Finite Group Databasebeta

← #6630454

Subgroups — #6630454

Order 512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#33147C212yesmaximal, complemented
4#33146C212yesmaximal, complemented
5#33109C212yesmaximal, complemented
6#33108C212yesmaximal, complemented
7#33100C214yesmaximal, complemented
8#32983C212yesmaximal
9#32982C212yesmaximal
10#21581C214yesmaximal, complemented
11#21579C214yesmaximal, complemented
12#21394C212yesmaximal, complemented
13#21365C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42—21no—
19C23×C42C2211yescharacteristic
20C2×#484C2214yes—
21C2×#483C2214yes—
22C2×#432C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855—21no—
28#2855C2212yes—
29#2851—21no—
30#2851—21no—
31#2848—21no—
32#2848C2212yes—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8C2212yes—
36C4×C22⋊Q8C2212yes—
37#2296—21no—
38#2285C2211yescharacteristic
39#2280—21no—
40C2×C4×C4⋊C4—21no—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4—21no—
44C2×C4×C4⋊C4—21no—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C22×#95—21no—
53#1988C2214yescomplemented
54#1981C2214yescomplemented
55#1948C2214yescomplemented
56#1947C2214yescomplemented
57#1928C2218yescomplemented
58#1927C2218yescomplemented
59#1919C2218yescomplemented
60#1918C2218yescomplemented
61#1872C2214yes—
62#1872C2214yes—
63#1872C2214yes—
64#1872C2214yes—
65#1853C2214yes—
66#1853C2214yes—
67#1852C2214yes—
68#1852C2214yes—
69#1824C2214yes—
70#1823C2214yes—
71#1817C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1779C2212yes—
75#1776C2218yescomplemented
76#1763C2218yescomplemented
77#1749C2212yes—
78#1749C2212yes—
79C24×C4—21no—
80C24×C4—21no—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84#533—28no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×#125—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×#117—21no—
97C2×#117—21no—
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—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C42—28no—
113C22×C42—22no—
114C22×C42—22no—
115C22×C42—22no—
116C22×C42—22no—
117C22×C42—22no—
118C22×C42—22no—
119C22×C42—48no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#484—216no—
133#483—216no—
134#432—28no—
135C2×#107C2318yes—
136C2×#106C2318yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2318yes—
139C2×#103C2318yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2314yes—
144#391—21no—
145#391—21no—
146#391C2312yes—
147#391C2312yes—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389C2312yes—
155#389C2312yes—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#385—22no—
161#385—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#385—22no—
166#384C2311yescharacteristic
167#384C2311yescharacteristic
168#382—21no—
169#381—22no—
170#380—21no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—21no—
175#380—21no—
176#380—21no—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#379—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—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C2×#95—22no—
204C2×#95—22no—
205C2×#95—22no—
206C2×#95—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×Q8C2412yes—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—28no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—42no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
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×C8C2418yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260#125—44no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C4×Q8—28no—
268C4×Q8—44no—
269#117—44no—
270#117—44no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
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—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4#4118yes—
303C2×C4⋊C4#4118yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2414yes—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
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—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—48no—
333C2×C42—48no—
334C2×C42—48no—
335C2×C42—48no—
336C2×C42—48no—
337C2×C42—48no—
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×C42C2×D411yescharacteristic
350C2×C42#4118yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—216no—
356#106—216no—
357C4⋊C8—28no—
358#103—216no—
359#103—216no—
360C22⋊C8—28no—
361C22⋊C8—28no—
362C22⋊C8—28no—
363#97—28no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
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—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24#13714yes—
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C22×C4—232no—
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—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—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—432no—
461C22×C4—432no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4#12314yes—
471C22×C4#12314yes—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4#13714yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4C2511yescharacteristic
489C2×C8—28no—
490C2×C8—28no—
491C2×C8—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4C2×#4118yes—
517C4⋊C4C2×#4118yes—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—48no—
537C42—48no—
538C42—44no—
539C42—44no—
540C42—48no—
541C42—48no—
542C42#12314yes—
543C42#12314yes—
544C42C2×#4114yes—
545C42C2×#4114yes—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—28no—
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—
565C23C2×#12312yes—
566C23#52512yes—
567C23#52512yes—
568C23C2×AGL(1,ℤ/8)12yes—
569C23C2×#13712yes—
570C23C2×#13712yes—
571C23C2×#13712yes—
572C23C2×#13712yes—
573C23#57512yes—
574C23C23×D411yescharacteristic
575Q8—48no—
576Q8—84no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
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—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—416no—
621C2×C4—416no—
622C2×C4—416no—
623C2×C4—44no—
624C2×C4—416no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4C2×#12312yes—
642C2×C4C2×#12312yes—
643C2×C4C2×#12312yes—
644C2×C4C2×#12312yes—
645C2×C4C2×#12312yes—
646C2×C4#52512yes—
647C2×C4#52512yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4C2×#13712yes—
651C2×C4C2×#13712yes—
652C2×C4C22×#4112yes—
653C2×C4C22×#4112yes—
654C2×C4C22×#4112yes—
655C8—48no—
656C22—232no—
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—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22#309514yes—
676C22#311014yes—
677C22#311314yes—
678C22C22×#12311yescharacteristic
679C22C2×#52511yescharacteristic
680C22C2×#52511yescharacteristic
681C22C22×AGL(1,ℤ/8)11yescharacteristic
682C22C22×#13711yescharacteristic
683C22C22×#13711yescharacteristic
684C22C2×#57511yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—41no—
694C4—41no—
695C4—48no—
696C4—48no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#309214yes—
702C4#344414yes—
703C2—24no—
704C2—24no—
705C2—24no—
706C2#3662611yescharacteristic
707C2C2×#309211yescharacteristic
708C2C2×#309511yescharacteristic
709C2C2×#311011yescharacteristic
710C2C2×#311311yescharacteristic
711C2#6055811yescharacteristic
712C2#6056011yescharacteristic
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. #33147C2semidirect
4. #33146C2semidirect
5. #33109C2semidirect
6. #33108C2semidirect
7. #33100C2semidirect
10. #21581C2semidirect
11. #21579C2semidirect
12. #21394C2semidirect
13. #21365C2semidirect
53. #1988C22semidirect
54. #1981C22semidirect
55. #1948C22semidirect
56. #1947C22semidirect
57. #1928C22semidirect
58. #1927C22semidirect
59. #1919C22semidirect
60. #1918C22semidirect
75. #1776C22semidirect
76. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×C4×C4⋊C42
#331472
#331462
#331092
#331082
#331002
#329832
#329822
#215812
#215792
#213942
#213652
#207152