Finite Group Databasebeta

← #900745

Subgroups — #900745

Order 512.

4038 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
1C2×#2684C212yesmaximal, complemented
2C2×#2679C211yesmaximal, characteristic
3C2×C4×#124C211yesmaximal, characteristic, complemented
4#17077C212yesmaximal, complemented
5#16684C212yesmaximal, complemented
6#15404C214yesmaximal, complemented
7#15281C214yesmaximal, complemented
8#14883C212yesmaximal, complemented
9#14802C214yesmaximal, complemented
10#14700C212yesmaximal, complemented
11#14440C211yesmaximal, characteristic, complemented
12#13995C214yesmaximal, complemented
13#13967C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×#120C2212yes—
16C22×#120C2214yes—
17C22×C4×Q8C2211yescharacteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C23×C22⋊C4C2212yescomplemented
20#2684—216no—
21#2679—28no—
22C2×#397C2214yescomplemented
23C2×#394C2212yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#394C2212yes—
27C2×#393C2214yescomplemented
28C4×#124—28no—
29C2×#391C2211yescharacteristic, complemented
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#389C2212yescomplemented
36C2×#388C2214yescomplemented
37C2×#388C2214yescomplemented
38C2×#388C2214yescomplemented
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#385C2214yescomplemented
43C2×#382C2211yescharacteristic, complemented
44C2×#379C2214yescomplemented
45C2×#379C2212yescomplemented
46C2×C4×C22⋊C4C2214yescomplemented
47C2×C43C2211yescharacteristic, complemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2212yescomplemented
51#1441C2212yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65C4⋊#95C2212yescomplemented
66C4⋊#95C2212yescomplemented
67#1434C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C22⋊#95C2212yescomplemented
70C26C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#124—24no—
79C2×#124—24no—
80C2×#124D418yescomplemented
81C2×#120—28no—
82C2×#120—28no—
83C2×#120—216no—
84C2×#120—28no—
85C2×#120—216no—
86C2×#120—48no—
87C2×C4×Q8—24no—
88C2×C4×Q8—42no—
89C2×C4×Q8—42no—
90C2×C4×D4—24no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—48no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#397—216no—
127#394—28no—
128#394—216no—
129#394—216no—
130#394—44no—
131#393—216no—
132#391—24no—
133#390—28no—
134#390—28no—
135#390—44no—
136#390—44no—
137#390D418yes—
138#389—44no—
139#388—216no—
140#388—216no—
141#388—48no—
142#386—28no—
143#386—216no—
144#386—48no—
145#385—48no—
146#382—24no—
147#379—216no—
148#379—44no—
149C4×C22⋊C4—216no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C43—24no—
165C25—28no—
166C25—24no—
167C25—24no—
168C25—24no—
169C25—42no—
170C25—44no—
171C25—42no—
172C25—44no—
173C25—44no—
174C25—42no—
175C25—42no—
176C25C2414yes—
177C25C2412yes—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—41no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×Q8—41no—
184C22×Q8C2×D414yes—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4C2×D414yes—
208C22×D4C2×D414yes—
209C22×D4#4118yes—
210C22×D4#4114yes—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232#124—232no—
233#120—464no—
234#120—432no—
235C4×Q8—416no—
236C4×D4—416no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
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—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—432no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D4116yes—
311C2×C22⋊C4C2×D4116yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—42no—
325C2×C42—41no—
326C2×C42—41no—
327C2×C42—42no—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—42no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—416no—
367C24—48no—
368C24—416no—
369C24—416no—
370C24—48no—
371C24—416no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—84no—
376C24—84no—
377C24—88no—
378C24—84no—
379C24—84no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24—84no—
385C24—84no—
386C24—88no—
387C24—88no—
388C24—88no—
389C24C22×D414yes—
390C24C22×D414yes—
391C24C2×#4118yes—
392C24C2×#4114yes—
393C24C2511yescharacteristic
394C2×Q8—28no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×D4—216no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—48no—
