Finite Group Databasebeta

← #6567471

Subgroups — #6567471

Order 512.

1899 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
1C2×#3398C212yesmaximal, complemented
2#36594C211yesmaximal, characteristic
3#34337C211yesmaximal, characteristic, complemented
4#29801C212yesmaximal
5#29799C211yesmaximal, characteristic, complemented
6#28447C212yesmaximal
7#28440C211yesmaximal, characteristic, complemented
8#28161C212yesmaximal, complemented
9#28157C211yesmaximal, characteristic
10#25379C214yesmaximal, complemented
11#25365C212yesmaximal, complemented
12#24741C214yesmaximal, complemented
13#24729C212yesmaximal, complemented
14#23685C212yesmaximal, complemented
15#23678C211yesmaximal, characteristic
16#22736C211yesmaximal, characteristic, complemented
17#22544C212yesmaximal, complemented
18C2×#557C2211yescharacteristic
19C2×#556C2212yes—
20#3398—216no—
21C2×#497C2214yes—
22C2×#494C2212yes—
23C2×#464C2214yes—
24C2×#464C2214yes—
25C2×#464C2214yes—
26C2×#462C2214yes—
27C2×#462C2212yes—
28C2×#462C2214yes—
29C2×#461C2212yes—
30C2×#446C2212yes—
31C2×#441C2214yes—
32C2×C4×SD16C2212yes—
33#2868C2211yescharacteristic
34#2858C2212yes—
35#2547C2212yes—
36#2545C2214yes—
37#2538—21no—
38#2538—21no—
39#2512—22no—
40#2512—22no—
41#2370—21no—
42#2370—22no—
43#2365C2211yescharacteristic
44#2365C2212yes—
45#2354—21no—
46#2354—21no—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4C2211yescharacteristic
49Q8×C4⋊C4C2211yescharacteristic
50Q8×C4⋊C4C2212yes—
51#2330—21no—
52#2330—22no—
53C4×C4⋊Q8C2212yes—
54C4×C4⋊Q8C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57#2295C2211yescharacteristic
58#2295C2212yes—
59#2276—21no—
60C42×Q8C2211yescharacteristic
61#2084C2211yescharacteristic
62#2072C2211yescharacteristic
63#2072C2212yes—
64#2066C2214yes—
65#2065C2212yes—
66#2056C2214yes—
67#2055C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2042C2214yes—
71#2041C2211yescharacteristic
72#2016C2212yes—
73#2016C2212yes—
74#2016C2212yes—
75#2016C2212yes—
76#2015C2214yes—
77#2014C2211yescharacteristic
78#2014C2212yes—
79#1983C2214yes—
80#1981C2212yes—
81#1959C2214yes—
82#1956C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1880C2214yes—
86#1878C2214yes—
87#1878C2214yes—
88#1831C2212yes—
89#1831C2211yescharacteristic
90#1803C2214yes—
91#1802C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#102C2212yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×SD16C2312yes—
100C22×SD16C2314yes—
101#557—24no—
102#557D418yescomplemented
103#556—28no—
104#556—28no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#126—24no—
112C2×#125—21no—
113C2×#125—22no—
114C2×#124C2314yes—
115C2×#124C2314yes—
116C2×C22⋊Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117—22no—
132#497—216no—
133#494—28no—
134#464—216no—
135#464—48no—
136#464—48no—
137#462—216no—
138#462—28no—
139#462—48no—
140#461—28no—
141#446—28no—
142#441—216no—
143C4×SD16—28no—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#397—24no—
163#396—24no—
164#396—24no—
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2314yes—
168#391C2311yescharacteristic
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390C2312yes—
177#389—21no—
178#389—22no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389C2311yescharacteristic
191#389C2312yes—
192#389C2311yescharacteristic
193#388—24no—
194#388—24no—
195#387—22no—
196#387—21no—
197#387—22no—
198#387—21no—
199#387—22no—
200#387—22no—
201#387—22no—
202#384—21no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2312yes—
211#380—22no—
212#380—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4D412yes—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D412yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2312yes—
233#376—22no—
234#376—21no—
235C43C2311yescharacteristic
236C43C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2312yes—
