—
208#376—21no—
209#376C2312yes—
210C2×#95—22no—
211C43—21no—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#336C2314yes—
219#336C2314yes—
220C25—21no—
221C2×#41—28no—
222C2×#41—44no—
223C2×#41—84no—
224C2×#41—82no—
225C2×#41—82no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—42no—
241C22×D4C2412yes—
242C22×D4C2412yes—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—44no—
251C23×C4—84no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×D8—28no—
256C2×D8—28no—
257C2×D8—28no—
258C2×D8—28no—
259C2×D8—44no—
260C2×D8—44no—
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—42no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8C2×D414yes—
271#127—28no—
272#127—22no—
273#127—22no—
274#127—22no—
275#127—44no—
276#127C2×D414yes—
277#124—28no—
278#124—44no—
279#121—28no—
280#121—24no—
281#121—24no—
282#121—24no—
283#121—44no—
284#121—84no—
285#121—84no—
286#121C2×D418yes—
287#120—84no—
288#120—84no—
289C4×Q8—84no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—84no—
304C4×D4C2×D418yes—
305#117—24no—
306#117—42no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D414yes—
345C2×C42#4114yes—
346C2×C42C2414yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—28no—
350#106—28no—
351C4⋊C8—28no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#102—28no—
355#102—28no—
356#102—24no—
357#102—24no—
358#102—28no—
359#102—28no—
360#102—24no—
361#102—28no—
362#102—44no—
363#102#4118yescomplemented
364C22⋊C8—24no—
365#97—24no—
366C4×C8—22no—
367C4×C8—22no—
368C4×C8—24no—
369#95—21no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D414yes—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—44no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—82no—
389C24C22×D411yescharacteristic
390#41—816no—
391#41—816no—
392C2×Q8—24no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4C22×D414yes—
436C2×D4C2×#4114yes—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—48no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—816no—
463C22×C4—82no—
464C22×C4#12114yes—
465C22×C4#12114yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2511yescharacteristic
476D8—416no—
477D8—416no—
478M16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8C2×#4114yes—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—84no—
502C4⋊C4C22×D414yes—
503C4⋊C4C2×#4114yes—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—44no—
523C42—44no—
524C42—81no—
525C42—81no—
526C42—84no—
527C42#12114yes—
528C42#12114yes—
529C42C22×D412yes—
530C42C22×D412yes—
531C42C2×#4112yes—
532C42C2×#4112yes—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—84no—
556C23—84no—
557C23—88no—
558C23—88no—
559C23C2×#12112yes—
560C23C23×D411yescharacteristic
561Q8—48no—
562Q8—84no—
563D4—24no—
564D4—24no—
565D4—28no—
566D4—48no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—48no—
575D4—84no—
576D4—84no—
577D4—88no—
578D4—88no—
579D4—84no—
580D4—168no—
581D4—168no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—88no—
614C2×C4—88no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4—168no—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4#57614yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4112yes—
628C2×C4#58512yes—
629C8—24no—
630C8—42no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—816no—
644C22—82no—
645C22—82no—
646C22—84no—
647C22—82no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—82no—
655C22C22×#12111yescharacteristic
656C22D4×#4112yes—
657C22C2×#57612yes—
658C22#358511yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#304114yes—
672C4#344411yescharacteristic
673C4#345511yescharacteristic
674C4D4×#4112yes—
675C2—42no—
676C2—84no—
677C2—84no—
678C2—162no—
679C2#3474311yescharacteristic
680C2C2×#304111yescharacteristic
681C2#6042611yescharacteristic
682C2#6153912yes—
683C2#6155512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3184C2semidirect
2. C2×#3100C2semidirect
3. #35237C2semidirect
4. #34232C2semidirect
5. #25452C2semidirect
6. #25364C2semidirect
7. #25321C2semidirect
8. #25272C2semidirect
9. #25227C2semidirect
10. #25097C2semidirect
11. #23688C2semidirect
12. #21513C2semidirect
13. #21390C2semidirect
14. #21354C2semidirect
15. C4×#418C2semidirect
20. C2×#498C22semidirect
21. C2×#493C22semidirect
22. C2×#486C22semidirect
23. C2×#486C22semidirect
31. C2×#442C22semidirect
45. #2366C22semidirect
46. #2365C22semidirect
53. #2319C22semidirect
56. #2275C22semidirect
58. #2085C22semidirect
59. #2073C22semidirect
60. #2067C22semidirect
63. #2054C22semidirect
66. #1987C22semidirect
68. #1982C22semidirect
69. #1949C22semidirect
70. #1939C22semidirect
71. #1939C22semidirect
73. #1925C22semidirect
75. #1879C22semidirect
78. #1804C22semidirect
79. #1768C22semidirect
84. C4×#102C22semidirect
85. C4×#102C22semidirect
97. #532D4semidirect
363. #102#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31842
C2×#31002
#352372
#342322
#254522
#253642
#253212
#252722
#252272
#250972
#236882
#215132
#213902
#213542
C4×#4182