Finite Group Databasebeta

← #6458909

Subgroups — #6458909

Order 512.

2104 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#27615C214yesmaximal, complemented
4#27614C212yesmaximal, complemented
5#27613C212yesmaximal, complemented
6#27612C214yesmaximal, complemented
7#27611C212yesmaximal, complemented
8#27610C212yesmaximal, complemented
9#23745C212yesmaximal, complemented
10#23658C212yesmaximal, complemented
11#21033C214yesmaximal, complemented
12#20846C212yesmaximal
13#20702C212yesmaximal
14#20682C211yesmaximal, characteristic
15C22×#117—21no—
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#428C2214yes—
20C2×#407C2212yes—
21#2296—21no—
22#2296—21no—
23#2296—21no—
24#2296—21no—
25#2296C2212yes—
26#2296C2212yes—
27#2296C2212yes—
28#2296C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#385—21no—
34C2×#385—21no—
35#2285—21no—
36#2285—21no—
37#2285—21no—
38#2285—21no—
39#2285C2212yes—
40#2285C2212yes—
41#2285C2212yes—
42#2285C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#382—21no—
46C2×#380—21no—
47C2×#380C2212yes—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2212yes—
52#2100C2214yescomplemented
53#2100C2214yescomplemented
54#2099C2214yescomplemented
55#2099C2214yescomplemented
56#2084C2214yescomplemented
57#2084C2214yescomplemented
58#2083C2214yescomplemented
59#2083C2214yescomplemented
60#1921C2218yescomplemented
61#1920C2218yescomplemented
62#1919C2218yescomplemented
63#1918C2218yescomplemented
64#1854C2218yes—
65#1853C2214yes—
66#1852C2214yes—
67#1825C2218yes—
68#1824C2214yes—
69#1823C2214yes—
70#1817C2212yes—
71#1817C2212yes—
72#1817C2212yes—
73#1817C2212yes—
74#1736C2214yes—
75#1736C2214yes—
76#1736C2214yes—
77#1728C2212yes—
78#1728C2212yes—
79C24×C4D411yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C22×M16C2314yes—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117—28no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—28no—
93C2×#117—48no—
94C2×#117C2312yes—
95C2×#117C2312yes—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42—28no—
117C22×C42—22no—
118C22×C42—21no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#428—216no—
132C2×#107C2318yes—
133C2×#106C2318yes—
134#407—28no—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#385—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#385—22no—
160#385—22no—
161#384—21no—
162#384—21no—
163#384—21no—
164#384—21no—
165#384C2312yes—
166#384C2312yes—
167#384C2312yes—
168#384C2312yes—
169#382—22no—
170#382—22no—
171#382—22no—
172#382—22no—
173#380—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380C2314yes—
180#380C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2314yes—
197C2×#95—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—22no—
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
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×C4#4118yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×M16—28no—
248C2×M16—28no—
249C2×M16—28no—
250C22×C8C2418yes—
251#117—28no—
252#117—28no—
253#117—24no—
254#117—24no—
255#117—24no—
256#117—28no—
257#117—28no—
258#117—48no—
259#117—48no—
260#117—48no—
261#117—44no—
262#117—44no—
263#117—48no—
264#117#4118yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
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—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—44no—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—48no—
336C2×C42—44no—
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×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—216no—
354#106—216no—
355C22⋊C8—28no—
356C22⋊C8—28no—
357#95—24no—
358#95—24no—
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—24no—
368#95—24no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—41no—
374C24—41no—
375C24#12314yes—
376C24C22×D411yescharacteristic
377C24C22×D411yescharacteristic
378C24C22×D411yescharacteristic
379C22×C4—232no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
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—232no—
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—24no—
411C22×C4—24no—
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—24no—
426C22×C4—24no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—416no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4#12314yes—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4AGL(1,ℤ/8)14yes—
454C22×C4#13714yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C42−1+414yes—
471C22×C4C2511yescharacteristic
472M16—416no—
473C2×C8—28no—
474C2×C8—28no—
475C2×C8—28no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—28no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
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—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4C2×#4114yes—
505C4⋊C4C2×#4114yes—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C42—28no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—28no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—48no—
531C42—44no—
532C42—44no—
533C42—48no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—28no—
543C23—21no—
544C23—21no—
545C23—28no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23C2×#12312yes—
556C23C2×#12312yes—
557C23C2×#12312yes—
558C23#52512yes—
559C23#53612yes—
560C23C2×AGL(1,ℤ/8)12yes—
561C23C2×#13712yes—
562C23C23×D411yescharacteristic
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
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—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4C2×#12312yes—
627C2×C4C2×#12312yes—
628C2×C4C2×#12312yes—
629C2×C4#52512yes—
630C2×C4#53612yes—
631C2×C4#552116yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×#13712yes—
635C2×C4C2×#13712yes—
636C2×C4#57512yes—
637C2×C4#57512yes—
638C2×C4C22×#4112yes—
639C2×C4C22×#4112yes—
640C2×C4C2×2−1+412yes—
641C8—48no—
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—22no—
657C22—416no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22#248414yes—
663C22#309214yes—
664C22C22×#12311yescharacteristic
665C22C2×#52511yescharacteristic
666C22C2×#53611yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57511yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—44no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—84no—
686C4—84no—
687C4#248414yes—
688C4C2×#55214yes—
689C2—24no—
690C2—24no—
691C2—42no—
692C2C2×#248411yescharacteristic
693C2C2×#309211yescharacteristic
694C2#6051111yescharacteristic
695C2#6056111yescharacteristic
696C2#6056211yescharacteristic
697C2#6086411yescharacteristic
698C2#6086511yescharacteristic
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. #27615C2semidirect
4. #27614C2semidirect
5. #27613C2semidirect
6. #27612C2semidirect
7. #27611C2semidirect
8. #27610C2semidirect
9. #23745C2semidirect
10. #23658C2semidirect
11. #21033C2semidirect
52. #2100C22semidirect
53. #2100C22semidirect
54. #2099C22semidirect
55. #2099C22semidirect
56. #2084C22semidirect
57. #2084C22semidirect
58. #2083C22semidirect
59. #2083C22semidirect
60. #1921C22semidirect
61. #1920C22semidirect
62. #1919C22semidirect
63. #1918C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#22852
#276152
#276142
#276132
#276122
#276112
#276102
#237452
#236582
#210332
#208462
#207022
#206822