Finite Group Databasebeta

← #6469949

Subgroups — #6469949

Order 512.

4920 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#3195C212yesmaximal, complemented
3C2×#2848C211yesmaximal, characteristic
4#32982C212yesmaximal, complemented
5#32882C212yesmaximal, complemented
6#29599C214yesmaximal, complemented
7#24386C218yesmaximal, complemented
8#24003C214yesmaximal, complemented
9#23921C214yesmaximal, complemented
10#21765C212yesmaximal, complemented
11#20579C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C22×C22⋊Q8—22no—
15C22×C22⋊Q8—21no—
16C22×C22⋊Q8—22no—
17C22×C22⋊Q8—21no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2214yes—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2212yes—
25C23×C4⋊C4—21no—
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4—21no—
28#3195—216no—
29C2×#483C2214yescomplemented
30C2×#483C2214yescomplemented
31C2×#470C2218yescomplemented
32C2×#451C2214yescomplemented
33C2×#451C2214yescomplemented
34C2×#451C2218yescomplemented
35C2×#407C2212yes—
36C2×#407C2214yes—
37#2848—21no—
38#2848—22no—
39#2848—21no—
40#2848C2214yes—
41#2848C2212yes—
42#2848C2212yes—
43C2×#385—22no—
44C2×#385—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×#379—21no—
48#2035C22116yescomplemented
49#2001C22116yescomplemented
50#1894C2218yescomplemented
51#1872C2218yescomplemented
52#1872C2218yescomplemented
53#1853C2218yescomplemented
54#1823C2214yescomplemented
55#1787C2214yescomplemented
56#1787C2218yescomplemented
57#1779C2214yes—
58#1779C2212yes—
59#1779C2212yes—
60C26D411yescharacteristic
61C23×Q8—28no—
62C23×Q8—22no—
63C23×Q8—21no—
64C23×Q8—48no—
65C23×Q8C2314yes—
66C23×Q8C2311yescharacteristic
67C23×Q8C2312yes—
68C24×C4—21no—
69C24×C4—21no—
70C24×C4—28no—
71C24×C4—48no—
72C24×C4D411yescharacteristic
73C24×C4D411yescharacteristic
74C24×C4D412yes—
75C24×C4D411yescharacteristic
76C24×C4D412yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16C2318yescomplemented
83C22×M16C2314yes—
84C2×C22⋊Q8—232no—
85C2×C22⋊Q8—232no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—232no—
98C2×C22⋊Q8—232no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—48no—
106C2×C22⋊Q8D4116yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2318yes—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C22×C4⋊C4—216no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—28no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—216no—
129C22×C22⋊C4—24no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—24no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137#483—216no—
138#483—216no—
139#470—232no—
140#451—216no—
141#451—216no—
142#451—416no—
143C2×#106C2318yescomplemented
144C2×#103C2318yescomplemented
145C2×#103C2318yescomplemented
146C2×#103C23116yescomplemented
147#407—28no—
148#407—48no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2318yes—
152C2×C22⋊C8C2318yes—
153#385—24no—
154#385—22no—
155#385—24no—
156#385—24no—
157#385—24no—
158#385—22no—
159#385—24no—
160#380—22no—
161#380—22no—
162#380—21no—
163#380—21no—
164#380C2312yes—
165#380C2314yes—
166#380C2312yes—
167#380C2314yes—
168#379—22no—
169#379—24no—
170#379—22no—
171C2×#95—22no—
172C2×#95—21no—
173#340C23116yescomplemented
174C25—24no—
175C25—28no—
176C25—24no—
177C25—44no—
178C25—42no—
179C25C2×D411yescharacteristic
180C25C2×D412yes—
181C25C2×D411yescharacteristic
182C25C2×D411yescharacteristic
183C25C2×D412yes—
184C22×Q8—28no—
185C22×Q8—216no—
186C22×Q8—216no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—21no—
191C22×Q8—24no—
192C22×Q8—216no—
193C22×Q8—48no—
194C22×Q8—48no—
195C22×Q8—44no—
196C22×Q8—464no—
197C22×Q8—48no—
198C22×Q8—48no—
199C22×Q8—48no—
200C22×Q8C2×D418yes—
201C22×Q8C2412yes—
202C22×Q8C2414yes—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—464no—
236C23×C4—44no—
237C23×C4—48no—
238C23×C4—44no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D418yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C2×Q16—216no—
259C2×Q16—416no—
260C2×M16—28no—
261C2×M16—48no—
262C22×C8C2418yes—
263C22⋊Q8—232no—
264C22⋊Q8—232no—
265C22⋊Q8—232no—
266C22⋊Q8—232no—
267C22⋊Q8—232no—
268C22⋊Q8—216no—
269C22⋊Q8—216no—
270C22⋊Q8—216no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—216no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—216no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8C2×D4116yes—
283C22⋊Q8C2×D4116yes—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—216no—
286C2×C4⋊C4—232no—
287C2×C4⋊C4—232no—
288C2×C4⋊C4—216no—
289C2×C4⋊C4—216no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—216no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D418yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C22⋊C4—216no—
