Finite Group Databasebeta

← #6495603

Subgroups — #6495603

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#33108C212yesmaximal, complemented
4#32905C212yesmaximal
5#29981C212yesmaximal, complemented
6#29798C212yesmaximal, complemented
7#29554C214yesmaximal, complemented
8#29368C214yesmaximal, complemented
9#29329C212yesmaximal, complemented
10#28302C212yesmaximal
11#24063C212yesmaximal, complemented
12#23933C212yesmaximal
13#22146C214yesmaximal, complemented
14#20777C211yesmaximal, characteristic, complemented
15C2×#533C2212yes—
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18C2×#483C2214yes—
19C2×#461C2214yes—
20C2×#450C2214yes—
21#2868C2211yescharacteristic, complemented
22#2868C2211yescharacteristic, complemented
23#2862C2212yescomplemented
24#2859—21no—
25#2859—21no—
26#2857—21no—
27#2857—21no—
28#2857C2212yes—
29#2857C2212yes—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2212yescomplemented
33C4×C4⋊Q8C2212yescomplemented
34#2293—21no—
35#2293—21no—
36#2293—21no—
37#2293—21no—
38#2293C2212yes—
39#2293C2212yes—
40#2293C2212yes—
41#2293C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#388—21no—
48#2285—21no—
49#2285—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52C2×C43C2211yescharacteristic
53#2100C2214yescomplemented
54#2094C2214yes—
55#2058C2214yescomplemented
56#2043C2214yescomplemented
57#2043C2214yescomplemented
58#2023C2214yes—
59#2013C2214yes—
60#2013C2214yes—
61#1981C2214yescomplemented
62#1968C2218yes—
63#1948C2214yescomplemented
64#1933C2218yescomplemented
65#1927C2218yescomplemented
66#1880C2218yes—
67#1877C2218yes—
68#1872C2214yes—
69#1872C2214yes—
70#1865C2218yescomplemented
71#1839C2212yescomplemented
72#1831C2212yescomplemented
73#1817C2212yes—
74#1802C2214yes—
75#1797C2212yes—
76C4×C4⋊C8C2212yescomplemented
77C4×C22⋊C8C2212yes—
78#1744C2212yescomplemented
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16C2318yes—
82#533—48no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8C2314yes—
92C2×C4×Q8—22no—
93C2×C4×Q8C2314yes—
94C2×C4×D4—28no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2312yes—
98C2×C4×D4C2312yes—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—48no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#483—48no—
124#461—216no—
125#450—48no—
126C2×#106C2318yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#102C2318yes—
132C2×C22⋊C8C2314yes—
133C2×C4×C8C2314yes—
134#391—22no—
135#391—22no—
136#391C2314yes—
137#391C2314yes—
138#390—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390C2312yes—
143#390C2312yes—
144#390C2312yes—
145#390C2312yes—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384—21no—
174#384—21no—
175#384—21no—
176#384C2312yes—
177#384C2312yes—
178#382—22no—
179#382—22no—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25—21no—
214C22×Q8—22no—
215C22×Q8C2414yes—
216C22×D4—24no—
217C22×D4—21no—
218C22×D4—24no—
219C22×D4—42no—
220C22×D4C2412yes—
221C23×C4—28no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C2×SD16—216no—
237C2×SD16—48no—
238C2×SD16—48no—
239C22×C8C2418yes—
240C4⋊Q8—28no—
241C4⋊Q8—44no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246#125—48no—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C22⋊Q8—48no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×Q8—48no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4#4118yes—
259#117—48no—
260#117—48no—
261#117—48no—
262#117—48no—
263#117—48no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42SD1618yescomplemented
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#106—48no—
349C4⋊C8—28no—
350#103—216no—
351#103—48no—
352#103—48no—
353#102—48no—
354C22⋊C8—28no—
355C4×C8—44no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366C24—24no—
367C24—22no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24C22×D411yescharacteristic
375C2×Q8—28no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—28no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—416no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4C2×#4114yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—432no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4C2×SD1614yes—
428C22×C4C2×SD1614yes—
429C22×C4C2×SD1614yes—
430C22×C4#13714yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4H(5,𝔽2)14yes—
445C22×C4C2511yescharacteristic
446SD16—816no—
447C2×C8—28no—
448C2×C8—28no—
449C2×C8—28no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4C2×#4114yes—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—48no—
511C42—44no—
512C42—44no—
513C42C2×SD1614yes—
514C42C2×SD1614yes—
515C42C2×SD1614yes—
516C42C2×SD1614yes—
517C42C2×#4114yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—24no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23#53512yes—
537C23C22×SD1612yes—
538C23C2×#13712yes—
539C23C23×D411yescharacteristic
540Q8—48no—
541Q8—48no—
542Q8—88no—
543D4—28no—
544D4—48no—
545D4—44no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—416no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4#52512yes—
600C2×C4#52512yes—
601C2×C4#53512yes—
602C2×C4C22×SD1612yes—
603C2×C4C22×SD1612yes—
604C2×C4C22×SD1612yes—
605C2×C4C22×SD1612yes—
606C2×C4C22×SD1612yes—
607C2×C4C22×SD1612yes—
608C2×C4C2×#13712yes—
609C2×C4C2×#13712yes—
610C2×C4#57512yes—
611C2×C4#57512yes—
612C2×C4C22×#4112yes—
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4#58512yes—
615C8—48no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22#320614yes—
631C22C2×#52511yescharacteristic
632C22C2×#53511yescharacteristic
633C22#344911yescharacteristic
634C22C23×SD1611yescharacteristic
635C22C22×#13711yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57511yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—44no—
649C4—41no—
650C4—41no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4—84no—
655C4#310114yes—
656C4#349414yes—
657C2—24no—
658C2—84no—
659C2#3530111yescharacteristic
660C2C2×#310111yescharacteristic
661C2#6056211yescharacteristic
662C2C2×#320611yescharacteristic
663C2#6080511yescharacteristic
664C2#6094111yescharacteristic
665C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33108C2semidirect
5. #29981C2semidirect
6. #29798C2semidirect
7. #29554C2semidirect
8. #29368C2semidirect
9. #29329C2semidirect
11. #24063C2semidirect
13. #22146C2semidirect
14. #20777C2semidirect
21. #2868C22semidirect
22. #2868C22semidirect
23. #2862C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×C4⋊Q8C22semidirect
53. #2100C22semidirect
55. #2058C22semidirect
56. #2043C22semidirect
57. #2043C22semidirect
61. #1981C22semidirect
63. #1948C22semidirect
64. #1933C22semidirect
65. #1927C22semidirect
70. #1865C22semidirect
71. #1839C22semidirect
72. #1831C22semidirect
76. C4×C4⋊C8C22semidirect
78. #1744C22semidirect
319. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×#22932
#331082
#329052
#299812
#297982
#295542
#293682
#293292
#283022
#240632
#239332
#221462
#207772