Finite Group Databasebeta

← #6660464

Subgroups — #6660464

Order 512.

1954 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
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#32774C212yesmaximal, complemented
4#32762C212yesmaximal, complemented
5#30318C214yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#29664C212yesmaximal, complemented
8#29597C214yesmaximal
9#29105C212yesmaximal, complemented
10#28304C212yesmaximal, complemented
11#27648C212yesmaximal, complemented
12#24652C214yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#20714C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19C2×#484C2214yes—
20C2×#462C2214yescomplemented
21C2×#451C2214yes—
22C2×#431C2212yes—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2212yes—
30#2857C2212yes—
31#2857C2212yes—
32#2857C2212yes—
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×#391C2212yes—
47C2×#389—21no—
48C2×#389—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2280—21no—
52C2×#380—21no—
53C2×#380—21no—
54#2095C2214yescomplemented
55#2083C2214yescomplemented
56#2072C2214yescomplemented
57#2072C2214yescomplemented
58#2043C2214yescomplemented
59#2036C2218yes—
60#2036C2218yes—
61#2024C2214yescomplemented
62#2016C2214yescomplemented
63#2016C2214yescomplemented
64#2001C2218yes—
65#2001C2218yes—
66#1981C2214yescomplemented
67#1969C2218yescomplemented
68#1947C2214yescomplemented
69#1933C2218yescomplemented
70#1824C2214yes—
71#1817C2212yes—
72#1803C2214yescomplemented
73#1797C2212yes—
74#1797C2212yes—
75#1787C2214yes—
76#1779C2212yes—
77C4×C22⋊C8C2212yes—
78C4×C22⋊C8C2212yes—
79C23×Q8C2312yes—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×Q16C2318yes—
84#533—44no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
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—48no—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
106C2×#117—21no—
107C2×#117—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
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#484—48no—
129#462—48no—
130#451—48no—
131#431—44no—
132C2×#107C2318yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×#103C2318yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C4×C8C2314yes—
142#391—21no—
143#391—21no—
144#391—22no—
145#391—22no—
146#391C2312yes—
147#391C2314yes—
148#391C2312yes—
149#391C2314yes—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#384—21no—
177#384—21no—
178#384C2312yes—
179#384C2311yescharacteristic
180#384C2312yes—
181#384C2311yescharacteristic
182#382—21no—
183#381—22no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202#379—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C2×#95—22no—
210C2×#95—22no—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C25C2×D411yescharacteristic
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—28no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×Q8C2412yes—
222C22×Q8C2414yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4Q1618yescomplemented
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
252C2×Q16—48no—
253C2×Q16—48no—
254C2×Q16—48no—
255C22×C8C2418yes—
256C4⋊Q8—28no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—42no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262#125—44no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—48no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C4×Q8—28no—
275C4×Q8—44no—
276#117—44no—
277#117—44no—
278C2×C4⋊C4—28no—
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—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—48no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
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×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—48no—
352C4⋊C8—28no—
353#103—48no—
354#103—48no—
355#103—48no—
356#103—48no—
357C22⋊C8—28no—
358C22⋊C8—44no—
359C22⋊C8—44no—
360C4×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—41no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24C2×Q1614yes—
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—816no—
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—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—416no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
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—416no—
454C22×C4—416no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4C2×Q1614yes—
462C22×C4C2×Q1614yes—
463C22×C4C2×Q1614yes—
464C22×C4C2×Q1614yes—
465C22×C4C2×Q1614yes—
466C22×C4C2×Q1614yes—
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×C4H(5,𝔽2)14yes—
480C22×C4C2511yescharacteristic
481Q16—816no—
482C2×C8—28no—
483C2×C8—28no—
484C2×C8—28no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
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—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—44no—
538C42—44no—
539C42H(5,𝔽2)14yes—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—28no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23#53512yes—
564C23C22×Q1612yes—
565C23C22×Q1612yes—
566C23C22×Q1612yes—
567C23C2×#13712yes—
568C23C23×D411yescharacteristic
569Q8—44no—
570Q8—48no—
571Q8—44no—
572Q8—48no—
573Q8—84no—
574Q8—88no—
575C2×C4—28no—
576C2×C4—21no—
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—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
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—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
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—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4#53412yes—
636C2×C4#53412yes—
637C2×C4#53512yes—
638C2×C4C22×Q1612yes—
639C2×C4C22×Q1612yes—
640C2×C4C22×Q1612yes—
641C2×C4C22×Q1612yes—
642C2×C4C2×#13712yes—
643C2×C4C2×#13712yes—
644C2×C4#57512yes—
645C2×C4#57512yes—
646C2×C4C2×H(5,𝔽2)12yes—
647C2×C4C2×H(5,𝔽2)12yes—
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—416no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22#320914yes—
672C22C2×#53411yescharacteristic
673C22C2×#53511yescharacteristic
674C22#344911yescharacteristic
675C22C23×Q1611yescharacteristic
676C22C22×#13711yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57511yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—41no—
688C4—41no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—84no—
695C4#318914yes—
696C4#354214yes—
697C2—24no—
698C2—42no—
699C2—42no—
700C2#3743711yescharacteristic
701C2C2×#318911yescharacteristic
702C2#6075111yescharacteristic
703C2C2×#320911yescharacteristic
704C2#6080211yescharacteristic
705C2#6094311yescharacteristic
706C2#6095211yescharacteristic
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. #32774C2semidirect
4. #32762C2semidirect
5. #30318C2semidirect
6. #29923C2semidirect
7. #29664C2semidirect
9. #29105C2semidirect
10. #28304C2semidirect
11. #27648C2semidirect
12. #24652C2semidirect
13. #24212C2semidirect
20. C2×#462C22semidirect
54. #2095C22semidirect
55. #2083C22semidirect
56. #2072C22semidirect
57. #2072C22semidirect
58. #2043C22semidirect
61. #2024C22semidirect
62. #2016C22semidirect
63. #2016C22semidirect
66. #1981C22semidirect
67. #1969C22semidirect
68. #1947C22semidirect
69. #1933C22semidirect
72. #1803C22semidirect
233. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#327742
#327622
#303182
#299232
#296642
#295972
#291052
#283042
#276482
#246522
#242122
#207142