Finite Group Databasebeta

← #6469948

Subgroups — #6469948

Order 512.

5190 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#3193C212yesmaximal, complemented
3C2×#2847C211yesmaximal, characteristic
4#32978C212yesmaximal, complemented
5#32879C212yesmaximal, complemented
6#29595C214yesmaximal, complemented
7#24387C218yesmaximal, complemented
8#24006C214yesmaximal, complemented
9#23919C214yesmaximal, complemented
10#21765C212yesmaximal, complemented
11#20579C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic, complemented
13C25×C4C2211yescharacteristic
14C22×C22⋊Q8—22no—
15C22×C22⋊Q8—21no—
16C22×C22⋊Q8—22no—
17C22×C22⋊Q8—21no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2214yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2214yescomplemented
22C22×C22⋊Q8C2212yescomplemented
23C22×#121C2211yescharacteristic
24C22×#121C2212yes—
25C22×#120—21no—
26C23×C4⋊C4—21no—
27C23×C4⋊C4C2211yescharacteristic, complemented
28C23×C22⋊C4—21no—
29#3193—216no—
30C2×#484C2214yescomplemented
31C2×#480C2214yescomplemented
32C2×#468C2218yescomplemented
33C2×#448C2214yescomplemented
34C2×#448C2218yescomplemented
35C2×#450C2214yescomplemented
36C2×#407C2212yes—
37C2×#407C2214yescomplemented
38#2847—21no—
39#2847—22no—
40#2847—21no—
41#2847C2214yes—
42#2847C2212yes—
43#2847C2212yes—
44C2×#386—21no—
45C2×#386—22no—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48#2036C22116yescomplemented
49#2000C22116yescomplemented
50#1894C2218yescomplemented
51#1874C2218yescomplemented
52#1872C2218yescomplemented
53#1852C2218yescomplemented
54#1823C2214yescomplemented
55#1787C2214yescomplemented
56#1787C2218yescomplemented
57#1779C2214yes—
58#1779C2212yescomplemented
59#1779C2212yescomplemented
60C26D411yescharacteristic
61C23×Q8—28no—
62C23×Q8—22no—
63C23×Q8—21no—
64C23×Q8—48no—
65C23×Q8C2314yes—
66C23×Q8C2312yes—
67C23×D4—22no—
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C24×C4—21no—
72C24×C4—28no—
73C24×C4—48no—
74C24×C4D411yescharacteristic
75C24×C4D411yescharacteristic
76C24×C4D412yes—
77C24×C4D411yescharacteristic
78C24×C4D412yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×SD16C2318yescomplemented
85C22×M16C2314yes—
86C2×C22⋊Q8—232no—
87C2×C22⋊Q8—232no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—232no—
100C2×C22⋊Q8—232no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8D4116yescomplemented
103C2×C22⋊Q8C2318yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C2×#121—22no—
107C2×#121—22no—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121—28no—
111C2×#121—48no—
112C2×#121C2312yes—
113C2×#121C2314yes—
114C2×#121C2312yes—
115C2×#121C2314yes—
116C2×#120—24no—
117C2×#120—22no—
118C2×#120—22no—
119C2×#120—48no—
120C22×C4⋊C4—216no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—28no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C22⋊C4—216no—
133C22×C22⋊C4—24no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—24no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4—21no—
141#484—216no—
142#480—216no—
143#468—232no—
144#450—216no—
145#448—216no—
146#448—416no—
147C2×#107C2318yescomplemented
148C2×#103C2318yescomplemented
149C2×#103C23116yescomplemented
150C2×#102C2318yescomplemented
151#407—28no—
152#407—48no—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2318yes—
156C2×C22⋊C8C2318yes—
157#386—24no—
158#386—22no—
159#386—24no—
160#386—24no—
161#386—24no—
162#386—22no—
163#386—24no—
164#380—22no—
165#380—22no—
166#380—21no—
167#380—21no—
168#380C2312yes—
169#380C2314yes—
170#380C2312yes—
171#380C2314yes—
172C2×#95—22no—
173C2×#95—21no—
174#340C23116yescomplemented
175C25—24no—
176C25—28no—
177C25—24no—
178C25—21no—
179C25—44no—
180C25—42no—
181C25C2×D411yescharacteristic
182C25C2×D412yes—
183C25C2×D411yescharacteristic
