Finite Group Databasebeta

← #6459043

Subgroups — #6459043

Order 512.

2024 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#27972C211yesmaximal, characteristic, complemented
4#27960C211yesmaximal, characteristic, complemented
5#27948C211yesmaximal, characteristic, complemented
6#27905C211yesmaximal, characteristic, complemented
7#27889C212yesmaximal, complemented
8#27875C212yesmaximal, complemented
9#27855C214yesmaximal, complemented
10#24551C214yesmaximal
11#23756C214yesmaximal, complemented
12C4×#483C214yesmaximal, complemented
13#21400C212yesmaximal, complemented
14#20816C212yesmaximal, complemented
15C2×C4×C22⋊C8C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#117—21no—
18C23×C42C2211yescharacteristic
19C2×#483C2214yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855—21no—
27#2855C2212yes—
28#2851—21no—
29#2851—21no—
30#2848—21no—
31#2848C2212yes—
32C4×C22⋊Q8—21no—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8C2212yes—
35C4×C22⋊Q8C2212yes—
36#2296—21no—
37#2285—21no—
38#2285—21no—
39#2285—21no—
40#2285—21no—
41#2285C2211yescharacteristic
42#2285C2212yes—
43#2285C2212yes—
44#2285C2212yes—
45#2285C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49#2280—21no—
50C2×#380—21no—
51C2×#380C2212yes—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2212yes—
54#2099C2214yescomplemented
55#2096C2214yescomplemented
56#2084C2214yescomplemented
57#2077C2218yes—
58#2072C2214yescomplemented
59#2035C2218yes—
60#1990C2212yescomplemented
61#1990C2212yescomplemented
62#1956C2212yescomplemented
63#1956C2212yescomplemented
64#1927C2218yescomplemented
65#1920C2218yescomplemented
66#1892C2218yes—
67#1872C2214yes—
68#1853C2214yes—
69#1844C2212yescomplemented
70#1844C2212yescomplemented
71#1834C2212yescomplemented
72#1834C2212yescomplemented
73#1823C2214yes—
74#1807C2218yes—
75C4×#106C2218yescomplemented
76C4×#103C2218yescomplemented
77C4×C22⋊C8C2211yescharacteristic
78C4×C22⋊C8C2211yescharacteristic
79C4×C22⋊C8C2212yes—
80C4×C22⋊C8C2212yes—
81C4×C22⋊C8C2212yes—
82C2×#336C2212yes—
83C24×C4D411yescharacteristic
84C24×C4D411yescharacteristic
85C24×C4C2311yescharacteristic
86C23×C8C2314yes—
87#533—28no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#125—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8—21no—
98C2×C4×Q8C2312yes—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—48no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113C22×C42—28no—
114C22×C42—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#483—216no—
128C2×#106C2318yes—
129C2×#106C2318yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132C2×C22⋊C8—28no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C4×C8—28no—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2314yes—
140#391—21no—
141#391—21no—
142#391C2312yes—
143#391C2312yes—
144#389—21no—
145#389—21no—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389C2312yes—
151#389C2312yes—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#385—22no—
160#385—22no—
161#385—22no—
162#384—21no—
163#384—21no—
164#384—21no—
165#384—21no—
166#384C2312yes—
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#384C2312yes—
170#384C2312yes—
171#384C2312yes—
172#382—21no—
173#382—22no—
174#382—22no—
175#382—22no—
176#382—22no—
177#381—22no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2314yes—
193#380C2312yes—
194#380C2314yes—
195#380C2312yes—
196#379—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C2×#95—22no—
212C2×#95—22no—
213#340C2318yes—
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25C2×D411yescharacteristic
221C22×Q8—21no—
222C22×Q8C2412yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4SD1618yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C22×C8—28no—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259#125—44no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C4×Q8—28no—
267C4×Q8—44no—
268#117—44no—
269#117—48no—
270#117—48no—
271#117—48no—
272#117—48no—
273#117—44no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
