Finite Group Databasebeta

← #369436

Subgroups — #369436

Order 512.

1904 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×C4×#124C211yesmaximal, characteristic
3C2×#1985C211yesmaximal, characteristic, complemented
4C2×#1985C211yesmaximal, characteristic, complemented
5C2×#1955C211yesmaximal, characteristic
6#11147C212yesmaximal, complemented
7#11147C212yesmaximal, complemented
8#10832C211yesmaximal, characteristic, complemented
9#10715C212yesmaximal, complemented
10#10714C212yesmaximal, complemented
11#10693C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#124C411yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#435—24no—
18C2×#435—24no—
19C2×#435—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C4×#124—24no—
29C4×#124—24no—
30C2×#391C411yescharacteristic
31C2×#390C411yescharacteristic
32C2×#388C412yes—
33C2×#388C412yes—
34C2×#382C411yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C43C2211yescharacteristic
38#1985—24no—
39#1985—24no—
40#1955—24no—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×Q8C2×C411yescharacteristic
50C23×D4C2×C411yescharacteristic
51C24×C4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—42no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—24no—
62C23×C8—42no—
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2312yes—
68C23×C8C2311yescharacteristic
69C2×#124—24no—
70C2×#124—28no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4D414yes—
82C2×C4×D4D414yes—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C411yescharacteristic
90C22×C22⋊C4C2×C411yescharacteristic
91C22×C22⋊C4C2×C412yes—
92C22×C42C2×C411yescharacteristic
93C22×C42C2×C412yes—
94C22×C42C2×C411yescharacteristic
95C22×C42C2312yes—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98#435—232no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8D414yes—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8—42no—
116C2×C22⋊C8—42no—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97—22no—
120C2×#97—22no—
121C2×#97—22no—
122C2×#97D414yes—
123#391—24no—
124#390—24no—
125#388—28no—
126#388—28no—
127#382—24no—
128C4×C22⋊C4—24no—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C43—22no—
137C43—22no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—24no—
142#336—24no—
143#336—24no—
144#336—42no—
145#336—42no—
146C25C22×C411yescharacteristic
147C25C22×C411yescharacteristic
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—42no—
157C22×D4C22⋊C414yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4C22×C412yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2411yescharacteristic
187C2×M16—28no—
188C2×M16—48no—
189C2×M16—48no—
190C2×M16—44no—
191C2×M16—44no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—28no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—48no—
222C22×C8—48no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8#4114yes—
230C22×C8#4114yes—
231#124—232no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×Q8—44no—
235C4×Q8—44no—
236C4×D4—28no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C22⋊C414yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C22⋊C414yes—
265C2×C22⋊C4C22⋊C414yes—
266C2×C22⋊C4M1618yescomplemented
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292C4⋊C8—28no—
293C4⋊C8—44no—
294C22⋊C8—28no—
295C22⋊C8—28no—
296C22⋊C8—48no—
297C22⋊C8—48no—
298#97—28no—
299#97—44no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24C2×C22⋊C412yes—
320C24C2×C22⋊C412yes—
321C24C2×M1614yes—
322C24C23×C411yescharacteristic
323C2×Q8—28no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C4×D412yes—
411C22×C4C4×D412yes—
412C22×C4C4×D412yes—
413C22×C4C4×D412yes—
414C22×C4C2×M1614yes—
415C22×C4C2×M1614yes—
416C22×C4#13114yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427M16—416no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—416no—
443C2×C8—42no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8#12414yescomplemented
463C4⋊C4—28no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C42—28no—
473C42—48no—
474C42—44no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42#12014yes—
484C42#12114yes—
485C42#12114yes—
486C42#12314yes—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23#40714yes—
529C23C22×C22⋊C411yescharacteristic
530C23C2×C4×D411yescharacteristic
531C23C2×C4×D411yescharacteristic
532C23C4×#4112yes—
533C23#52012yes—
534C23C22×M1612yes—
535C23C2×#13112yes—
536C23#56812yes—
537C23#56812yes—
538Q8—416no—
539D4—416no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4#38614yes—
599C2×C4#38614yes—
600C2×C4C2×#12011yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12311yescharacteristic
604C2×C4C2×#12411yescharacteristic
605C2×C4#52912yes—
606C2×C4#53211yescharacteristic
607C2×C4#53511yescharacteristic
608C2×C4#53611yescharacteristic
609C2×C4#54212yes—
610C2×C4#54612yes—
611C2×C4#54612yes—
612C2×C4#54612yes—
613C2×C4#54712yes—
614C2×C4#54712yes—
615C2×C4#54812yes—
616C2×C4#55612yes—
617C2×C4#55912yes—
618C8—24no—
619C8—44no—
620C8—42no—
621C8—42no—
622C8—42no—
623C8—82no—
624C22—24no—
625C22—28no—
626C22—44no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—42no—
637C22—44no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C2×#38611yescharacteristic
643C22C4×#12411yescharacteristic
644C22#232611yescharacteristic
645C22#233912yes—
646C22#236712yes—
647C22C2×#40711yescharacteristic
648C22#287911yescharacteristic
649C22#288012yes—
650C22#293111yescharacteristic
651C22#293211yescharacteristic
652C22D4×M1612yes—
653C22D4×M1612yes—
654C22#293512yes—
655C22#296512yes—
656C22#296712yes—
657C22#296812yes—
658C22#297012yes—
659C22#298912yes—
660C22#299012yes—
661C22#299212yes—
662C22#299512yes—
663C4—24no—
664C4—24no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—82no—
671C4#242712yes—
672C4#243211yescharacteristic
673C4#243511yescharacteristic
674C4#244712yes—
675C4#254411yescharacteristic
676C4#254711yescharacteristic
677C2—44no—
678C2—82no—
679C2#1402711yescharacteristic
680C2#2089811yescharacteristic
681C2#2090011yescharacteristic
682C2#2163212yes—
683C2#2221612yes—
684C2#2247112yes—
685C2#2247512yes—
686C2#2320612yes—
687C2#2320712yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1985C2semidirect
4. C2×#1985C2semidirect
6. #11147C2semidirect
7. #11147C2semidirect
8. #10832C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10693C2semidirect
266. C2×C22⋊C4M16semidirect
462. C2×C8#124semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×C4×#1242
C2×#19852
C2×#19552
#111472
#108322
#107152
#107142
#106932