Finite Group Databasebeta

← #394765

Subgroups — #394765

Order 512.

2084 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic, complemented
2C2×#2417C211yesmaximal, characteristic
3C2×#2036C212yesmaximal, complemented
4C2×#1892C211yesmaximal, characteristic, complemented
5C2×#1872C211yesmaximal, characteristic, complemented
6C2×#1869C211yesmaximal, characteristic
7#13872C211yesmaximal, characteristic
8#11219C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11096C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×#107—28no—
19C22×#107—24no—
20C22×#107C2212yes—
21C22×#107C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26#2417—24no—
27#2417—24no—
28C2×#398—21no—
29C2×#396—21no—
30C2×#391C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#386—21no—
33C2×#386—21no—
34C2×#380—21no—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38#2036—28no—
39#1892—24no—
40#1872—24no—
41#1869—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47#1441—21no—
48#1439—21no—
49C4⋊#95C2212yes—
50C22⋊#95—21no—
51C23×Q8C2311yescharacteristic
52C23×D4D411yescharacteristic
53C24×C4—21no—
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—24no—
60C23×C8—24no—
61C23×C8D414yes—
62C23×C8C2312yes—
63C23×C8C2312yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—42no—
73C2×C22⋊Q8D414yes—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4D411yescharacteristic
93C22×C4⋊C4D414yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4D411yescharacteristic
105C22×C22⋊C4D411yescharacteristic
106C22×C42C2311yescharacteristic
107C2×#107—232no—
108C2×#107—24no—
109C2×#107—24no—
110C2×#107—24no—
111C2×#107—24no—
112C2×#107—24no—
113C2×#107D418yescomplemented
114C2×#103—24no—
115C2×#103—24no—
116C2×#103—24no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103—42no—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8—24no—
124C2×C22⋊C8—24no—
125C2×C22⋊C8—24no—
126#398—28no—
127#396—28no—
128#391—22no—
129#391—22no—
130#389—22no—
131#389—22no—
132#386—28no—
133#386—44no—
134#380—28no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—24no—
140#380—42no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—22no—
149#340—24no—
150#340—24no—
151#340—24no—
152#340—42no—
153#336—24no—
154C25C2×D411yescharacteristic
155C25C2×D411yescharacteristic
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8C2×D412yes—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4Q1614yes—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—82no—
202C23×C4—82no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4#4112yes—
211C23×C4#4112yes—
212C23×C4#4112yes—
213C23×C4C2411yescharacteristic
214C22×C8—216no—
215C22×C8—28no—
216C22×C8—28no—
217C22×C8—28no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—216no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8C2×D414yes—
231C22×C8C2×D414yes—
232C22×C8#4114yes—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C22⋊Q8—48no—
236C22⋊Q8—44no—
237C22⋊Q8—44no—
238C22⋊Q8—44no—
239C22⋊Q8—44no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—44no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D412yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4—82no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315#107—216no—
316#107—216no—
317#103—48no—
318#103—48no—
319C22⋊C8—44no—
320C22⋊C8—48no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—82no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24#12012yes—
344C24#12112yes—
345C24#12312yes—
346C24C2×Q1612yes—
347C24C2×Q1612yes—
348C24#13512yes—
349C24AGL(1,ℤ/8)12yes—
350C24#13712yes—
351C24C22×D411yescharacteristic
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—48no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—84no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4#12012yes—
472C22×C4#12114yes—
473C22×C4#12114yes—
474C22×C4#12112yes—
475C22×C4#12314yes—
476C22×C4#12312yes—
477C22×C4#12414yes—
478C22×C4C2×Q1612yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4111yescharacteristic
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+412yes—
489C2×C8—232no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8#12118yes—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—84no—
527C4⋊C4#12014yes—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—84no—
533C22⋊C4—88no—
534C22⋊C4—84no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—22no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—41no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—42no—
578C23#39414yes—
579C23#44914yes—
580C23#45114yes—
581C23#46714yes—
582C23#46814yes—
583C23#48414yes—
584C23#48514yes—
585C23C2×#12011yescharacteristic
586C23C2×#12111yescharacteristic
587C23C2×#12311yescharacteristic
588C23D4212yes—
589C23#54712yes—
590C23C22×Q1611yescharacteristic
591C23C2×#13511yescharacteristic
592C23C2×AGL(1,ℤ/8)11yescharacteristic
593C23C2×#13711yescharacteristic
594C23#57612yes—
595C23#57812yes—
596Q8—44no—
597Q8—44no—
598Q8—81no—
599Q8—81no—
600Q8—81no—
601Q8—81no—
602D4—416no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—816no—
648C2×C4—816no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—82no—
663C2×C4—81no—
664C2×C4—82no—
665C2×C4—82no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4#39414yes—
673C2×C4C2×#12011yescharacteristic
674C2×C4C2×#12111yescharacteristic
675C2×C4C2×#12112yes—
676C2×C4C2×#12111yescharacteristic
677C2×C4C2×#12311yescharacteristic
678C2×C4#52511yescharacteristic
679C2×C4#52512yes—
680C2×C4C2×#12411yescharacteristic
681C2×C4#53411yescharacteristic
682C2×C4#53611yescharacteristic
683C2×C4#53712yes—
684C2×C4#53912yes—
685C2×C4D4212yes—
686C2×C4#54712yes—
687C2×C4#54812yes—
688C2×C4#54812yes—
689C2×C4#55312yes—
690C2×C4#55312yes—
691C8—28no—
692C8—82no—
693C22—28no—
694C22—28no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—28no—
702C22—28no—
703C22—44no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—82no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—82no—
714C22#203618yescomplemented
715C22C2×#39411yescharacteristic
716C22#240311yescharacteristic
717C22#240611yescharacteristic
718C22#241612yes—
719C22#244312yes—
720C22C2×#45111yescharacteristic
721C22C2×#44911yescharacteristic
722C22#300811yescharacteristic
723C22#301111yescharacteristic
724C22C2×#46711yescharacteristic
725C22C2×#46811yescharacteristic
726C22#306112yes—
727C22C2×#48511yescharacteristic
728C22C2×#48411yescharacteristic
729C22#309812yes—
730C22#326412yes—
731C22#327012yes—
732C22#328012yes—
733C22D4×Q1612yes—
734C22#333612yes—
735C22#333812yes—
736C22#334312yes—
737C22#334612yes—
738C4—24no—
739C4—44no—
740C4—42no—
741C4—82no—
742C4—84no—
743C4—81no—
744C4—82no—
745C4—81no—
746C4—82no—
747C4—162no—
748C4#241612yes—
749C4#243611yescharacteristic
750C4#244312yes—
751C4#259111yescharacteristic
752C4#261811yescharacteristic
753C4#284711yescharacteristic
754C2—28no—
755C2—82no—
756C2#1471911yescharacteristic
757C2C2×#203611yescharacteristic
758C2#2348811yescharacteristic
759C2#2385111yescharacteristic
760C2#2385611yescharacteristic
761C2#2409411yescharacteristic
762C2#2409811yescharacteristic
763C2#2478112yes—
764C2#2478712yes—
765C2#2613212yes—
766C2#2613412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#107C2semidirect
3. C2×#2036C2semidirect
4. C2×#1892C2semidirect
5. C2×#1872C2semidirect
8. #11219C2semidirect
9. #11124C2semidirect
10. #11096C2semidirect
11. #10730C2semidirect
113. C2×#107D4semidirect
714. C22#2036semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C2×#24172
C2×#20362
C2×#18922
C2×#18722
C2×#18692
#138722
#112192
#111242
#110962
#107302