Finite Group Databasebeta

← #6661062

Subgroups — #6661062

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#32741C214yesmaximal, complemented
4#30296C212yesmaximal, complemented
5#30199C212yesmaximal, complemented
6#29966C212yesmaximal, complemented
7#29938C212yesmaximal, complemented
8#29756C212yesmaximal, complemented
9#29691C212yesmaximal, complemented
10#27647C212yesmaximal, complemented
11#24846C214yesmaximal, complemented
12#24559C214yesmaximal
13#24302C212yesmaximal, complemented
14#20626C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C42C2211yescharacteristic
19C2×#483C2214yes—
20C2×#477C2214yescomplemented
21C2×#423C2212yes—
22C2×#407C2212yes—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2857C2212yes—
30#2857C2212yes—
31#2857C2212yes—
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2855—21no—
36#2855C2212yes—
37#2851—21no—
38#2851—21no—
39#2848—21no—
40#2848C2212yes—
41C4×C22⋊Q8—21no—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8C2212yes—
44C4×C22⋊Q8C2212yes—
45#2296—21no—
46C2×#391—21no—
47C2×#389—21no—
48C2×#389C2212yes—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2280—21no—
52C2×#380—21no—
53C2×#380C2212yes—
54#2099C2214yescomplemented
55#2086C2218yescomplemented
56#2084C2214yescomplemented
57#2084C2214yescomplemented
58#2077C2218yes—
59#2072C2214yescomplemented
60#2066C2218yescomplemented
61#2057C2214yescomplemented
62#2057C2214yescomplemented
63#2043C2214yescomplemented
64#2036C2218yes—
65#2024C2214yescomplemented
66#1987C2214yescomplemented
67#1948C2214yescomplemented
68#1894C2218yes—
69#1872C2214yes—
70#1852C2214yes—
71#1823C2214yes—
72#1809C2218yes—
73#1803C2214yescomplemented
74#1797C2212yes—
75#1797C2212yes—
76#1797C2212yes—
77#1797C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84#533—44no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×#125—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—48no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
106C2×#117—21no—
107C2×#117—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C42—44no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#483—48no—
129#477—48no—
130C2×#107C2318yes—
131C2×#106C2318yes—
132#423—44no—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137#407—44no—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140#391—21no—
141#391—21no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391C2312yes—
147#391C2312yes—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#389C2312yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#384—21no—
175#384—21no—
176#384C2312yes—
177#384C2312yes—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#382—21no—
181#381—22no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200#379—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C2×#95—22no—
208C2×#95—22no—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C25C2×D411yescharacteristic
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×Q8—44no—
222C22×Q8C2412yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
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×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—28no—
254C2×M16—44no—
255C2×M16—44no—
256C22×C8C2418yes—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263#125—44no—
264C22⋊Q8—24no—
265C22⋊Q8—28no—
266C22⋊Q8—24no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270C22⋊Q8—48no—
271C22⋊Q8—48no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C4×Q8—28no—
276C4×Q8—44no—
277#117—44no—
278#117—44no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—48no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
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—48no—
354#106—48no—
355C4⋊C8—28no—
356C4⋊C8—44no—
357#103—48no—
358#103—48no—
359C22⋊C8—28no—
360C22⋊C8—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
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—
374#95—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—41no—
381C24—41no—
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—816no—
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—21no—
409C22×C4—22no—
410C22×C4—21no—
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—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4#13714yes—
468C22×C4#13714yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+414yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484M16—88no—
485C2×C8—28no—
486C2×C8—28no—
487C2×C8—28no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—44no—
542C42—44no—
543C42H(5,𝔽2)14yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—28no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23#53612yes—
568C23#53712yes—
569C23C2×AGL(1,ℤ/8)12yes—
570C23C2×#13712yes—
571C23C2×#13712yes—
572C23C23×D411yescharacteristic
573Q8—44no—
574Q8—44no—
575Q8—84no—
576Q8—88no—
577Q8—88no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
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—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4#53412yes—
642C2×C4#53412yes—
643C2×C4#53612yes—
644C2×C4#53712yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4C2×#13712yes—
648C2×C4C2×#13712yes—
649C2×C4C2×#13712yes—
650C2×C4C2×#13712yes—
651C2×C4#57512yes—
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4C2×2−1+412yes—
655C2×C4C2×2−1+411yescharacteristic
656C8—48no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—416no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22#325414yes—
679C22C2×#53411yescharacteristic
680C22C2×#53611yescharacteristic
681C22C2×#53711yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C22×#13711yescharacteristic
684C22C22×#13711yescharacteristic
685C22C2×#57511yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4#319514yes—
703C4#353612yes—
704C4#353812yes—
705C2—24no—
706C2—42no—
707C2—42no—
708C2#3744211yescharacteristic
709C2C2×#319511yescharacteristic
710C2#6075011yescharacteristic
711C2#6086011yescharacteristic
712C2#6086511yescharacteristic
713C2C2×#325411yescharacteristic
714C2#6091811yescharacteristic
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. #32741C2semidirect
4. #30296C2semidirect
5. #30199C2semidirect
6. #29966C2semidirect
7. #29938C2semidirect
8. #29756C2semidirect
9. #29691C2semidirect
10. #27647C2semidirect
11. #24846C2semidirect
13. #24302C2semidirect
20. C2×#477C22semidirect
54. #2099C22semidirect
55. #2086C22semidirect
56. #2084C22semidirect
57. #2084C22semidirect
59. #2072C22semidirect
60. #2066C22semidirect
61. #2057C22semidirect
62. #2057C22semidirect
63. #2043C22semidirect
65. #2024C22semidirect
66. #1987C22semidirect
67. #1948C22semidirect
73. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#327412
#302962
#301992
#299662
#299382
#297562
#296912
#276472
#248462
#245592
#243022
#206262