184C25C2×D411yescharacteristic
185C25C2×D412yes—
186C22×Q8—28no—
187C22×Q8—216no—
188C22×Q8—216no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—216no—
193C22×Q8—48no—
194C22×Q8—48no—
195C22×Q8—464no—
196C22×Q8—48no—
197C22×Q8—48no—
198C22×Q8—48no—
199C22×Q8C2×D418yescomplemented
200C22×Q8C2414yes—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—21no—
207C22×D4—44no—
208C22×D4—48no—
209C22×D4—44no—
210C22×D4—48no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—48no—
214C22×D4C2412yes—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—28no—
228C23×C4—28no—
229C23×C4—28no—
230C23×C4—24no—
231C23×C4—28no—
232C23×C4—28no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—464no—
247C23×C4—44no—
248C23×C4—48no—
249C23×C4—44no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D418yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2412yes—
269C2×SD16—216no—
270C2×SD16—416no—
271C2×M16—28no—
272C2×M16—48no—
273C22×C8C2418yes—
274C22⋊Q8—232no—
275C22⋊Q8—232no—
276C22⋊Q8—232no—
277C22⋊Q8—232no—
278C22⋊Q8—232no—
279C22⋊Q8—216no—
280C22⋊Q8—216no—
281C22⋊Q8—216no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—216no—
285C22⋊Q8C2×D4116yes—
286C22⋊Q8C2×D4116yescomplemented
287#121—28no—
288#121—28no—
289#121—216no—
290#121—48no—
291#121—48no—
292#121—48no—
293#121—44no—
294#121—44no—
295#120—416no—
296#120—48no—
297#120—48no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—216no—
300C2×C4⋊C4—232no—
301C2×C4⋊C4—232no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—216no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D418yescomplemented
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—216no—
320C2×C22⋊C4—232no—
321C2×C22⋊C4—232no—
322C2×C22⋊C4—216no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—48no—
335C2×C22⋊C4—48no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338#107—216no—
339#103—216no—
340#103—416no—
341#102—216no—
342C22⋊C8—28no—
343C22⋊C8—28no—
344C22⋊C8—216no—
345C22⋊C8—48no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—24no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—232no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—432no—
379C24—416no—
380C24—416no—
381C24—432no—
382C24—44no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—44no—
387C24#12014yes—
388C24AGL(1,ℤ/8)14yes—
389C24#13718yes—
390C24C22×D411yescharacteristic
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C24C22×D412yes—
395C2×Q8—216no—
396C2×Q8—28no—
397C2×Q8—24no—
398C2×Q8—216no—
399C2×Q8—416no—
400C2×Q8—464no—
401C2×Q8—464no—
402C2×Q8—432no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—416no—
406C2×Q8—48no—
407C2×Q8C22×D418yes—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—44no—
411C2×D4—48no—
412C2×D4—44no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—44no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—88no—
425C2×D4—832no—
426C2×D4—816no—
427C22×C4—232no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—416no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—432no—
477C22×C4—432no—
478C22×C4—416no—
479C22×C4—432no—
480C22×C4—432no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—432no—
484C22×C4—48no—
485C22×C4—416no—
486C22×C4—432no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4#12014yes—
500C22×C4#12014yes—
501C22×C4#12014yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D414yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4H(5,𝔽2)18yes—
511C22×C4C2511yescharacteristic
512SD16—816no—
513M16—88no—
514C2×C8—28no—
515C2×C8—216no—
516C4⋊C4—216no—
517C4⋊C4—232no—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—28no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4C22×D418yes—
527C22⋊C4—232no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C23—28no—
