Finite Group Databasebeta

← #366280

Subgroups — #366280

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#408C211yesmaximal, characteristic
2C2×#1924C212yesmaximal, complemented
3C2×#1850C211yesmaximal, characteristic, complemented
4C2×#1780C211yesmaximal, characteristic
5C2×#1753C211yesmaximal, characteristic, complemented
6#13851C211yesmaximal, characteristic
7#11147C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10#10715C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#408—28no—
17C2×#408—24no—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1924—28no—
31#1850—24no—
32#1780—24no—
33#1753—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1446C412yes—
42#1438C412yes—
43#1438C412yes—
44#1438C412yes—
45C22×#41—24no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—24no—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41—22no—
52C22×#41—22no—
53C22×#41—42no—
54C22×#41—42no—
55C23×Q8C2×C411yescharacteristic
56C23×D4C2×C412yes—
57C23×D4C2×C411yescharacteristic
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—24no—
63C22×M16—42no—
64C22×M16D414yes—
65C23×C8—22no—
66C23×C8—24no—
67C23×C8—22no—
68C23×C8—22no—
69C23×C8—42no—
70C23×C8C2312yes—
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2312yes—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2312yes—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—42no—
89#408—232no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8—42no—
106C2×C22⋊C8D414yes—
107C2×C22⋊C8D418yes—
108C2×#97—24no—
109C2×#97—22no—
110C2×C4×C8—22no—
111C2×C4×C8—42no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—24no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—42no—
119C4×C22⋊C4—42no—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126#336—24no—
127#336—24no—
128#336—24no—
129#336—22no—
130#336—22no—
131#336—42no—
132#336—42no—
133C25C22×C412yes—
134C25C22×C411yescharacteristic
135C2×#41—28no—
136C2×#41—216no—
137C2×#41—28no—
138C2×#41—48no—
139C2×#41—48no—
140C2×#41—48no—
141C2×#41—48no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—44no—
147C2×#41—44no—
148C2×#41—48no—
149C2×#41—44no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—44no—
153C22×Q8—22no—
154C22×Q8—24no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—42no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—22no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4C22×C412yes—
187C23×C4C22×C412yes—
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C22×C412yes—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D414yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4#4114yes—
196C23×C4#4112yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C2×M16—28no—
202C2×M16—28no—
203C2×M16—44no—
204C2×M16—48no—
205C2×M16—48no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—28no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—48no—
236C22×C8—48no—
237C22×C8—44no—
238C22×C8—44no—
239C22×C8C2×D414yes—
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D414yes—
244C22×C8C2×D412yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C4⋊C8—48no—
279C22⋊C8—216no—
280C22⋊C8—28no—
281C22⋊C8—216no—
282C22⋊C8—48no—
283C22⋊C8—48no—
284C22⋊C8—44no—
285#97—44no—
286C4×C8—44no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—42no—
292#95—42no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24#11712yes—
314C24C4×D414yes—
315C24C4×D412yes—
316C24C4×D412yes—
317C24C23×C411yescharacteristic
318#41—416no—
319#41—416no—
320#41—432no—
321#41—432no—
322#41—416no—
323#41—416no—
324C2×Q8—28no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×D4—28no—
330C2×D4—216no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—44no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
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—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
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—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4#11712yes—
418C22×C4C4×D414yes—
419C22×C4C4×D412yes—
420C22×C4C4×D412yes—
421C22×C4#12114yes—
422C22×C4#12314yes—
423C22×C4#12414yes—
424C22×C4#12614yes—
425C22×C4#13114yes—
426C22×C4#13114yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)12yes—
437M16—416no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—416no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
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—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8#12014yes—
473C22⋊C4—28no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C42—41no—
481C42—42no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—42no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
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—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23#38814yes—
546C23C2×#11711yescharacteristic
547C23C2×C4×D411yescharacteristic
548C23C2×C4×D412yes—
549C23C2×C4×D411yescharacteristic
550C23#51812yes—
551C23C2×#13112yes—
552C23C2×#13112yes—
553C23#56812yes—
554C23#56812yes—
555Q8—416no—
556D4—416no—
557D4—432no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
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—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4#38814yes—
612C2×C4C2×#12011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×#12311yescharacteristic
615C2×C4#52512yes—
616C2×C4#52511yescharacteristic
617C2×C4C2×#12411yescharacteristic
618C2×C4C2×#12611yescharacteristic
619C2×C4#53411yescharacteristic
620C2×C4#53511yescharacteristic
621C2×C4#53812yes—
622C2×C4D4212yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4#54612yes—
627C2×C4#55112yes—
628C2×C4#55112yes—
629C2×C4#55112yes—
630C8—24no—
631C8—42no—
632C8—42no—
633C8—44no—
634C8—42no—
635C8—82no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C2×#38811yescharacteristic
661C22C4×#12011yescharacteristic
662C22#232111yescharacteristic
663C22#234212yes—
664C22#235812yes—
665C22#292211yescharacteristic
666C22#292311yescharacteristic
667C22#292512yes—
668C22#292911yescharacteristic
669C22#293012yes—
670C22#293011yescharacteristic
671C22#293112yes—
672C22#293312yes—
673C22#293312yes—
674C22#293512yes—
675C22#293512yes—
676C22#293512yes—
677C22#297612yes—
678C22#297712yes—
679C22#298012yes—
680C22#298112yes—
681C4—24no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4#240511yescharacteristic
688C4#240811yescharacteristic
689C4#247212yes—
690C4#248012yes—
691C4#248211yescharacteristic
692C4#253411yescharacteristic
693C2—44no—
694C2—84no—
695C2#1399111yescharacteristic
696C2#2104811yescharacteristic
697C2#2105211yescharacteristic
698C2#2144512yes—
699C2#2203712yes—
700C2#2255712yes—
701C2#2256012yes—
702C2#2284512yes—
703C2#2284712yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×#41C22
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
2. C2×#1924C2semidirect
3. C2×#1850C2semidirect
5. C2×#1753C2semidirect
7. #11147C2semidirect
8. #11042C2semidirect
9. #10921C2semidirect
10. #10715C2semidirect
11. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4082
C2×#19242
C2×#18502
C2×#17802
C2×#17532
#138512
#111472
#110422
#109212
#107152
#107142