Finite Group Databasebeta

← #6489130

Subgroups — #6489130

Order 512.

2074 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#32906C212yesmaximal, complemented
4#32905C212yesmaximal
5#30046C214yesmaximal, complemented
6#29148C212yesmaximal, complemented
7#28449C212yesmaximal, complemented
8#28159C214yesmaximal, complemented
9#27983C212yesmaximal, complemented
10#25730C212yesmaximal
11#23934C212yesmaximal, complemented
12#23933C212yesmaximal
13#22273C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C2×#533C2212yes—
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#497C2214yes—
20C2×#462C2214yescomplemented
21C2×#461C2214yes—
22C2×#451C2214yes—
23C2×#450C2214yes—
24#2868C2211yescharacteristic, complemented
25#2868C2211yescharacteristic, complemented
26#2862C2212yescomplemented
27#2859—21no—
28#2859—21no—
29#2857—21no—
30#2857—21no—
31#2857C2212yes—
32#2857C2212yes—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2212yescomplemented
36C4×C4⋊Q8C2212yescomplemented
37C4×#124—21no—
38C4×#124—21no—
39C4×#124—21no—
40C4×#124—21no—
41C4×#124C2212yes—
42C4×#124C2212yes—
43C4×#124C2212yes—
44C4×#124C2212yes—
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49#2285—21no—
50#2285—21no—
51C2×#382C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4—21no—
54C2×C43C2211yescharacteristic
55#2074C2218yescomplemented
56#2024C2214yescomplemented
57#2023C2214yes—
58#2016C2214yescomplemented
59#2015C2218yes—
60#2013C2214yes—
61#1960C2214yescomplemented
62#1959C2214yes—
63#1880C2218yes—
64#1878C2218yescomplemented
65#1878C2218yescomplemented
66#1877C2218yes—
67#1872C2214yes—
68#1872C2214yes—
69#1839C2212yescomplemented
70#1836C2214yescomplemented
71#1831C2212yescomplemented
72#1817C2212yes—
73#1803C2214yescomplemented
74#1802C2214yes—
75#1797C2212yes—
76#1767C2212yescomplemented
77#1749C2212yes—
78#1745C2212yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×Q16C2318yes—
83C22×SD16C2318yes—
84#533—48no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#125—22no—
92C2×#124—28no—
93C2×#124—21no—
94C2×#124—21no—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8C2314yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—44no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C42—21no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#497—216no—
129#462—48no—
130#461—48no—
131#451—48no—
132#450—48no—
133C2×C4⋊C8C2314yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×#102C2318yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140#391—22no—
141#391—22no—
142#391—21no—
143#391—21no—
144#391C2314yes—
145#391C2314yes—
146#391C2312yes—
147#391C2312yes—
148#390—21no—
149#390—21no—
150#390C2312yes—
151#390C2312yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#384—21no—
169#384—21no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384—21no—
174#384C2312yes—
175#384C2312yes—
176#382—22no—
177#382—22no—
178#382—21no—
179#382—21no—
180#382C2312yes—
181#382C2312yes—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25—21no—
217C22×Q8—24no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—42no—
222C22×Q8C2414yes—
223C22×Q8C2412yes—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—24no—
227C22×D4—42no—
228C22×D4C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×Q16—216no—
245C2×Q16—48no—
246C2×Q16—48no—
247C2×SD16—216no—
248C2×SD16—48no—
249C2×SD16—48no—
250C22×C8C2418yes—
251C4⋊Q8—28no—
252C4⋊Q8—28no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257#125—48no—
258#124—28no—
259#124—28no—
260#124—24no—
261#124—24no—
262#124—24no—
263#124#4118yes—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—48no—
267C4×Q8—28no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×D4—28no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—44no—
278#117—48no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—44no—
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×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C4⋊C8—28no—
358#103—48no—
359#103—48no—
360#103—48no—
361#102—48no—
362C22⋊C8—28no—
363#97—28no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376C24—24no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—416no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8C2×#4114yes—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—416no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C2×#4114yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—816no—
447C22×C4#13714yes—
448C22×C4#13714yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4H(5,𝔽2)14yes—
463C22×C4C2511yescharacteristic
464Q16—816no—
465SD16—816no—
466C2×C8—28no—
467C2×C8—28no—
468C2×C8—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—44no—
528C42—48no—
529C42—44no—
530C42C2×#4114yes—
531C42H(5,𝔽2)14yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23#53512yes—
551C23C2×#13712yes—
552C23C2×#13712yes—
553C23C23×D411yescharacteristic
554Q8—28no—
555Q8—48no—
556Q8—44no—
557Q8—48no—
558Q8—48no—
559Q8—88no—
560D4—28no—
561D4—48no—
562D4—44no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4#53512yes—
621C2×C4#53512yes—
622C2×C4#53512yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4C2×#13712yes—
627C2×C4#57512yes—
628C2×C4#57512yes—
629C2×C4C22×#4112yes—
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4C2×H(5,𝔽2)12yes—
632C8—48no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—816no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22#321814yes—
648C22C2×#52511yescharacteristic
649C22C2×#53511yescharacteristic
650C22C2×#53511yescharacteristic
651C22C22×#13711yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57511yescharacteristic
654C22C2×#57511yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—44no—
668C4—82no—
669C4—82no—
670C4—84no—
671C4—84no—
672C4#321514yes—
673C4#351914yes—
674C2—24no—
675C2—84no—
676C2#3523711yescharacteristic
677C2#6056211yescharacteristic
678C2#6056211yescharacteristic
679C2C2×#321511yescharacteristic
680C2C2×#321811yescharacteristic
681C2#6080411yescharacteristic
682C2#6080511yescharacteristic
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. #32906C2semidirect
5. #30046C2semidirect
6. #29148C2semidirect
7. #28449C2semidirect
8. #28159C2semidirect
9. #27983C2semidirect
11. #23934C2semidirect
13. #22273C2semidirect
14. #20778C2semidirect
20. C2×#462C22semidirect
24. #2868C22semidirect
25. #2868C22semidirect
26. #2862C22semidirect
35. C4×C4⋊Q8C22semidirect
36. C4×C4⋊Q8C22semidirect
55. #2074C22semidirect
56. #2024C22semidirect
58. #2016C22semidirect
61. #1960C22semidirect
64. #1878C22semidirect
65. #1878C22semidirect
69. #1839C22semidirect
70. #1836C22semidirect
71. #1831C22semidirect
73. #1803C22semidirect
76. #1767C22semidirect
78. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×C4×#1242
#329062
#329052
#300462
#291482
#284492
#281592
#279832
#257302
#239342
#239332
#222732
#207782