Finite Group Databasebeta

← #6606570

Subgroups — #6606570

Order 512.

1839 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×#3342C212yesmaximal
2#36188C211yesmaximal, characteristic
3C4×#556C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic, complemented
5#33114C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28283C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25456C212yesmaximal, complemented
10#25201C214yesmaximal, complemented
11#24752C212yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#24584C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic
15#22490C212yesmaximal
16C4×#462C211yesmaximal, characteristic, complemented
17#21169C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#556C2211yescharacteristic
20#3342—216no—
21C2×#500C2212yes—
22C2×#485C2214yes—
23C2×#485C2214yes—
24C2×#475C2212yes—
25C2×#475C2212yes—
26C2×#467C2214yes—
27C2×#462C2211yescharacteristic
28C2×C4×Q16C2212yes—
29C2×C8×D4C2212yes—
30C2×#427C2214yes—
31C22×C4⋊C8C2212yes—
32#2868C2211yescharacteristic
33#2857C2212yes—
34#2568C2212yes—
35#2547—21no—
36#2547—21no—
37#2499C2214yes—
38#2496—22no—
39#2496—22no—
40#2373—22no—
41#2367—21no—
42#2365—21no—
43#2365—21no—
44#2365C2211yescharacteristic
45#2362C2212yes—
46#2356—21no—
47#2355C2211yescharacteristic
48#2354C2211yescharacteristic
49#2347C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#126—22no—
53C4×#124—21no—
54C4×C22⋊Q8C2212yes—
55#2285C2212yes—
56C42×Q8—21no—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C4×#117C2212yes—
60#2085C2214yes—
61#2083C2214yes—
62#2074C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#1990C2212yes—
67#1988C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1947C2211yescharacteristic
78#1939C2212yes—
79#1928C2212yes—
80#1928C2214yes—
81#1927C2214yes—
82#1892C2214yes—
83#1878C2212yes—
84#1873C2214yes—
85#1850C2214yes—
86#1842C2211yescharacteristic
87#1825C2214yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1735C2214yes—
96C22×Q16C2312yes—
97C23×C8C2314yes—
98#556—28no—
99#556—24no—
100#556—28no—
101#556D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—22no—
108C2×#126—24no—
109C2×#124—21no—
110C2×#124—22no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117C2314yes—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C42C2312yes—
128#500—28no—
129#485—216no—
130#485—216no—
131#475—28no—
132#475—28no—
133#467—216no—
134#462—24no—
135C4×Q16—28no—
136C8×D4—28no—
137#427—216no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—28no—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2312yes—
159#400—24no—
160#397—24no—
161#397—24no—
162#391—22no—
163#391—22no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#391C2314yes—
169#390—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—24no—
187#387—24no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#383—22no—
196#382—21no—
197#382C2314yes—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231#376—22no—
232C43—22no—
233C43—21no—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—42no—
255C22×C8—28no—
256C22×C8—28no—
257C22×C8—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8C2×D414yes—
269#126—216no—
270#126—416no—
271#124—28no—
272#124—48no—
273C22⋊Q8—24no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8C2×D418yes—
277C4×Q8—28no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—41no—
285C4×Q8—44no—
286C4×Q8—48no—
287C4×Q8—44no—
288C4×Q8C2×D414yes—
289C4×D4—24no—
290C4×D4—42no—
291C4×D4C2×D414yes—
292C4×D4#4114yes—
293#117—24no—
294#117—28no—
295#117—28no—
296#117—24no—
297#117—42no—
298#117C2×D418yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—28no—
365#107—28no—
366#106—28no—
367#106—28no—
368C4⋊C8—28no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—22no—
372C4⋊C8—42no—
373C4⋊C8#4114yes—
374#103—24no—
375#103—28no—
376#103—24no—
377#103—24no—
378#103—28no—
379#103—24no—
380#103—44no—
381C22⋊C8—28no—
382C4×C8—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24C22×D412yes—
404C2×Q8—24no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—42no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8C22×D412yes—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4#12114yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D414yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4114yes—
455C22×C4C2511yescharacteristic
456Q16—48no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—416no—
466C2×C8C2×#4114yes—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4#12114yes—
501C4⋊C4C22×D412yes—
502C4⋊C4C22×D414yes—
503C4⋊C42−1+412yes—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—82no—
512C22⋊C4#12118yes—
513C42—28no—
514C42—24no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—21no—
523C42—41no—
524C42—44no—
525C42—42no—
526C42—44no—
527C42—48no—
528C42—44no—
529C42—42no—
530C42—44no—
531C42—42no—
532C42—42no—
533C42—81no—
534C42—82no—
535C42#13514yes—
536C42C22×D412yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C422−1+412yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23C2×#12114yes—
547C23C23×D411yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—48no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—88no—
558D4—28no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—84no—
609C2×C4C2×#12114yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4#54712yes—
613C2×C4#54712yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4#57512yes—
618C2×C4#57812yes—
619C2×C4C23×D411yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58511yescharacteristic
625C8—24no—
626C8—44no—
627C8—42no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22C22×#12111yescharacteristic
636C22C2×#54711yescharacteristic
637C22#348911yescharacteristic
638C22C22×#13511yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—44no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—84no—
658C4#303514yes—
659C4#334214yescomplemented
660C4D4×#4111yescharacteristic
661C4#348211yescharacteristic
662C4#349911yescharacteristic
663C4#351511yescharacteristic
664C2—44no—
665C2#3625911yescharacteristic
666C2C2×#303511yescharacteristic
667C2#6043211yescharacteristic
668C2C2×#334211yescharacteristic
669C2#6125711yescharacteristic
670C2#6210111yescharacteristic
671C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33196C2semidirect
5. #33114C2semidirect
6. #28689C2semidirect
7. #28283C2semidirect
8. #27649C2semidirect
9. #25456C2semidirect
10. #25201C2semidirect
11. #24752C2semidirect
12. #24683C2semidirect
13. #24584C2semidirect
16. C4×#462C2semidirect
17. #21169C2semidirect
659. C4#3342semidirect

Maximal subgroups

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

subgroupindex
C2×#33422
#361882
C4×#5562
#331962
#331142
#286892
#282832
#276492
#254562
#252012
#247522
#246832
#245842
D4×C4⋊C82
#224902
C4×#4622
#211692