Finite Group Databasebeta

← #366744

Subgroups — #366744

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1850C211yesmaximal, characteristic, complemented
5C2×#1817C211yesmaximal, characteristic, complemented
6C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
7C22×#336C211yesmaximal, characteristic, complemented
8#11346C212yesmaximal, complemented
9#11042C212yesmaximal, complemented
10#10921C212yesmaximal, complemented
11#10834C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#123C411yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2212yes—
24C4×#123—24no—
25C4×#123—24no—
26C2×#388C412yes—
27C2×#387C412yes—
28C2×#386C411yescharacteristic
29C2×#385C411yescharacteristic
30C2×#381C411yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1869—24no—
35#1869—24no—
36#1850—24no—
37#1817—24no—
38C8×C22⋊C4—24no—
39C2×#336—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2211yescharacteristic
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—42no—
62C23×C8—42no—
63C23×C8D414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69C23×C8C2311yescharacteristic
70C2×#123—24no—
71C2×#123—24no—
72C2×#123C818yescomplemented
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C412yes—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2×C411yescharacteristic
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42—22no—
92C22×C42—22no—
93C22×C42—22no—
94C22×C42C2×C412yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2312yes—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—42no—
112C2×C22⋊C8—42no—
113C2×C22⋊C8D414yes—
114C2×C4×C8—24no—
115C2×C4×C8—24no—
116C2×C4×C8—42no—
117#388—28no—
118#387—28no—
119#386—24no—
120#385—24no—
121#381—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4D414yes—
127C4×C22⋊C4D414yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133#336—24no—
134#336—24no—
135#336—24no—
136#336—22no—
137#336—22no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—24no—
142C25C22×C411yescharacteristic
143C25C22×C411yescharacteristic
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C2×C814yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—42no—
176C23×C4C2×C814yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C412yes—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—28no—
196C22×C8—28no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
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—28no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—44no—
225C22×C8—48no—
226C22×C8—48no—
227C22×C8—48no—
228C22×C8—48no—
229C22×C8—44no—
230C22×C8—44no—
231C22×C8C2×D412yes—
232C22×C8C2×D414yes—
233C22×C8C2×D412yes—
234C22×C8#4114yes—
235C22×C8#4114yes—
236#123—232no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×C818yescomplemented
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4C22⋊C414yes—
264C2×C22⋊C4C22⋊C414yes—
265C2×C22⋊C4C2×C814yes—
266C2×C22⋊C4C2×C818yescomplemented
267C2×C22⋊C4M1614yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C4⋊C8—48no—
294C22⋊C8—28no—
295C22⋊C8—48no—
296C22⋊C8—44no—
297C22⋊C8—44no—
298C22⋊C8—44no—
299C4×C8—48no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95C22⋊C418yes—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—22no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24C2×C22⋊C412yes—
320C24C4×D412yes—
321C24C4×D412yes—
322C24C22×C812yes—
323C24C22×C812yes—
324C24C2×M1612yes—
325C24#13112yes—
326C24#13112yes—
327C24C23×C411yescharacteristic
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4C2×C22⋊C414yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C414yes—
422C22×C4C4×D412yes—
423C22×C4C4×D412yes—
424C22×C4#12114yes—
425C22×C4#12114yes—
426C22×C4#12414yes—
427C22×C4#12714yes—
428C22×C4C22×C814yes—
429C22×C4C22×C812yes—
430C22×C4C22×C814yes—
431C22×C4C2×M1614yes—
432C22×C4#13114yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—416no—
457C2×C8—416no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8#12314yescomplemented
479C4⋊C4—216no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C22⋊C4—28no—
483C22⋊C4—216no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4C22⋊C8116yescomplemented
488C42—24no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—42no—
493C42—42no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—41no—
498C42—42no—
499C42—41no—
500C42—48no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23#39014yes—
541C23C2×C22⋊C814yes—
542C23#40814yes—
543C23C8×D414yes—
544C23C8×D414yes—
545C23#43614yes—
546C23#43614yes—
547C23C22×C22⋊C411yescharacteristic
548C23C2×C4×D411yescharacteristic
549C23C2×C4×D411yescharacteristic
550C23C4×#4112yes—
551C23#52012yes—
552C23C23×C811yescharacteristic
553C23C22×M1611yescharacteristic
554C23C2×#13111yescharacteristic
555C23C2×#13111yescharacteristic
556C23C2×#13112yes—
557C23#56812yes—
558D4—416no—
559C2×C4—22no—
560C2×C4—24no—
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—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4#39014yes—
622C2×C4C2×C22⋊C818yes—
623C2×C4C2×#12111yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4C2×#12411yescharacteristic
628C2×C4#52912yes—
629C2×C4C2×#12711yescharacteristic
630C2×C4#53411yescharacteristic
631C2×C4#53611yescharacteristic
632C2×C4#53912yes—
633C2×C4#54612yes—
634C2×C4#54612yes—
635C2×C4#54612yes—
636C2×C4#54712yes—
637C2×C4#54712yes—
638C2×C4#54712yes—
639C2×C4#55112yes—
640C2×C4#55212yes—
641C8—24no—
642C8—42no—
643C8—44no—
644C8—44no—
645C8—81no—
646C8—81no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—28no—
652C22—28no—
653C22—24no—
654C22—28no—
655C22—28no—
656C22—24no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—42no—
661C22—44no—
662C22—81no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22—82no—
667C22—81no—
668C22#195418yes—
669C22C2×#39011yescharacteristic
670C22C4×#12311yescharacteristic
671C22#232211yescharacteristic
672C22#233912yes—
673C22#235912yes—
674C22C22×C22⋊C811yescharacteristic
675C22C2×#40811yescharacteristic
676C22#288012yes—
677C22C2×C8×D411yescharacteristic
678C22C2×C8×D411yescharacteristic
679C22C2×#43611yescharacteristic
680C22C2×#43611yescharacteristic
681C22#292911yescharacteristic
682C22#293011yescharacteristic
683C22#293312yes—
684C22C8×#4112yes—
685C22#296612yes—
686C22#296812yes—
687C22#297012yes—
688C22#298812yes—
689C22#299012yes—
690C22#299212yes—
691C22#299412yes—
692C4—24no—
693C4—28no—
694C4—44no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—81no—
699C4—81no—
700C4#241211yescharacteristic
701C4#241311yescharacteristic
702C4#242512yes—
703C4#248812yes—
704C4#250111yescharacteristic
705C4#253911yescharacteristic
706C2—28no—
707C2—82no—
708C2#1399611yescharacteristic
709C2C2×#195411yescharacteristic
710C2#2119111yescharacteristic
711C2C8×#12311yescharacteristic
712C2#2156411yescharacteristic
713C2#2206311yescharacteristic
714C2#2206711yescharacteristic
715C2#2246912yes—
716C2#2247012yes—
717C2#2290012yes—
718C2#2291012yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ 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×#1850C2semidirect
5. C2×#1817C2semidirect
6. C2×C8×C22⋊C4C2semidirect
7. C22×#336C2semidirect
8. #11346C2semidirect
9. #11042C2semidirect
10. #10921C2semidirect
11. #10834C2semidirect
72. C2×#123C8semidirect
249. C2×C4⋊C4C2×C8semidirect
266. C2×C22⋊C4C2×C8semidirect
478. C2×C8#123semidirect
487. C22⋊C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#18692
C2×#18502
C2×#18172
C2×C8×C22⋊C42
C22×#3362
#113462
#110422
#109212
#108342