Finite Group Databasebeta

← #393218

Subgroups — #393218

Order 512.

1480 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1973C211yesmaximal, characteristic
2C2×#1973C211yesmaximal, characteristic
3C2×#1968C211yesmaximal, characteristic
4C2×#1943C211yesmaximal, characteristic, complemented
5C2×#1936C211yesmaximal, characteristic
6C2×#1935C211yesmaximal, characteristic, complemented
7C2×C4×#97C211yesmaximal, characteristic, complemented
8#14703C211yesmaximal, characteristic, complemented
9#14700C211yesmaximal, characteristic
10#10928C211yesmaximal, characteristic, complemented
11#10927C211yesmaximal, characteristic
12#10926C211yesmaximal, characteristic, complemented
13#10925C211yesmaximal, characteristic
14#10863C211yesmaximal, characteristic, complemented
15#10862C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#388—22no—
41C2×C43C2211yescharacteristic
42#1973—24no—
43#1973C418yescomplemented
44#1968—24no—
45#1943—24no—
46#1936—24no—
47#1935—24no—
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#340C2211yescharacteristic
52C2×#340C2211yescharacteristic
53C2×#340C2211yescharacteristic
54C4×#97—24no—
55#1437—22no—
56#1436—22no—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61#1434—22no—
62C23×Q8C2311yescharacteristic
63C23×D4C2311yescharacteristic
64C22×SD16—22no—
65C22×SD16—22no—
66C22×SD16—22no—
67C22×SD16—22no—
68C22×SD16—22no—
69C22×SD16—22no—
70C22×SD16C2×C414yes—
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107C2×C414yes—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103C2×C414yes—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102C2×C414yes—
119C2×#97—22no—
120C2×#97—22no—
121C2×#97—22no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—22no—
125C2×#97—22no—
126C2×#97—22no—
127C2×#97—22no—
128C2×#97C2×C414yes—
129C2×#97D414yes—
130C2×#97D414yes—
131C2×C4×C8—22no—
132C2×C4×C8—22no—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136C2×C4×C8D414yes—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391C2×C414yes—
143#390—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390C2×C414yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#388—48no—
154C2×#95—22no—
155C2×#95—21no—
156C2×#95—21no—
157C2×#95—22no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95—22no—
162C2×#95—21no—
163C2×#95—21no—
164C43—22no—
165C43—22no—
166#340—22no—
167#340—22no—
168#340—22no—
169#340—22no—
170#340—22no—
171#340—22no—
172C25—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8C22×C412yes—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4C22×C412yes—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2411yescharacteristic
213C2×SD16—28no—
214C2×SD16—28no—
215C2×SD16—28no—
216C2×SD16—44no—
217C2×SD16—44no—
218C2×SD16—44no—
219C2×SD16—44no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C2×D412yes—
235C22×C8C2×D412yes—
236C22×C8C2×D412yes—
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8C2×D412yes—
240C22×C8#4112yes—
241C22×C8#4112yes—
242C22×C8#4112yes—
243C22×C8#4112yes—
244C22×C8#4112yes—
245C22×C8#4112yes—
246C22×C8#4112yes—
247C22×C8#4112yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C412yes—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42C22×C412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42#4112yes—
321C2×C42#4112yes—
322#107—28no—
323#107—44no—
324#106—44no—
325#103—28no—
326#103—44no—
327#103—44no—
328#102—28no—
329#102—44no—
330#102—44no—
331#97—28no—
332#97—28no—
333#97—28no—
334#97—28no—
335C4×C8—28no—
336C4×C8—28no—
337#95—28no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—41no—
349C24—41no—
350C24—41no—
351C24—41no—
352C24—41no—
353C24—41no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24C22×D411yescharacteristic
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
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—22no—
435C22×C4—22no—
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—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4C4×D414yes—
471C22×C4#12114yes—
472C22×C4#12712yes—
473C22×C4#12712yes—
474C22×C4AGL(1,ℤ/8)14yes—
475C22×C4#13714yes—
476C22×C4C23×C411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4111yescharacteristic
481C22×C4C2×#4111yescharacteristic
482C22×C4C2×#4111yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4111yescharacteristic
487C22×C4C2×#4111yescharacteristic
488C22×C4C2×#4111yescharacteristic
