Finite Group Databasebeta

← #6497264

Subgroups — #6497264

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic
3#33153C214yesmaximal, complemented
4#29942C214yesmaximal, complemented
5#27962C212yesmaximal, complemented
6#27949C212yesmaximal, complemented
7#27877C212yesmaximal, complemented
8#27876C212yesmaximal, complemented
9#27617C214yesmaximal, complemented
10#27604C214yesmaximal
11#21982C211yesmaximal, characteristic, complemented
12#21981C211yesmaximal, characteristic, complemented
13#21834C211yesmaximal, characteristic, complemented
14#21834C211yesmaximal, characteristic, complemented
15#20714C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C42C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2857—21no—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29C2×#391—21no—
30C2×#389—21no—
31C2×#389C2212yes—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2283—21no—
40#2283—21no—
41#2283—21no—
42#2283—21no—
43#2283C2212yes—
44#2283C2212yes—
45#2280—21no—
46#2280—21no—
47#2280C2212yes—
48C2×#380—21no—
49C2×#380C2212yes—
50C4×#117—21no—
51C4×#117—21no—
52C4×#117C2212yes—
53#2101C2218yescomplemented
54#2085C2218yescomplemented
55#2077C2218yes—
56#2075C2218yes—
57#1990C2212yescomplemented
58#1990C2212yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1987C2214yescomplemented
62#1987C2214yescomplemented
63#1948C2214yescomplemented
64#1947C2214yescomplemented
65#1919C2218yescomplemented
66#1918C2218yescomplemented
67#1854C2218yes—
68#1853C2214yes—
69#1852C2214yes—
70#1845C2212yescomplemented
71#1844C2212yescomplemented
72#1834C2212yescomplemented
73#1834C2212yescomplemented
74#1825C2218yes—
75#1824C2214yes—
76#1824C2214yes—
77#1817C2212yes—
78#1797C2212yes—
79#1797C2212yes—
80#1779C2212yes—
81C4×C22⋊C8C2212yes—
82C4×C22⋊C8C2212yes—
83C23×Q8—21no—
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—48no—
98C2×#117—21no—
99C2×#117—21no—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—22no—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#431—24no—
126#431—24no—
127C2×#107C2318yes—
128C2×#106C2318yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389C2314yes—
148#389C2314yes—
149#384—21no—
150#384—21no—
151#384C2312yes—
152#384C2312yes—
153#384C2311yescharacteristic
154#384C2311yescharacteristic
155#384C2311yescharacteristic
156#384C2311yescharacteristic
157#383—22no—
158#383—22no—
159#382—21no—
160#382—21no—
161#382—21no—
162#382—21no—
163#382C2312yes—
164#382C2312yes—
165#381—22no—
166#381—22no—
167#381—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380—21no—
175#380—21no—
176#380—21no—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380C2312yes—
183#380C2314yes—
184#380C2312yes—
185#380C2312yes—
186#380C2312yes—
187#380C2314yes—
188#379—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207#376—22no—
208#376—22no—
209#376—22no—
210C2×#95—22no—
211C2×#95—22no—
212C43—22no—
213#340C2318yes—
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#340C2318yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—44no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—42no—
238C23×C4—42no—
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×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269#117—28no—
270#117—44no—
271#117—44no—
272#117—44no—
273#117—44no—
274#117—44no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
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—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—48no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
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×C42#4114yes—
350C2×C42#4114yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—48no—
356#106—48no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—28no—
362C22⋊C8—44no—
363C4×C8—24no—
364C4×C8—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—816no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
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—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4#12314yes—
464C22×C4#12314yes—
465C22×C4#13514yes—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4#13714yes—
468C22×C4C22×D411yescharacteristic
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×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C42−1+414yes—
482C22×C4C2511yescharacteristic
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42—44no—
542C42#12314yes—
543C42#12314yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—28no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23C2×#12312yes—
572C23#53612yes—
573C23C2×#13512yes—
574C23C2×AGL(1,ℤ/8)12yes—
575C23C2×#13712yes—
576C23C23×D411yescharacteristic
577Q8—88no—
578Q8—88no—
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—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—44no—
606C2×C4—44no—
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—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4#485116yes—
640C2×C4C2×#12312yes—
641C2×C4C2×#12312yes—
642C2×C4C2×#12312yes—
643C2×C4C2×#12312yes—
644C2×C4C2×#12312yes—
645C2×C4#53612yes—
646C2×C4C2×#13512yes—
647C2×C4C2×#13512yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4C2×#13712yes—
651C2×C4C2×#13712yes—
652C2×C4#57512yes—
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×2−1+412yes—
656C2×C4#58512yes—
657C8—44no—
658C8—44no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—416no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22C2×#48514yes—
681C22C22×#12311yescharacteristic
682C22C2×#53611yescharacteristic
683C22#344911yescharacteristic
684C22C22×#13511yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C22×#13711yescharacteristic
687C22C2×#57511yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4C2×#48514yes—
703C4C2×#48514yes—
704C4#309414yes—
705C4#349512yes—
706C4#349512yes—
707C2—24no—
708C2—42no—
709C2—42no—
710C2#3531111yescharacteristic
711C2C22×#48511yescharacteristic
712C2C2×#309411yescharacteristic
713C2#6084611yescharacteristic
714C2#6086011yescharacteristic
715C2#6094611yescharacteristic
716C2#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. #33153C2semidirect
4. #29942C2semidirect
5. #27962C2semidirect
6. #27949C2semidirect
7. #27877C2semidirect
8. #27876C2semidirect
9. #27617C2semidirect
11. #21982C2semidirect
12. #21981C2semidirect
13. #21834C2semidirect
14. #21834C2semidirect
53. #2101C22semidirect
54. #2085C22semidirect
57. #1990C22semidirect
58. #1990C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1987C22semidirect
62. #1987C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
65. #1919C22semidirect
66. #1918C22semidirect
70. #1845C22semidirect
71. #1844C22semidirect
72. #1834C22semidirect
73. #1834C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#340122
#331532
#299422
#279622
#279492
#278772
#278762
#276172
#276042
#219822
#219812
#218342
#207142