539C23—28no—
540C23—216no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—216no—
546C23—28no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—28no—
553C23—216no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—216no—
559C23—28no—
560C23—216no—
561C23—28no—
562C23—22no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—21no—
567C23—22no—
568C23—44no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—48no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23—44no—
583C23—4128no—
584C23—48no—
585C23—48no—
586C23—44no—
587C23—42no—
588C23—44no—
589C23—42no—
590C23—44no—
591C23—42no—
592C23—88no—
593C23—84no—
594C23—84no—
595C23—88no—
596C23—84no—
597C23C2×#12012yes—
598C23C2×#12012yes—
599C23C2×#12012yes—
600C23#53414yes—
601C23C2×AGL(1,ℤ/8)14yes—
602C23C2×AGL(1,ℤ/8)12yes—
603C23C2×#13714yes—
604C23C2×#13714yes—
605C23C2×#13714yes—
606C23#57512yes—
607C23C23×D411yescharacteristic
608Q8—216no—
609Q8—416no—
610Q8—432no—
611Q8—48no—
612D4—48no—
613D4—84no—
614D4—816no—
615D4—88no—
616D4—816no—
617D4—88no—
618C2×C4—216no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—216no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—48no—
644C2×C4—432no—
645C2×C4—44no—
646C2×C4—416no—
647C2×C4—432no—
648C2×C4—48no—
649C2×C4—416no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—88no—
660C2×C4—84no—
661C2×C4C2×#12012yes—
662C2×C4C2×#12012yes—
663C2×C4C2×#12012yes—
664C2×C4#53414yes—
665C2×C4C23×D412yes—
666C2×C4C2×H(5,𝔽2)14yes—
667C8—48no—
668C22—232no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—22no—
677C22—24no—
678C22—22no—
679C22—22no—
680C22—24no—
681C22—24no—
682C22—22no—
683C22—24no—
684C22—24no—
685C22—22no—
686C22—22no—
687C22—24no—
688C22—22no—
689C22—432no—
690C22—416no—
691C22—432no—
692C22—416no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—84no—
700C22—84no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—168no—
705C22#300414yes—
706C22#300614yes—
707C22#319318yescomplemented
708C22C22×#12011yescharacteristic
709C22C2×#53412yes—
710C22C22×AGL(1,ℤ/8)11yescharacteristic
711C22C22×#13712yes—
712C22C2×#57511yescharacteristic
713C4—22no—
714C4—24no—
715C4—22no—
716C4—22no—
717C4—42no—
718C4—48no—
719C4—48no—
720C4—84no—
721C4—82no—
722C4#348414yes—
723C2—28no—
724C2—24no—
725C2—24no—
726C2—44no—
727C2—42no—
728C2—162no—
729C2#3498011yescharacteristic
730C2C2×#300411yescharacteristic
731C2C2×#300611yescharacteristic
732C2C2×#319312yes—
733C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2847C2
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
1. C23×C22⋊Q8C2semidirect
2. C2×#3193C2semidirect
4. #32978C2semidirect
5. #32879C2semidirect
6. #29595C2semidirect
7. #24387C2semidirect
8. #24006C2semidirect
9. #23919C2semidirect
10. #21765C2semidirect
11. #20579C2semidirect
12. C24×Q8C22semidirect
21. C22×C22⋊Q8C22semidirect
22. C22×C22⋊Q8C22semidirect
27. C23×C4⋊C4C22semidirect
30. C2×#484C22semidirect
31. C2×#480C22semidirect
32. C2×#468C22semidirect
33. C2×#448C22semidirect
34. C2×#448C22semidirect
35. C2×#450C22semidirect
37. C2×#407C22semidirect
48. #2036C22semidirect
49. #2000C22semidirect
50. #1894C22semidirect
51. #1874C22semidirect
52. #1872C22semidirect
53. #1852C22semidirect
54. #1823C22semidirect
55. #1787C22semidirect
56. #1787C22semidirect
58. #1779C22semidirect
59. #1779C22semidirect
84. C22×SD16C23semidirect
102. C2×C22⋊Q8D4semidirect
147. C2×#107C23semidirect
148. C2×#103C23semidirect
149. C2×#103C23semidirect
150. C2×#102C23semidirect
174. #340C23semidirect
199. C22×Q8C2×D4semidirect
286. C22⋊Q8C2×D4semidirect
315. C2×C4⋊C4C2×D4semidirect
707. C22#3193semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#31932
C2×#28472
#329782
#328792
#295952
#243872
#240062
#239192
#217652
#205792