489C22×C4C2×#4111yescharacteristic
490SD16—416no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8#11714yes—
510C2×C8C4×D414yes—
511C2×C8C4×D414yes—
512C2×C8C4×D414yes—
513C2×C8#12114yes—
514C2×C8#12114yes—
515C2×C8#12314yes—
516C2×C8#12414yes—
517C2×C8#12414yes—
518C2×C8#12614yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C22⋊C4—416no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—88no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—48no—
562C42—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—81no—
569C23—81no—
570C23—81no—
571C23—81no—
572C23—81no—
573C23—81no—
574C23—81no—
575C23—81no—
576C23—82no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23—82no—
583C23—82no—
584C23—82no—
585C23—82no—
586C23C2×C4×D411yescharacteristic
587C23C2×#12111yescharacteristic
588C23C2×#12711yescharacteristic
589C23#53211yescharacteristic
590C23#53211yescharacteristic
591C23#53211yescharacteristic
592C23#53211yescharacteristic
593C23C2×AGL(1,ℤ/8)12yes—
594C23C2×#13712yes—
595C23#57512yes—
596C23#57512yes—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—81no—
602Q8—81no—
603Q8—81no—
604Q8—81no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—81no—
610D4—81no—
611D4—81no—
612D4—81no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—42no—
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—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—82no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4#46818yes—
674C2×C4C2×#11711yescharacteristic
675C2×C4C2×C4×D411yescharacteristic
676C2×C4C2×C4×D411yescharacteristic
677C2×C4C2×C4×D411yescharacteristic
678C2×C4C4×#4111yescharacteristic
679C2×C4C4×#4111yescharacteristic
680C2×C4C2×#12111yescharacteristic
681C2×C4C2×#12111yescharacteristic
682C2×C4C2×#12311yescharacteristic
683C2×C4#52511yescharacteristic
684C2×C4#52511yescharacteristic
685C2×C4C2×#12411yescharacteristic
686C2×C4C2×#12411yescharacteristic
687C2×C4C2×#12611yescharacteristic
688C2×C4#52911yescharacteristic
689C2×C4#52911yescharacteristic
690C2×C4#52911yescharacteristic
691C2×C4#52911yescharacteristic
692C2×C4#52911yescharacteristic
693C2×C4#52911yescharacteristic
694C2×C4#52911yescharacteristic
695C2×C4#52911yescharacteristic
696C2×C4#53211yescharacteristic
697C2×C4#53211yescharacteristic
698C2×C4#53211yescharacteristic
699C2×C4#53211yescharacteristic
700C2×C4#53211yescharacteristic
701C2×C4#53211yescharacteristic
702C8—24no—
703C8—24no—
704C8—24no—
705C8#38818yescomplemented
706C22—84no—
707C22—84no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—162no—
712C22—161no—
713C22—161no—
714C22—161no—
715C22—161no—
716C22—162no—
717C22#229311yescharacteristic
718C22C4×#12111yescharacteristic
719C22C4×#12711yescharacteristic
720C22#285411yescharacteristic
721C22#285611yescharacteristic
722C22#285811yescharacteristic
723C22#286011yescharacteristic
724C22C4×AGL(1,ℤ/8)12yes—
725C22C4×#13712yes—
726C22C2×#46812yes—
727C22#305412yes—
728C22#315111yescharacteristic
729C22#315211yescharacteristic
730C22#315311yescharacteristic
731C22#315311yescharacteristic
732C22#317112yes—
733C22#317212yes—
734C22#318012yes—
735C22#318012yes—
736C22#318112yes—
737C22#318112yes—
738C22#318212yes—
739C22#318312yes—
740C4—28no—
741C4—44no—
742C4—42no—
743C4—42no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—161no—
748C4—161no—
749C4—161no—
750C4—161no—
751C4C2×#38811yescharacteristic
752C4#229311yescharacteristic
753C4C4×#12111yescharacteristic
754C4C4×#12411yescharacteristic
755C4#285011yescharacteristic
756C4#285411yescharacteristic
757C4#286011yescharacteristic
758C4#286111yescharacteristic
759C2—162no—
760C2#1470011yescharacteristic
761C2#2128812yes—
762C2C4×#46812yes—
763C2#2170111yescharacteristic
764C2#2170211yescharacteristic
765C2#3313412yes—
766C2#3317411yescharacteristic
767C2#3317711yescharacteristic
768C2#3325011yescharacteristic
769C2#3325111yescharacteristic
770C2#3326812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1943C2semidirect
6. C2×#1935C2semidirect
7. C2×C4×#97C2semidirect
8. #14703C2semidirect
10. #10928C2semidirect
12. #10926C2semidirect
14. #10863C2semidirect
15. #10862C2semidirect
43. #1973C4semidirect
705. C8#388semidirect

Maximal subgroups

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

subgroupindex
C2×#19732
C2×#19682
C2×#19432
C2×#19362
C2×#19352
C2×C4×#972
#147032
#147002
#109282
#109272
#109262
#109252
#108632
#108622