Finite Group Databasebeta

← #6567475

Subgroups — #6567475

Order 512.

1883 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3398C212yesmaximal
2#36593C211yesmaximal, characteristic, complemented
3#34322C211yesmaximal, characteristic
4#29799C212yesmaximal, complemented
5#29798C211yesmaximal, characteristic
6#28449C212yesmaximal
7#28440C211yesmaximal, characteristic, complemented
8#28164C212yesmaximal
9#28161C211yesmaximal, characteristic, complemented
10#25382C214yesmaximal
11#25370C212yesmaximal, complemented
12#24746C214yesmaximal, complemented
13#24730C212yesmaximal
14#23685C212yesmaximal, complemented
15#23677C211yesmaximal, characteristic
16#22736C211yesmaximal, characteristic, complemented
17#22547C212yesmaximal, complemented
18C2×#557C2212yes—
19C2×#556C2211yescharacteristic
20#3398—216no—
21C2×#497C2214yes—
22C2×#494C2212yes—
23C2×#464C2214yes—
24C2×#462C2212yes—
25C2×#462C2214yes—
26C2×#462C2214yes—
27C2×#462C2214yes—
28C2×#461C2212yes—
29C2×#461C2214yes—
30C2×#446C2212yes—
31C2×#441C2214yes—
32C2×C4×SD16C2212yes—
33#2868C2211yescharacteristic
34#2868C2212yes—
35#2552—22no—
36#2552—21no—
37#2552—22no—
38#2552—21no—
39#2546C2212yes—
40#2546C2214yes—
41#2373—22no—
42#2367C2212yes—
43#2365C2211yescharacteristic
44#2360—21no—
45#2355C2211yescharacteristic
46#2351C2212yes—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4—21no—
49D4×C4⋊C4C2211yescharacteristic
50#2346—21no—
51#2334—22no—
52#2323—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2212yes—
56C4×#124C2212yes—
57#2295C2212yes—
58#2293C2211yescharacteristic
59#2274—21no—
60C42×Q8C2211yescharacteristic
61#2084C2211yescharacteristic
62#2072C2212yes—
63#2071C2211yescharacteristic
64#2066C2212yes—
65#2066C2214yes—
66#2056C2214yes—
67#2055C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2211yescharacteristic
71#2043C2214yes—
72#2016C2212yes—
73#2016C2212yes—
74#2016C2212yes—
75#2016C2214yes—
76#2013C2212yes—
77#2013C2211yescharacteristic
78#2013C2212yes—
79#1981C2212yes—
80#1981C2214yes—
81#1960C2214yes—
82#1956C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1878C2214yes—
86#1878C2214yes—
87#1877C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1805C2214yes—
91#1803C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#103C2212yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×SD16C2312yes—
100C22×SD16C2314yes—
101#557—28no—
102#557—28no—
103#556—24no—
104#556D418yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×#126—22no—
113C2×#125—22no—
114C2×#125—24no—
115C2×#124C2312yes—
116C2×C22⋊Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117—21no—
130C22×C4⋊C4—21no—
131#497—216no—
132#494—28no—
133#464—216no—
134#462—28no—
135#462—216no—
136#462—48no—
137#462—48no—
138#461—28no—
139#461—48no—
140#446—28no—
141#441—216no—
142C4×SD16—28no—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2312yes—
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#398—24no—
162#398—24no—
163#398—24no—
164#391—21no—
165#391C2312yes—
166#391C2314yes—
167#391C2312yes—
168#391C2312yes—
169#390C2311yescharacteristic
170#390C2311yescharacteristic
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389—24no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—21no—
184#389C2312yes—
185#389C2311yescharacteristic
186#389C2311yescharacteristic
187#388—22no—
188#388—22no—
189#388—21no—
190#388—22no—
191#388—21no—
192#388—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#385—22no—
198#385—22no—
199#384—22no—
200#384—21no—
201#384—22no—
202#384—21no—
203#384—22no—
204#384—21no—
205#384—22no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382—22no—
210#382C2312yes—
211#380—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D412yes—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234#376—22no—
235C43C2311yescharacteristic
236C43C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2314yes—
244#336C2312yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2414yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C2×Q16—28no—
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—44no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×SD16—24no—
262C2×SD16—28no—
263C2×SD16—24no—
264C2×SD16—42no—
265C2×SD16—44no—
266C2×SD16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—28no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—28no—
275C4⋊Q8—42no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8C2×D414yes—
281#126—216no—
282#125—48no—
283#125—416no—
284#124—24no—
285#124C2×D418yes—
286C22⋊Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—28no—
292C4×Q8—24no—
293C4×Q8—28no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—48no—
297C4×Q8C2×D418yes—
298C4×Q8C2×D414yes—
299C4×Q8#4114yes—
300C4×D4—22no—
301C4×D4—44no—
302C4×D4C2×D414yes—
303#117—28no—
304#117—44no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2412yes—
371#106—24no—
372C4⋊C8—28no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—28no—
376#103—28no—
377#103—28no—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—44no—
382#103—42no—
383#103—44no—
384#103—44no—
385#103—44no—
386#102—24no—
387#97—28no—
388C4×C8—24no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—24no—
399#95—22no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8#12114yes—
426C2×Q8C22×D414yes—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4C22×D412yes—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456Q16—416no—
457Q16—88no—
458Q16—88no—
459SD16—48no—
460SD16—88no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8H(5,𝔽2)14yes—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4#12114yes—
506C4⋊C4#12118yes—
507C4⋊C4#13714yes—
508C4⋊C4C22×D412yes—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—28no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—82no—
517C42—28no—
518C42—24no—
519C42—22no—
520C42—24no—
521C42—22no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—42no—
527C42—41no—
528C42—48no—
529C42—44no—
530C42—42no—
531C42—82no—
532C42—82no—
533C42—82no—
534C42#13714yes—
535C42C22×D412yes—
536C42C22×D414yes—
537C42C2×#4112yes—
538C42C2×#4114yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C23×D411yescharacteristic
547Q8—28no—
548Q8—28no—
549Q8—44no—
550Q8—42no—
551Q8—44no—
552Q8—44no—
553Q8—42no—
554Q8—44no—
555Q8—42no—
556Q8—42no—
557Q8—44no—
558D4—24no—
559D4—42no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—164no—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4C2×#12112yes—
609C2×C4C2×#12112yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13714yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13712yes—
615C2×C4#57812yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4#58512yes—
620C8—24no—
621C8—44no—
622C8—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C22×#12111yescharacteristic
631C22#348711yescharacteristic
632C22#349811yescharacteristic
633C22C22×#13711yescharacteristic
634C22C22×#13711yescharacteristic
635C22C2×#57811yescharacteristic
636C22#358611yescharacteristic
637C4—24no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—24no—
643C4—24no—
644C4—44no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4—81no—
652C4—81no—
653C4—82no—
654C4#303914yes—
655C4#348812yes—
656C4#351712yes—
657C2—82no—
658C2#3602911yescharacteristic
659C2C2×#303911yescharacteristic
660C2#6043211yescharacteristic
661C2#6189311yescharacteristic
662C2#6190011yescharacteristic
663C2#6242411yescharacteristic
664C2#6244811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
2. #36593C2semidirect
4. #29799C2semidirect
7. #28440C2semidirect
9. #28161C2semidirect
11. #25370C2semidirect
12. #24746C2semidirect
14. #23685C2semidirect
16. #22736C2semidirect
17. #22547C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33982
#365932
#343222
#297992
#297982
#284492
#284402
#281642
#281612
#253822
#253702
#247462
#247302
#236852
#236772
#227362
#225472