Finite Group Databasebeta

← #6567478

Subgroups — #6567478

Order 512.

1935 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3400C212yesmaximal
2#36593C211yesmaximal, characteristic, complemented
3#34331C211yesmaximal, characteristic
4#29799C211yesmaximal, characteristic, complemented
5#29798C212yesmaximal, complemented
6#28449C212yesmaximal, complemented
7#28438C211yesmaximal, characteristic
8#28164C212yesmaximal, complemented
9#28160C211yesmaximal, characteristic
10#25382C214yesmaximal, complemented
11#25368C212yesmaximal
12#24748C214yesmaximal, complemented
13#24732C212yesmaximal, complemented
14#23683C212yesmaximal, complemented
15#23679C211yesmaximal, characteristic, complemented
16#22736C211yesmaximal, characteristic, complemented
17#22547C212yesmaximal, complemented
18C2×#559C2211yescharacteristic
19C2×#557C2212yes—
20#3400—216no—
21C2×#497C2214yes—
22C2×#492C2212yes—
23C2×#464C2214yes—
24C2×#462C2212yes—
25C2×#462C2214yes—
26C2×#460C2212yes—
27C2×#461C2214yes—
28C2×#461C2214yes—
29C2×#461C2214yes—
30C2×#446C2212yes—
31C2×#440C2214yes—
32C2×C4×Q16C2212yes—
33#2868C2211yescharacteristic
34#2868C2212yes—
35#2552—22no—
36#2552—21no—
37#2552—22no—
38#2552—21no—
39#2546C2212yes—
40#2546C2214yes—
41#2367—21no—
42#2367C2212yes—
43#2366C2211yescharacteristic
44#2359—22no—
45#2355C2211yescharacteristic
46#2354—21no—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4—21no—
49Q8×C4⋊C4C2211yescharacteristic
50#2349C2212yes—
51#2335—22no—
52#2326—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2212yes—
55C4×#127C2211yescharacteristic
56C4×#124C2212yes—
57#2295C2211yescharacteristic
58#2293C2212yes—
59#2275—21no—
60C42×Q8C2211yescharacteristic
61#2083C2211yescharacteristic
62#2072C2211yescharacteristic
63#2071C2212yes—
64#2068C2212yes—
65#2068C2214yes—
66#2056C2212yes—
67#2055C2214yes—
68#2043C2211yescharacteristic
69#2043C2212yes—
70#2043C2214yes—
71#2043C2212yes—
72#2016C2212yes—
73#2016C2211yescharacteristic
74#2016C2212yes—
75#2013C2212yes—
76#2013C2212yes—
77#2013C2212yes—
78#2013C2214yes—
79#1981C2212yes—
80#1981C2214yes—
81#1960C2214yes—
82#1956C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1878C2214yes—
86#1877C2214yes—
87#1877C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1805C2214yes—
91#1803C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#103C2212yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×SD16C2314yes—
101#559—24no—
102#559D418yes—
103#557—28no—
104#557—28no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×#127C2311yescharacteristic
112C2×#125—22no—
113C2×#125—24no—
114C2×#124—21no—
115C2×#124C2312yes—
116C2×#123—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117—22no—
133#497—216no—
134#492—28no—
135#464—216no—
136#462—28no—
137#462—48no—
138#461—216no—
139#461—48no—
140#461—48no—
141#460—28no—
142#446—28no—
143#440—216no—
144C4×Q16—28no—
145C2×#107C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#97C2314yes—
162C2×C4×C8C2312yes—
163#398—24no—
164#398—24no—
165#398—24no—
166#391—21no—
167#391C2314yes—
168#391C2312yes—
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#390—21no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—22no—
177#389—24no—
178#389—24no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—21no—
184#389—22no—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2311yescharacteristic
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—21no—
196#387—21no—
197#386—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—22no—
203#384—21no—
204#384—22no—
205#384—21no—
206#384—22no—
207#384—21no—
208#384—22no—
209#384—21no—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#382C2312yes—
213#380—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D412yes—
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233#376—21no—
234C2×#95—22no—
235C43C2311yescharacteristic
236C43C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2312yes—
241#336C2311yescharacteristic
242#336C2311yescharacteristic
243#336C2312yes—