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—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—48no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—48no—
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×C42#4118yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#106—216no—
351#106—48no—
352#103—216no—
353#103—48no—
354C22⋊C8—24no—
355C22⋊C8—28no—
356C22⋊C8#4118yes—
357C4×C8—28no—
358C4×C8—24no—
359C4×C8—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
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—22no—
373#95—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—41no—
379C24—41no—
380C24C2×SD1614yes—
381C24C22×D411yescharacteristic
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—232no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
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—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4#12314yes—
448C22×C4#12314yes—
449C22×C4C2×SD1614yes—
450C22×C4C2×SD1614yes—
451C22×C4C2×SD1614yes—
452C22×C4C2×SD1614yes—
453C22×C4C2×SD1614yes—
454C22×C4C2×SD1614yes—
455C22×C4#13514yes—
456C22×C4#13714yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2511yescharacteristic
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—416no—
477C2×C8C2×#4118yes—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—44no—
527C42—48no—
528C42—48no—
529C42—44no—
530C42#12314yes—
531C42#12314yes—
532C42C2×#4114yes—
533C42C2×#4114yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—28no—
543C23—28no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23C2×#12312yes—
556C23#53212yes—
557C23C22×SD1612yes—
558C23C22×SD1612yes—
559C23C22×SD1612yes—
560C23C2×#13512yes—
561C23C2×#13712yes—
562C23C23×D411yescharacteristic
563Q8—48no—
564Q8—84no—
565C2×C4—28no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
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—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4#483116yescomplemented
621C2×C4C2×#12312yes—
622C2×C4C2×#12312yes—
623C2×C4C2×#12312yes—
624C2×C4C2×#12312yes—
625C2×C4C2×#12312yes—
626C2×C4#53212yes—
627C2×C4C22×SD1612yes—
628C2×C4C22×SD1612yes—
629C2×C4C22×SD1612yes—
630C2×C4C22×SD1612yes—
631C2×C4C2×#13512yes—
632C2×C4C2×#13512yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4#57512yes—
636C2×C4C22×#4112yes—
637C2×C4C22×#4112yes—
638C2×C4#58512yes—
639C8—28no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—416no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22C2×#48314yes—
662C22#316714yes—
663C22C22×#12311yescharacteristic
664C22C2×#53211yescharacteristic
665C22#344911yescharacteristic
666C22C23×SD1611yescharacteristic
667C22C22×#13511yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57511yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—41no—
677C4—41no—
678C4—84no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4C2×#48314yes—
685C4C2×#48314yes—
686C4#309114yes—
687C4#347514yes—
688C2—24no—
689C2—24no—
690C2—42no—
691C2#3481011yescharacteristic
692C2C22×#48311yescharacteristic
693C2C2×#309111yescharacteristic
694C2C2×#316711yescharacteristic
695C2#6070111yescharacteristic
696C2#6094111yescharacteristic
697C2#6094711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27972C2semidirect
4. #27960C2semidirect
5. #27948C2semidirect
6. #27905C2semidirect
7. #27889C2semidirect
8. #27875C2semidirect
9. #27855C2semidirect
11. #23756C2semidirect
12. C4×#483C2semidirect
13. #21400C2semidirect
14. #20816C2semidirect
54. #2099C22semidirect
55. #2096C22semidirect
56. #2084C22semidirect
58. #2072C22semidirect
60. #1990C22semidirect
61. #1990C22semidirect
62. #1956C22semidirect
63. #1956C22semidirect
64. #1927C22semidirect
65. #1920C22semidirect
69. #1844C22semidirect
70. #1844C22semidirect
71. #1834C22semidirect
72. #1834C22semidirect
75. C4×#106C22semidirect
76. C4×#103C22semidirect
620. C2×C4#483semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#22852
#279722
#279602
#279482
#279052
#278892
#278752
#278552
#245512
#237562
C4×#4832
#214002
#208162
C2×C4×C22⋊C82