312C2×C22⋊C4—232no—
313C2×C22⋊C4—232no—
314C2×C22⋊C4—216no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—48no—
330#106—216no—
331#103—216no—
332#103—216no—
333#103—416no—
334C22⋊C8—28no—
335C22⋊C8—28no—
336C22⋊C8—216no—
337C22⋊C8—48no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—232no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—432no—
370C24—416no—
371C24—416no—
372C24—432no—
373C24#12014yes—
374C24#13714yes—
375C24#13718yes—
376C24C22×D411yescharacteristic
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379C24C22×D411yescharacteristic
380C24C22×D412yes—
381C2×Q8—216no—
382C2×Q8—28no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—216no—
386C2×Q8—28no—
387C2×Q8—416no—
388C2×Q8—464no—
389C2×Q8—464no—
390C2×Q8—432no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—416no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—88no—
398C2×Q8C22×D418yes—
399C22×C4—232no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—416no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—432no—
450C22×C4—432no—
451C22×C4—416no—
452C22×C4—432no—
453C22×C4—432no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—432no—
457C22×C4—48no—
458C22×C4—416no—
459C22×C4—432no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4#12014yes—
474C22×C4#12014yes—
475C22×C4#12014yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4H(5,𝔽2)18yes—
485C22×C4C2511yescharacteristic
486Q16—816no—
487M16—88no—
488C2×C8—28no—
489C2×C8—216no—
490C4⋊C4—216no—
491C4⋊C4—232no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4C22×D418yes—
505C22⋊C4—232no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C23—28no—
517C23—28no—
518C23—216no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—216no—
524C23—28no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—28no—
531C23—216no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—216no—
537C23—28no—
538C23—216no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—44no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—44no—
561C23—4128no—
562C23—48no—
563C23—48no—
564C23—44no—
565C23C2×#12012yes—
566C23C2×#12012yes—
567C23C2×#12012yes—
568C23#53414yes—
569C23C2×#13714yes—
570C23C2×#13714yes—
571C23C2×#13714yes—
572C23C2×#13714yes—
573C23C2×#13712yes—
574C23#57512yes—
575C23C23×D411yescharacteristic
576Q8—216no—
577Q8—416no—
578Q8—432no—
579Q8—48no—
580Q8—48no—
581Q8—84no—
582C2×C4—216no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—216no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—432no—
609C2×C4—44no—
610C2×C4—416no—
611C2×C4—432no—
612C2×C4—48no—
613C2×C4—416no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4C2×#12012yes—
628C2×C4C2×#12012yes—
629C2×C4C2×#12012yes—
630C2×C4#53414yes—
631C2×C4C23×D412yes—
632C2×C4C2×H(5,𝔽2)14yes—
633C8—48no—
634C22—232no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—22no—
643C22—24no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—24no—
648C22—22no—
649C22—24no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—432no—
656C22—416no—
657C22—432no—
658C22—416no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22#300414yes—
666C22#300714yes—
667C22#319518yescomplemented
668C22C22×#12011yescharacteristic
669C22C2×#53412yes—
670C22C22×#13711yescharacteristic
671C22C22×#13712yes—
672C22C2×#57511yescharacteristic
673C4—22no—
674C4—24no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—48no—
679C4—48no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4#348414yes—
684C2—28no—
685C2—24no—
686C2—24no—
687C2—44no—
688C2—42no—
689C2#3498011yescharacteristic
690C2C2×#300411yescharacteristic
691C2C2×#300711yescharacteristic
692C2C2×#319512yes—
693C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#3195C2semidirect
4. #32982C2semidirect
5. #32882C2semidirect
6. #29599C2semidirect
7. #24386C2semidirect
8. #24003C2semidirect
9. #23921C2semidirect
10. #21765C2semidirect
29. C2×#483C22semidirect
30. C2×#483C22semidirect
31. C2×#470C22semidirect
32. C2×#451C22semidirect
33. C2×#451C22semidirect
34. C2×#451C22semidirect
48. #2035C22semidirect
49. #2001C22semidirect
50. #1894C22semidirect
51. #1872C22semidirect
52. #1872C22semidirect
53. #1853C22semidirect
54. #1823C22semidirect
55. #1787C22semidirect
56. #1787C22semidirect
82. C22×Q16C23semidirect
143. C2×#106C23semidirect
144. C2×#103C23semidirect
145. C2×#103C23semidirect
146. C2×#103C23semidirect
173. #340C23semidirect
667. C22#3195semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#31952
C2×#28482
#329822
#328822
#295992
#243862
#240032
#239212
#217652
#205792