Finite Group Databasebeta

← #616471

Subgroups — #616471

Order 512.

3418 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2851C211yesmaximal, characteristic, complemented
3C22×#382C211yesmaximal, characteristic, complemented
4C2×C42×D4C211yesmaximal, characteristic
5#15221C212yesmaximal, complemented
6#15220C212yesmaximal, complemented
7#15075C214yesmaximal, complemented
8#14923C214yesmaximal, complemented
9#14709C211yesmaximal, characteristic, complemented
10#14707C211yesmaximal, characteristic, complemented
11#14682C211yesmaximal, characteristic, complemented
12#14047C214yesmaximal, complemented
13#13975C214yesmaximal
14#13907C212yesmaximal, complemented
15#13872C212yesmaximal
16C22×#125C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C42C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2851—28no—
23#2851—28no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#387C2214yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#382—28no—
33C2×#382C2211yescharacteristic
34C2×#382C2214yes—
35C2×#382C2211yescharacteristic
36C2×#382C2212yes—
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2214yes—
41C2×#380C2212yes—
42C42×D4—28no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2214yes—
50C2×C43C2211yescharacteristic
51#1445C2214yes—
52#1444C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2314yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C2×#125—24no—
82C2×#125—24no—
83C2×#125—24no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—28no—
89C2×C4×D4Q818yes—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42—24no—
116C22×C42D418yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#389—28no—
127#389—28no—
128#389—44no—
129#387—216no—
130#387—216no—
131#386—216no—
132#385—216no—
133#385—216no—
134#382—28no—
135#382—24no—
136#382—24no—
137#382—216no—
138#382D418yescomplemented
139#380—28no—
140#380—28no—
141#380—216no—
142#380—44no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—24no—
149C2×#95—216no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C43—24no—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4C2×Q814yes—
179C22×D4#4118yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×Q814yes—
213C23×C4C2×Q814yes—
214C23×C4#4114yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#125—416no—
229C4×D4—264no—
230C4×D4—232no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—432no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×Q814yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C2×Q814yes—
284C2×C22⋊C4C2×Q814yes—
285C2×C22⋊C4#4118yes—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—232no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—416no—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×Q814yes—
314C2×C42#4114yes—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95C2×D4116yes—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24C22×D412yes—
353C24C22×Q812yes—
354C24C22×Q812yes—
355C24C2×#4114yes—
356C24C2×#4112yes—
357C24C2×#4114yes—
358C24C2×#4114yes—
359C24C2511yescharacteristic
360C2×D4—28no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—28no—
364C2×D4#125116yescomplemented
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—216no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—416no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4C22⋊Q8116yes—
466C22×C4C22×D418yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C4⋊C4—216no—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C22⋊C4—216no—
499C22⋊C4—28no—
500C22⋊C4—216no—
501C22⋊C4—28no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—88no—
505C42—216no—
506C42—216no—
507C42—28no—
508C42—28no—
509C42—48no—
510C42—48no—
511C42—44no—
512C42—44no—
513C42#121116yes—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C2×C22⋊Q814yes—
555C23#52518yes—
556C23C2×#12514yes—
557C23C2×#12514yes—
558C23#52918yes—
559C23#53314yes—
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C23×Q811yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4112yes—
566C23C22×#4112yes—
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58512yes—
571D4—232no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—232no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—432no—
608C2×C4—432no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—416no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—832no—
654C2×C4C2×#12118yes—
655C2×C4C2×#12114yes—
656C2×C4C2×C22⋊Q814yes—
657C2×C4C2×C22⋊Q814yes—
658C2×C4C2×#12514yes—
659C2×C4#53714yes—
660C2×C4#54614yes—
661C2×C4D4×Q814yes—
662C2×C4#55214yes—
663C2×C4#55414yes—
664C22—24no—
665C22—24no—
666C22—28no—
667C22—24no—
668C22—28no—
669C22—24no—
670C22—28no—
671C22—24no—
672C22—24no—
673C22—28no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22#285118yescomplemented
686C22C22×#12111yescharacteristic
687C22C22×C22⋊Q811yescharacteristic
688C22C2×#52512yes—
689C22C22×#12511yescharacteristic
690C22C2×#52912yes—
691C22C2×#53311yescharacteristic
692C22C2×#53711yescharacteristic
693C22#344912yes—
694C22C2×#54611yescharacteristic
695C22C2×D4×Q811yescharacteristic
696C22D4×#4112yes—
697C22D4×#4112yes—
698C22C2×#55211yescharacteristic
699C22C2×#55411yescharacteristic
700C22#347512yes—
701C22#347512yes—
702C22#350411yescharacteristic
703C22#350411yescharacteristic
704C22#351012yes—
705C22#351412yes—
706C22#351412yes—
707C22#351512yes—
708C22#351512yes—
709C4—24no—
710C4—24no—
711C4—28no—
712C4—42no—
713C4—48no—
714C4—44no—
715C4—42no—
716C4—44no—
717C4—44no—
718C4—84no—
719C4—84no—
720C4—84no—
721C4#241718yes—
722C2—28no—
723C2—44no—
724C2C2×#241711yescharacteristic
725C2#3473712yes—
726C2#3513012yes—
727C2#3533411yescharacteristic
728C2D4×#12511yescharacteristic
729C2#3611112yes—
730C2#3613111yescharacteristic
731C2#3616511yescharacteristic
732C2#3617112yes—
733C2#3619111yescharacteristic
734C2C2×#285111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2851C2semidirect
2. C2×#2851C2semidirect
3. C22×#382C2semidirect
5. #15221C2semidirect
6. #15220C2semidirect
7. #15075C2semidirect
8. #14923C2semidirect
9. #14709C2semidirect
10. #14707C2semidirect
11. #14682C2semidirect
12. #14047C2semidirect
14. #13907C2semidirect
138. #382D4semidirect
364. C2×D4#125semidirect
685. C22#2851semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C22×#3822
C2×C42×D42
#152212
#152202
#150752
#149232
#147092
#147072
#146822
#140472
#139752
#139072
#138722