Finite Group Databasebeta

← #364644

Subgroups — #364644

Order 512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2395C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4C2×#1818C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11147C212yesmaximal, complemented
8#10874C212yesmaximal, complemented
9#10715C211yesmaximal, characteristic, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10714C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×C8×D4—24no—
18C2×C8×D4—24no—
19C2×C8×D4—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2395—24no—
27#2395—24no—
28C2×#400C411yescharacteristic
29C2×#399C411yescharacteristic
30C2×#398C411yescharacteristic
31C2×#397C412yes—
32C2×#397C411yescharacteristic
33C2×#394C412yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1870—24no—
38#1818—24no—
39#1729—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—42no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—24no—
63C23×C8—24no—
64C23×C8—42no—
65C23×C8—42no—
66C23×C8D414yes—
67C23×C8C2312yes—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2312yes—
72C23×C8C2311yescharacteristic
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4D414yes—
78C2×C4×D4D414yes—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C22×C4⋊C4C2×C412yes—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C22⋊C4C2×C412yes—
91C22×C22⋊C4C2×C412yes—
92C22×C22⋊C4C2×C412yes—
93C22×C22⋊C4C2×C412yes—
94C22×C22⋊C4C2×C411yescharacteristic
95C22×C22⋊C4C2×C411yescharacteristic
96C22×C42C2312yes—
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99C8×D4—232no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—42no—
114C2×C22⋊C8D414yes—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118#400—24no—
119#399—24no—
120#398—24no—
121#397—28no—
122#397—24no—
123#394—28no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136#336—24no—
137#336—24no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—24no—
142#336—24no—
143#336—42no—
144#336—42no—
145C25C22×C411yescharacteristic
146C25C22×C411yescharacteristic
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4C22⋊C414yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183C2×M16—28no—
184C2×M16—48no—
185C2×M16—48no—
186C2×M16—44no—
187C2×M16—44no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—28no—
192C22×C8—28no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
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—28no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—44no—
220C22×C8—48no—
221C22×C8—48no—
222C22×C8—48no—
223C22×C8—48no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230C22×C8#4114yes—
231C22×C8#4114yes—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—28no—
236#117—44no—
237#117—44no—
238#117—44no—
239#117—44no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4C22⋊C414yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4C22⋊C414yes—
272C2×C22⋊C4C22⋊C414yes—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C4⋊C8—28no—
292C22⋊C8—28no—
293C22⋊C8—28no—
294C22⋊C8—48no—
295C22⋊C8—44no—
296C4×C8—28no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—24no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24C2×C22⋊C412yes—
320C24C2×C22⋊C412yes—
321C24C4×D412yes—
322C24C4×D412yes—
323C24#13112yes—
324C24#13112yes—
325C24C23×C411yescharacteristic
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
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—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
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×C4#12014yes—
413C22×C4#12114yes—
414C22×C4#12114yes—
415C22×C4#12314yes—
416C22×C4#13114yes—
417C22×C4#13112yes—
418C22×C4#13112yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+412yes—
429M16—416no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—416no—
445C2×C8—416no—
446C2×C8—42no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—42no—
451C2×C8—41no—
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—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8#12314yes—
466C4⋊C4—28no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23#38614yes—
534C23#40814yes—
535C23C22×C22⋊C411yescharacteristic
536C23C2×C4×D411yescharacteristic
537C23C2×C4×D411yescharacteristic
538C23C4×#4112yes—
539C23#52012yes—
540C23C2×#13111yescharacteristic
541C23C2×#13111yescharacteristic
542C23C2×#13111yescharacteristic
543C23C2×#13111yescharacteristic
544C23C2×#13112yes—
545C23#56812yes—
546D4—232no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
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—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4#38614yes—
608C2×C4#40814yes—
609C2×C4C2×#12011yescharacteristic
610C2×C4C2×#12111yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4C2×#12311yescharacteristic
613C2×C4C2×#12311yescharacteristic
614C2×C4#52511yescharacteristic
615C2×C4#52912yes—
616C2×C4#53411yescharacteristic
617C2×C4#53611yescharacteristic
618C2×C4#53912yes—
619C2×C4#54612yes—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4#54712yes—
623C2×C4#54712yes—
624C2×C4#54712yes—
625C2×C4#55112yes—
626C2×C4#55212yes—
627C8—24no—
628C8—42no—
629C8—42no—
630C8—42no—
631C8—44no—
632C8—82no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—28no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22C2×#38611yescharacteristic
656C22C4×#12311yescharacteristic
657C22#232211yescharacteristic
658C22#233912yes—
659C22#235912yes—
660C22C2×#40811yescharacteristic
661C22C2×#40811yescharacteristic
662C22#288012yes—
663C22#292911yescharacteristic
664C22#292911yescharacteristic
665C22#292911yescharacteristic
666C22#292911yescharacteristic
667C22#293111yescharacteristic
668C22#293211yescharacteristic
669C22#293512yes—
670C22#296612yes—
671C22#296612yes—
672C22#296812yes—
673C22#296912yes—
674C22#299212yes—
675C22#299312yes—
676C22#299412yes—
677C22#299412yes—
678C4—24no—
679C4—24no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#240511yescharacteristic
687C4#240611yescharacteristic
688C4#241912yes—
689C4#242512yes—
690C4#248611yescharacteristic
691C4#253411yescharacteristic
692C2—44no—
693C2—44no—
694C2#1396811yescharacteristic
695C2#2089511yescharacteristic
696C2#2090011yescharacteristic
697C2#2156611yescharacteristic
698C2#2156711yescharacteristic
699C2#2206411yescharacteristic
700C2#2206611yescharacteristic
701C2#2247012yes—
702C2#2247412yes—
703C2#2290112yes—
704C2#2290912yes—
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
4. C2×#1818C2semidirect
5. C2×#1729C2semidirect
6. #11202C2semidirect
7. #11147C2semidirect
8. #10874C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#23952
C2×#18702
C2×#18182
C2×#17292
#112022
#111472
#108742
#107152
#107142