413C2×D4—416no—
414C2×D4—416no—
415C2×D4—416no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—88no—
422C2×D4—88no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—84no—
426C2×D4#121116yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—48no—
485C22×C4—416no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—84no—
489C22×C4—88no—
490C22×C4—84no—
491C22×C4—88no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4C22×D414yes—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4114yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)14yes—
504C22×C4H(5,𝔽2)14yes—
505C22×C4H(5,𝔽2)12yes—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—88no—
513C4⋊C4—84no—
514C22⋊C4—232no—
515C22⋊C4—232no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—84no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C42—28no—
532C42—28no—
533C42—44no—
534C42—48no—
535C42—48no—
536C42—48no—
537C23—24no—
538C23—24no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—432no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—832no—
596C23—816no—
597C23—816no—
598C23—816no—
599C23—832no—
600C23—832no—
601C23—832no—
602C23—816no—
603C23—816no—
604C23—816no—
605C23C2×#12118yes—
606C23C23×D411yescharacteristic
607C23C23×D411yescharacteristic
608C23C22×#4111yescharacteristic
609C23C22×#4112yes—
610C23C2×H(5,𝔽2)11yescharacteristic
611C23C2×H(5,𝔽2)12yes—
612C23C2×H(5,𝔽2)11yescharacteristic
613C23C2×H(5,𝔽2)12yes—
614C23#58511yescharacteristic
615C23#58511yescharacteristic
616C23#58512yes—
617Q8—416no—
618D4—416no—
619D4—816no—
620D4—816no—
621D4—832no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—816no—
689C2×C4—816no—
690C2×C4C2×#12114yes—
691C2×C4#54618yes—
692C2×C4H(3,𝔽4)14yes—
693C22—216no—
694C22—28no—
695C22—48no—
696C22—44no—
697C22—44no—
698C22—48no—
699C22—48no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—48no—
705C22—48no—
706C22—48no—
707C22—48no—
708C22—48no—
709C22—48no—
710C22—48no—
711C22—864no—
712C22—84no—
713C22—84no—
714C22—84no—
715C22—84no—
716C22—84no—
717C22—84no—
718C22—84no—
719C22—84no—
720C22C22×#12111yescharacteristic
721C22#345912yes—
722C22#345911yescharacteristic
723C22#345911yescharacteristic
724C22C2×#54612yes—
725C22D4×#4112yes—
726C22#348112yes—
727C22#348111yescharacteristic
728C22#348411yescharacteristic
729C22#348511yescharacteristic
730C22#348812yes—
731C22#349112yes—
732C22#349611yescharacteristic
733C22#349611yescharacteristic
734C22#349812yes—
735C22#351412yes—
736C22#351912yes—
737C22C2×H(3,𝔽4)11yescharacteristic
738C22#354511yescharacteristic
739C22#354911yescharacteristic
740C22#355812yes—
741C22#355812yes—
742C22#356012yes—
743C4—28no—
744C4—24no—
745C4—24no—
746C4—42no—
747C4—44no—
748C4—44no—
749C4—44no—
750C4—84no—
751C4—82no—
752C4—84no—
753C4—84no—
754C2—48no—
755C2—88no—
756C2#3498411yescharacteristic
757C2#3625812yes—
758C2#3715011yescharacteristic
759C2#3774612yes—
760C2#3876812yes—
761C2#3944412yes—
762C2#4597711yescharacteristic
763C2#4599611yescharacteristic
764C2#5230111yescharacteristic
765C2#5904111yescharacteristic
766C2#5983611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
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×#2684C2semidirect
3. C2×C4×#124C2semidirect
4. #17077C2semidirect
5. #16684C2semidirect
6. #15404C2semidirect
7. #15281C2semidirect
8. #14883C2semidirect
9. #14802C2semidirect
10. #14700C2semidirect
11. #14440C2semidirect
12. #13995C2semidirect
13. #13967C2semidirect
17. C22×C4×Q8C22semidirect
19. C23×C22⋊C4C22semidirect
22. C2×#397C22semidirect
27. C2×#393C22semidirect
29. C2×#391C22semidirect
35. C2×#389C22semidirect
36. C2×#388C22semidirect
37. C2×#388C22semidirect
38. C2×#388C22semidirect
42. C2×#385C22semidirect
43. C2×#382C22semidirect
44. C2×#379C22semidirect
45. C2×#379C22semidirect
46. C2×C4×C22⋊C4C22semidirect
47. C2×C43C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
68. C22⋊#95C22semidirect
69. C22⋊#95C22semidirect
80. C2×#124D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26842
C2×#26792
C2×C4×#1242
#170772
#166842
#154042
#152812
#148832
#148022
#147002
#144402
#139952
#139672