241#336C2311yescharacteristic
242#336C2311yescharacteristic
243#336C2314yes—
244#336C2312yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2414yes—
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—22no—
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×SD16—24no—
263C2×SD16—28no—
264C2×SD16—24no—
265C2×SD16—42no—
266C2×SD16—44no—
267C2×SD16—44no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8C2×D414yes—
278C4⋊Q8C2×D418yes—
279#127—42no—
280#126—416no—
281#125—28no—
282#125—216no—
283#124—28no—
284#124—28no—
285#124—44no—
286#124—44no—
287C22⋊Q8—48no—
288C4×Q8—28no—
289C4×Q8—24no—
290C4×Q8—22no—
291C4×Q8—24no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—42no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8C2×D414yes—
300C4×Q8C2×D418yes—
301C4×Q8C2×D414yes—
302C4×Q8#4114yes—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—44no—
306C4×D4—44no—
307#117—48no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372#106—24no—
373C4⋊C8—24no—
374C4⋊C8—28no—
375C4⋊C8—24no—
376C4⋊C8—28no—
377#103—24no—
378#103—28no—
379#103—28no—
380#103—28no—
381#103—44no—
382#103—44no—
383#103—44no—
384#102—24no—
385#102—24no—
386#102—42no—
387#102—44no—
388#97—28no—
389C4×C8—24no—
390#95—21no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95—22no—
402#95—21no—
403#95—22no—
404#95—21no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8#12114yes—
424C2×Q8C22×D412yes—
425C2×Q8C22×D414yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460Q16—416no—
461Q16—88no—
462Q16—88no—
463SD16—48no—
464SD16—88no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8H(5,𝔽2)14yes—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—82no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4#12114yes—
510C4⋊C4#12118yes—
511C4⋊C4C22×D412yes—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C42—28no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—42no—
529C42—41no—
530C42—42no—
531C42—44no—
532C42—48no—
533C42—42no—
534C42—82no—
535C42—82no—
536C42—82no—
537C42#13714yes—
538C42C22×D412yes—
539C42C22×D414yes—
540C42C2×#4112yes—
541C42C2×#4114yes—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C23×D411yescharacteristic
550Q8—28no—
551Q8—24no—
552Q8—28no—
553Q8—42no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—84no—
564D4—84no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—88no—
611C2×C4C2×#12114yes—
612C2×C4C2×#12114yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4C2×#13712yes—
618C2×C4#57512yes—
619C2×C4#57812yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4#58512yes—
624C8—24no—
625C8—44no—
626C8—42no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C22×#12111yescharacteristic
635C22#348711yescharacteristic
636C22#349811yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57511yescharacteristic
639C22C2×#57811yescharacteristic
640C22#358611yescharacteristic
641C4—24no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—84no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4#303714yes—
659C4#348812yes—
660C4#351712yes—
661C2—162no—
662C2#3602911yescharacteristic
663C2C2×#303711yescharacteristic
664C2#6043211yescharacteristic
665C2#6188911yescharacteristic
666C2#6190511yescharacteristic
667C2#6242411yescharacteristic
668C2#6245211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3398C2semidirect
3. #34337C2semidirect
5. #29799C2semidirect
7. #28440C2semidirect
8. #28161C2semidirect
10. #25379C2semidirect
11. #25365C2semidirect
12. #24741C2semidirect
13. #24729C2semidirect
14. #23685C2semidirect
16. #22736C2semidirect
17. #22544C2semidirect
102. #557D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33982
#365942
#343372
#298012
#297992
#284472
#284402
#281612
#281572
#253792
#253652
#247412
#247292
#236852
#236782
#227362
#225442