Finite Group Databasebeta

← #6497271

Subgroups — #6497271

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#33150C214yesmaximal, complemented
4#29944C214yesmaximal, complemented
5#27961C211yesmaximal, characteristic, complemented
6#27961C211yesmaximal, characteristic, complemented
7#27950C211yesmaximal, characteristic, complemented
8#27948C211yesmaximal, characteristic, complemented
9#27877C212yesmaximal, complemented
10#27875C212yesmaximal, complemented
11#27612C214yesmaximal, complemented
12#27604C214yesmaximal
13#21984C212yesmaximal, complemented
14#21835C212yesmaximal, complemented
15#20714C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#117—21no—
18C23×C42C2211yescharacteristic
19C2×#431C2212yes—
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2855—21no—
25#2855—21no—
26#2851C2212yes—
27#2851C2212yes—
28#2848—21no—
29#2848—21no—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32C4×C22⋊Q8—21no—
33C4×C22⋊Q8—21no—
34#2296C2212yes—
35#2285—21no—
36#2285—21no—
37#2285—21no—
38#2285—21no—
39#2285C2212yes—
40#2285C2212yes—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47#2280C2212yes—
48C2×#380—21no—
49C2×#380C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2212yes—
52#2100C2214yescomplemented
53#2100C2214yescomplemented
54#2084C2214yescomplemented
55#2083C2214yescomplemented
56#2077C2218yes—
57#2075C2218yes—
58#1990C2212yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1990C2212yescomplemented
62#1989C2218yescomplemented
63#1949C2218yescomplemented
64#1920C2218yescomplemented
65#1918C2218yescomplemented
66#1854C2218yes—
67#1853C2214yes—
68#1852C2214yes—
69#1845C2212yescomplemented
70#1844C2212yescomplemented
71#1834C2212yescomplemented
72#1834C2212yescomplemented
73#1825C2218yes—
74#1824C2214yes—
75#1824C2214yes—
76#1817C2212yes—
77#1797C2212yes—
78#1797C2212yes—
79#1779C2212yes—
80C4×C22⋊C8C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82C4×C22⋊C8C2212yes—
83C24×C4D411yescharacteristic
84C24×C4D411yescharacteristic
85C24×C4C2311yescharacteristic
86#533—28no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#125C2312yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—48no—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C42—28no—
113C22×C42—22no—
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#431—28no—
127C2×#107C2318yes—
128C2×#106C2318yes—
129C2×C4⋊C8C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2312yes—
136#391—21no—
137#391—21no—
138#391—21no—
139#391—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389C2312yes—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#385—22no—
153#385—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#384—21no—
159#384—21no—
160#384—21no—
161#384—21no—
162#384C2312yes—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2312yes—
166#384C2312yes—
167#384C2312yes—
168#382—22no—
169#382—22no—
170#382—22no—
171#382—22no—
172#382C2312yes—
173#381—22no—
174#380—21no—
175#380—21no—
176#380—21no—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380C2312yes—
187#380C2314yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#380C2314yes—
192#379—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C2×#95—22no—
208C2×#95—22no—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#340C2318yes—
215#336C2312yes—
216#336C2312yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C22×Q8—21no—
220C22×Q8—21no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
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×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254#125—28no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263#117—28no—
264#117—44no—
265#117—48no—
266#117—48no—
267#117—48no—
268#117—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
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—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—48no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
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—22no—
326C2×C42—22no—
327C2×C42—48no—
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×C42#4118yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—48no—
346#106—48no—
347C4⋊C8—28no—
348C22⋊C8—28no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351C22⋊C8—44no—
352C4×C8—24no—
353C4×C8—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—41no—
374C24—41no—
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C24C22×D411yescharacteristic
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C22×C4—232no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
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—22no—
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—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4#12314yes—
447C22×C4#12314yes—
448C22×C4#13514yes—
449C22×C4AGL(1,ℤ/8)14yes—
450C22×C4#13714yes—
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×C4C2×#4114yes—
463C22×C42−1+414yes—
464C22×C4C2511yescharacteristic
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
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—48no—
520C42—44no—
521C42—44no—
522C42—48no—
523C42#12314yes—
524C42#12314yes—
525C42C2×#4114yes—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—28no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23C2×#12312yes—
551C23#53612yes—
552C23C2×#13512yes—
553C23C2×AGL(1,ℤ/8)12yes—
554C23C2×#13712yes—
555C23C23×D411yescharacteristic
556Q8—84no—
557Q8—84no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
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—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—88no—
615C2×C4—88no—
616C2×C4C2×#12312yes—
617C2×C4C2×#12312yes—
618C2×C4C2×#12312yes—
619C2×C4C2×#12312yes—
620C2×C4C2×#12312yes—
621C2×C4#53612yes—
622C2×C4C2×#13512yes—
623C2×C4C2×#13512yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4#57512yes—
629C2×C4C22×#4112yes—
630C2×C4C2×2−1+412yes—
631C2×C4#58512yes—
632C8—44no—
633C8—44no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—416no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22#323114yes—
656C22C22×#12311yescharacteristic
657C22C2×#53611yescharacteristic
658C22#344911yescharacteristic
659C22C22×#13511yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57511yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—84no—
678C4#309114yes—
679C4#309214yes—
680C4#349514yes—
681C2—24no—
682C2—42no—
683C2—42no—
684C2#3531111yescharacteristic
685C2C2×#309111yescharacteristic
686C2C2×#309211yescharacteristic
687C2C2×#323111yescharacteristic
688C2#6086411yescharacteristic
689C2#6094611yescharacteristic
690C2#6096611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33150C2semidirect
4. #29944C2semidirect
5. #27961C2semidirect
6. #27961C2semidirect
7. #27950C2semidirect
8. #27948C2semidirect
9. #27877C2semidirect
10. #27875C2semidirect
11. #27612C2semidirect
13. #21984C2semidirect
14. #21835C2semidirect
52. #2100C22semidirect
53. #2100C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
58. #1990C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1990C22semidirect
62. #1989C22semidirect
63. #1949C22semidirect
64. #1920C22semidirect
65. #1918C22semidirect
69. #1845C22semidirect
70. #1844C22semidirect
71. #1834C22semidirect
72. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#22852
#331502
#299442
#279612
#279502
#279482
#278772
#278752
#276122
#276042
#219842
#218352
#207142