Finite Group Databasebeta

← #6660814

Subgroups — #6660814

Order 512.

1988 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
2C2×#2855C211yesmaximal, characteristic
3#32724C214yesmaximal, complemented
4#30318C214yesmaximal, complemented
5#29909C212yesmaximal, complemented
6#29757C212yesmaximal, complemented
7#29715C212yesmaximal, complemented
8#29627C212yesmaximal, complemented
9#29329C212yesmaximal, complemented
10#28304C212yesmaximal, complemented
11#24502C214yesmaximal, complemented
12#24006C212yesmaximal
13#23921C212yesmaximal
14#20714C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19C2×#484C2214yes—
20C2×#451C2214yes—
21C2×#451C2214yes—
22C2×#431C2212yes—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2212yes—
31#2855—21no—
32#2855—21no—
33#2855—21no—
34#2855—21no—
35#2855C2212yes—
36#2855C2212yes—
37#2855C2212yes—
38#2855C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2212yes—
42C2×#389—21no—
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#387—21no—
47C2×#387—21no—
48C2×#385—21no—
49C2×#384C2211yescharacteristic
50C2×#381—21no—
51C2×#380—21no—
52C2×#380—21no—
53C2×#380C2211yescharacteristic
54#2100C2214yescomplemented
55#2095C2214yescomplemented
56#2083C2214yescomplemented
57#2072C2214yescomplemented
58#2068C2218yescomplemented
59#2056C2218yescomplemented
60#2043C2214yescomplemented
61#2043C2214yescomplemented
62#2036C2218yes—
63#2016C2214yescomplemented
64#2016C2214yescomplemented
65#2001C2218yes—
66#1969C2218yescomplemented
67#1933C2218yescomplemented
68#1872C2214yes—
69#1872C2214yes—
70#1872C2214yes—
71#1852C2214yes—
72#1817C2212yes—
73#1797C2212yes—
74#1797C2212yes—
75#1787C2214yes—
76#1779C2212yes—
77C4×C22⋊C8C2212yes—
78C4×C22⋊C8C2212yes—
79C23×Q8C2312yes—
80C23×Q8C2311yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×Q16C2318yes—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—44no—
95C2×C22⋊Q8—48no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—22no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
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#451—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—21no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391C2312yes—
149#391C2314yes—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#391C2314yes—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2312yes—
186#381—22no—
187#381—22no—
188#381—22no—
189#381—22no—
190#380—21no—
191#380—21no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2312yes—
202C2×#95—22no—
203C2×#95—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C25C2×D411yescharacteristic
216C22×Q8—22no—
217C22×Q8—21no—
218C22×Q8—28no—
219C22×Q8—24no—
220C22×Q8—42no—
221C22×Q8—44no—
222C22×Q8—42no—
223C22×Q8—44no—
224C22×Q8C2414yes—
225C22×Q8C2412yes—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4Q1618yescomplemented
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×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×Q16—48no—
260C2×Q16—48no—
261C2×Q16—48no—
262C22×C8C2418yes—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C22⋊Q8—28no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
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—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—48no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—42no—
325C2×C42—42no—
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×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×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—48no—
351C4⋊C8—44no—
352#103—48no—
353#103—48no—
354#103—48no—
355#103—48no—
356C22⋊C8—28no—
357C22⋊C8—28no—
358C22⋊C8—44no—
359C4×C8—44no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
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—
375#95—22no—
376#95—22no—
377#95—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24C2×Q1614yes—
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—28no—
394C2×Q8—24no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—88no—
402C2×Q8—816no—
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—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
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—416no—
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—416no—
456C22×C4—416no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4C2×Q1614yes—
476C22×C4C2×Q1614yes—
477C22×C4C2×Q1614yes—
478C22×C4C2×Q1614yes—
479C22×C4C2×Q1614yes—
480C22×C4C2×Q1614yes—
481C22×C4#13714yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4H(5,𝔽2)14yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4C2511yescharacteristic
496Q16—816no—
497C2×C8—28no—
498C2×C8—28no—
499C2×C8—28no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—28no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—28no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23#53412yes—
569C23#53512yes—
570C23C22×Q1612yes—
571C23C22×Q1612yes—
572C23C22×Q1612yes—
573C23C2×#13712yes—
574C23#57512yes—
575C23C23×D411yescharacteristic
576Q8—48no—
577Q8—48no—
578Q8—48no—
579Q8—88no—
580Q8—84no—
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—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
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—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—88no—
644C2×C4—84no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4#53412yes—
648C2×C4#53512yes—
649C2×C4C22×Q1612yes—
650C2×C4C22×Q1612yes—
651C2×C4C22×Q1612yes—
652C2×C4C22×Q1612yes—
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4#57512yes—
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×H(5,𝔽2)12yes—
658C2×C4#58512yes—
659C8—48no—
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—416no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22#318914yes—
681C22#320914yes—
682C22C2×#53411yescharacteristic
683C22C2×#53511yescharacteristic
684C22#344911yescharacteristic
685C22C23×Q1611yescharacteristic
686C22C22×#13711yescharacteristic
687C22C2×#57511yescharacteristic
688C22C2×#57511yescharacteristic
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—41no—
699C4—41no—
700C4—82no—
701C4—82no—
702C4—84no—
703C4—82no—
704C4—84no—
705C4—82no—
706C4#353414yes—
707C2—24no—
708C2—24no—
709C2—42no—
710C2#3744011yescharacteristic
711C2C2×#318911yescharacteristic
712C2#6075111yescharacteristic
713C2C2×#320911yescharacteristic
714C2#6080511yescharacteristic
715C2#6094311yescharacteristic
716C2#6095011yescharacteristic
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. #32724C2semidirect
4. #30318C2semidirect
5. #29909C2semidirect
6. #29757C2semidirect
7. #29715C2semidirect
8. #29627C2semidirect
9. #29329C2semidirect
10. #28304C2semidirect
11. #24502C2semidirect
54. #2100C22semidirect
55. #2095C22semidirect
56. #2083C22semidirect
57. #2072C22semidirect
58. #2068C22semidirect
59. #2056C22semidirect
60. #2043C22semidirect
61. #2043C22semidirect
63. #2016C22semidirect
64. #2016C22semidirect
66. #1969C22semidirect
67. #1933C22semidirect
240. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28552
#327242
#303182
#299092
#297572
#297152
#296272
#293292
#283042
#245022
#240062
#239212
#207142