244#336C2314yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2412yes—
247C22×Q8C2414yes—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C2×Q16—24no—
256C2×Q16—28no—
257C2×Q16—24no—
258C2×Q16—42no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×SD16—28no—
262C2×SD16—28no—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—44no—
266C2×SD16—44no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—28no—
274C4⋊Q8—24no—
275C4⋊Q8—28no—
276C4⋊Q8—24no—
277C4⋊Q8—44no—
278C4⋊Q8—42no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282#127—22no—
283#127C2×D414yes—
284#125—48no—
285#125—416no—
286#124—28no—
287#124—24no—
288#124C2×D418yes—
289#123—216no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—22no—
293C4×Q8—24no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—48no—
299C4×Q8C2×D414yes—
300C4×Q8C2×D414yes—
301C4×Q8#4114yes—
302C4×D4—28no—
303C4×D4—24no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—82no—
309C4×D4C2×D418yes—
310#117—84no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373#107—24no—
374C4⋊C8—24no—
375C4⋊C8—28no—
376C4⋊C8—24no—
377C4⋊C8—28no—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—24no—
382#103—28no—
383#103—42no—
384#103—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388#102—28no—
389#97—28no—
390C4×C8—24no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95—24no—
402#95—22no—
403#95—24no—
404#95—22no—
405#95—21no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8#12114yes—
428C2×Q8C22×D412yes—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—81no—
437C2×D4—81no—
438C2×D4C22×D414yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—28no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467Q16—48no—
468Q16—88no—
469SD16—416no—
470SD16—88no—
471SD16—88no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8H(5,𝔽2)14yes—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4#12114yes—
514C4⋊C4#12118yes—
515C4⋊C4#13714yes—
516C4⋊C4C22×D412yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—22no—
535C42—42no—
536C42—41no—
537C42—48no—
538C42—44no—
539C42—42no—
540C42—82no—
541C42—82no—
542C42#13714yes—
543C42C22×D412yes—
544C42C22×D414yes—
545C42C2×#4114yes—
546C42C2×#4112yes—
547C23—24no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C23×D411yescharacteristic
555Q8—28no—
556Q8—24no—
557Q8—42no—
558Q8—44no—
559Q8—42no—
560Q8—44no—
561Q8—42no—
562Q8—42no—
563Q8—44no—
564Q8—44no—
565Q8—42no—
566D4—28no—
567D4—44no—
568D4—48no—
569D4—84no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—164no—
616C2×C4—164no—
617C2×C4C2×#12114yes—
618C2×C4C2×#12114yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13714yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4#57712yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4#58512yes—
631C8—24no—
632C8—42no—
633C8—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22C22×#12111yescharacteristic
642C22#348711yescharacteristic
643C22#349811yescharacteristic
644C22C22×#13711yescharacteristic
645C22C22×#13711yescharacteristic
646C22C2×#57711yescharacteristic
647C22#358611yescharacteristic
648C4—24no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—24no—
654C4—24no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—82no—
660C4—84no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4#303914yes—
666C4#348812yes—
667C4#351712yes—
668C2—84no—
669C2#3602911yescharacteristic
670C2C2×#303911yescharacteristic
671C2#6042911yescharacteristic
672C2#6189311yescharacteristic
673C2#6190611yescharacteristic
674C2#6242211yescharacteristic
675C2#6244811yescharacteristic
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
2. #36593C2semidirect
4. #29799C2semidirect
5. #29798C2semidirect
6. #28449C2semidirect
8. #28164C2semidirect
10. #25382C2semidirect
12. #24748C2semidirect
13. #24732C2semidirect
14. #23683C2semidirect
15. #23679C2semidirect
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×#34002
#365932
#343312
#297992
#297982
#284492
#284382
#281642
#281602
#253822
#253682
#247482
#247322
#236832
#236792
#227362
#225472