Finite Group Databasebeta

← #6458959

Subgroups — #6458959

Order 512.

2100 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#27681C212yesmaximal, complemented
4#27647C212yesmaximal, complemented
5#27631C214yesmaximal, complemented
6#27613C212yesmaximal, complemented
7#27600C212yesmaximal
8#27586C212yesmaximal, complemented
9#24003C212yesmaximal
10#23762C214yesmaximal, complemented
11#21579C214yesmaximal, complemented
12#21365C212yesmaximal, complemented
13#20740C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×#117—21no—
18C23×C42C2211yescharacteristic
19C2×#483C2214yes—
20C2×#432C2212yes—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8C2212yes—
26C4×C22⋊Q8C2212yes—
27C4×C22⋊Q8C2212yes—
28C4×C22⋊Q8C2212yes—
29C2×#391C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#387—21no—
32C2×#385—21no—
33#2285—21no—
34#2285—21no—
35#2285—21no—
36#2285—21no—
37#2285C2212yes—
38#2285C2212yes—
39#2285C2212yes—
40#2285C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382—21no—
44C2×#380—21no—
45C2×#380C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52#2100C2214yescomplemented
53#2095C2214yescomplemented
54#2084C2214yescomplemented
55#2075C2218yes—
56#2072C2214yescomplemented
57#2035C2218yes—
58#1948C2214yescomplemented
59#1947C2214yescomplemented
60#1928C2218yescomplemented
61#1919C2218yescomplemented
62#1872C2214yes—
63#1853C2214yes—
64#1852C2214yes—
65#1852C2214yes—
66#1823C2214yes—
67#1823C2214yes—
68#1817C2212yes—
69#1803C2214yescomplemented
70#1803C2214yescomplemented
71#1797C2212yes—
72#1797C2212yes—
73#1787C2214yes—
74#1779C2212yes—
75#1776C2218yescomplemented
76#1763C2218yescomplemented
77#1749C2212yes—
78#1749C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8C2312yes—
87C2×C22⋊Q8C2312yes—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—28no—
91C2×C4×Q8C2312yes—
92C2×C4×Q8C2312yes—
93C2×#117—22no—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117—48no—
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⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—21no—
113C22×C42—28no—
114C22×C42—22no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#483—216no—
130#432—28no—
131C2×#107C2318yes—
132C2×#106C2318yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140#391—21no—
141#391—21no—
142#391C2312yes—
143#391C2312yes—
144#389—21no—
145#389—21no—
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#384—21no—
157#384—21no—
158#384—21no—
159#384—21no—
160#384C2312yes—
161#384C2312yes—
162#384C2312yes—
163#384C2312yes—
164#382—22no—
165#382—22no—
166#382—22no—
167#382—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#380C2312yes—
177#380C2314yes—
178#380C2312yes—
179#380C2314yes—
180C4×C4⋊C4—22no—
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—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—22no—
206C2×#95—21no—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C22×Q8—24no—
215C22×Q8—21no—
216C22×Q8—24no—
217C22×Q8—42no—
218C22×Q8C2412yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—42no—
233C23×C4—42no—
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×C8C2418yes—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8#4118yes—
259C4×Q8—28no—
260C4×Q8—28no—
261C4×Q8—44no—
262C4×Q8—44no—
263#117—48no—
264#117—48no—
265#117—48no—
266#117—48no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
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—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—44no—
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—22no—
328C2×C42—22no—
329C2×C42—48no—
330C2×C42—44no—
331C2×C42—44no—
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×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4118yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—48no—
350#106—216no—
351C4⋊C8—28no—
352#103—216no—
353#103—48no—
354C22⋊C8—28no—
355C22⋊C8—28no—
356C22⋊C8—44no—
357#97—28no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—41no—
377C24—41no—
378C24AGL(1,ℤ/8)14yes—
379C24C22×D411yescharacteristic
380C24C22×D411yescharacteristic
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—416no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8C2×#4114yes—
390C22×C4—232no—
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—24no—
416C22×C4—24no—
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—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4#12314yes—
466C22×C4#12314yes—
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×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4C2511yescharacteristic
483C2×C8—28no—
484C2×C8—28no—
485C2×C8—28no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—22no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4C2×#4114yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—48no—
531C42—48no—
532C42—44no—
533C42—44no—
534C42—44no—
535C42#12314yes—
536C42#12314yes—
537C42C2×#4114yes—
538C42C2×#4114yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—28no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23C2×#12312yes—
561C23#52512yes—
562C23C2×AGL(1,ℤ/8)12yes—
563C23C2×AGL(1,ℤ/8)12yes—
564C23C2×AGL(1,ℤ/8)12yes—
565C23C2×#13712yes—
566C23C2×#13712yes—
567C23C23×D411yescharacteristic
568Q8—28no—
569Q8—48no—
570Q8—44no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—21no—
578C2×C4—21no—
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—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—24no—
600C2×C4—24no—
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—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—88no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4C2×#12312yes—
636C2×C4C2×#12312yes—
637C2×C4C2×#12312yes—
638C2×C4C2×#12312yes—
639C2×C4C2×#12312yes—
640C2×C4#52512yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4C2×#13712yes—
644C2×C4C2×#13712yes—
645C2×C4#57512yes—
646C2×C4C22×#4112yes—
647C2×C4C22×#4112yes—
648C2×C4#58512yes—
649C8—48no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—416no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22#309214yes—
671C22#311214yes—
672C22C22×#12311yescharacteristic
673C22C2×#52511yescharacteristic
674C22#344911yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C22×#13711yescharacteristic
677C22C22×#13711yescharacteristic
678C22C2×#57511yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—41no—
688C4—44no—
689C4—41no—
690C4—44no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4—84no—
695C4#309514yes—
696C4#347514yes—
697C2—24no—
698C2—24no—
699C2—42no—
700C2#3480911yescharacteristic
701C2C2×#309211yescharacteristic
702C2C2×#309511yescharacteristic
703C2C2×#311211yescharacteristic
704C2#6056211yescharacteristic
705C2#6095811yescharacteristic
706C2#6097111yescharacteristic
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. #27681C2semidirect
4. #27647C2semidirect
5. #27631C2semidirect
6. #27613C2semidirect
8. #27586C2semidirect
10. #23762C2semidirect
11. #21579C2semidirect
12. #21365C2semidirect
13. #20740C2semidirect
52. #2100C22semidirect
53. #2095C22semidirect
54. #2084C22semidirect
56. #2072C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1928C22semidirect
61. #1919C22semidirect
69. #1803C22semidirect
70. #1803C22semidirect
75. #1776C22semidirect
76. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#22852
#276812
#276472
#276312
#276132
#276002
#275862
#240032
#237622
#215792
#213652
#207402
#207152