Finite Group Databasebeta

← #4941754

Subgroups — #4941754

Order 512.

4474 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×H(3,𝔽4)C211yesmaximal, characteristic
2C2×#2731C212yesmaximal, complemented
3C2×#2545C211yesmaximal, characteristic, complemented
4#16559C212yesmaximal, complemented
5#16505C214yesmaximal, complemented
6#15281C212yesmaximal, complemented
7#15268C214yesmaximal, complemented
8#15268C214yesmaximal, complemented
9#15259C212yesmaximal, complemented
10#15070C212yesmaximal, complemented
11#14932C214yesmaximal, complemented
12#14492C211yesmaximal, characteristic, complemented
13#14026C212yesmaximal, complemented
14C2×H(3,𝔽4)—216no—
15C2×H(3,𝔽4)—28no—
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×#124C2214yes—
18C22×#124C2212yes—
19C22×#124C2211yescharacteristic
20C22×#124C2212yes—
21C22×#124C2211yescharacteristic
22C22×#120C2212yes—
23C22×#120C2214yes—
24C22×C4×Q8C2211yescharacteristic, complemented
25#2731—216no—
26#2545—28no—
27C2×#396C2214yescomplemented
28C2×#396C2214yescomplemented
29C2×#395C2212yescomplemented
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#394C2212yes—
33C2×#393C2214yescomplemented
34C2×#391C2211yescharacteristic, complemented
35C2×#391C2212yescomplemented
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#388C2214yescomplemented
40C2×#388C2214yescomplemented
41C2×#388C2214yescomplemented
42C2×#387C2214yescomplemented
43C2×#379C2212yescomplemented
44C2×#379C2212yescomplemented
45C2×#379C2214yescomplemented
46C2×C4×C4⋊C4C2211yescharacteristic, complemented
47#1446C2212yescomplemented
48#1446C2214yescomplemented
49#1440C2212yescomplemented
50#1440C2214yescomplemented
51#1440C2212yescomplemented
52#1440C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2212yes—
65#1437C2214yescomplemented
66#1437C2214yescomplemented
67#1437C2214yescomplemented
68C4⋊#95C2212yescomplemented
69#1434C2212yescomplemented
70C4×#95C2212yescomplemented
71C26C2312yes—
72C23×Q8C2311yescharacteristic
73C23×Q8C2312yes—
74C23×Q8C2311yescharacteristic
75C23×D4C2314yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79H(3,𝔽4)—2128no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×#124—28no—
83C2×#124—28no—
84C2×#124—28no—
85C2×#124—24no—
86C2×#124—216no—
87C2×#124—216no—
88C2×#124—216no—
89C2×#124—28no—
90C2×#124—216no—
91C2×#124—42no—
92C2×#124—42no—
93C2×#124D4116yes—
94C2×#124D418yes—
95C2×#120—28no—
96C2×#120—216no—
97C2×#120—216no—
98C2×#120—216no—
99C2×#120—216no—
100C2×C4×Q8—24no—
101C2×C4×Q8—42no—
102C2×C4×Q8—42no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#396—216no—
135#396—216no—
136#395—44no—
137#394—216no—
138#394—48no—
139#394—44no—
140#393—216no—
141#391—28no—
142#391—24no—
143#390—24no—
144#390—44no—
145#390—44no—
146#388—216no—
147#388—216no—
148#388—216no—
149#387—216no—
150#379—44no—
151#379—44no—
152#379—48no—
153C4×C4⋊C4—24no—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25—24no—
169C25—28no—
170C25—44no—
171C25—44no—
172C25—44no—
173C25—44no—
174C25—42no—
175C25—42no—
176C25—44no—
177C25C2414yes—
178C25C2412yes—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—42no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—44no—
191C22×Q8—42no—
192C22×Q8C2×D414yes—
193C22×Q8C2×D418yes—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—42no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—42no—
216C22×D4C2×D414yes—
217C22×D4C2×D418yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C4⋊Q8—416no—
230#124—264no—
231#124—2128no—
232#124—264no—
233#124—232no—
234#124—416no—
235#120—264no—
236#120—2128no—
237C4×Q8—416no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
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—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
308C2×C22⋊C4C2×D4116yes—
309C2×C22⋊C4C2×D4116yes—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42C2×D414yes—
325C2×C42C2×D418yes—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—42no—
340C24—216no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
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—48no—
364C24—48no—
365C24—416no—
366C24—84no—
367C24—84no—
368C24—88no—
369C24—88no—
370C24—88no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24—88no—
376C24—84no—
377C24—88no—
378C24—88no—
379C24—84no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24C22×D414yes—
384C24C22×D418yes—
385C24C2511yescharacteristic
386C2×Q8—216no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×Q8—88no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×Q8—88no—
400C2×Q8#120116yes—
401C2×D4—216no—
402C2×D4—232no—
403C2×D4—28no—
404C2×D4—216no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4—88no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—88no—
423C2×D4—88no—
424C2×D4—88no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4C22×D414yes—
478C22×C4C22×D418yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)14yes—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—416no—
492C4⋊C4—84no—
493C22⋊C4—232no—
494C22⋊C4—216no—
495C22⋊C4—232no—
496C22⋊C4—232no—
497C22⋊C4—232no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C42—28no—
513C42—28no—
514C42—216no—
515C42—48no—
516C42—44no—
517C42—84no—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—816no—
575C23—832no—
576C23—816no—
577C23—832no—
578C23—832no—
579C23—816no—
580C23—832no—
581C23—832no—
582C23—832no—
583C23C23×D411yescharacteristic
584C23C23×D412yes—
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)12yes—
588C23#58512yes—
589C23#58512yes—
590C23#58514yes—
591Q8—416no—
592Q8—88no—
593Q8—816no—
594D4—816no—
595D4—88no—
596D4—832no—
597D4—816no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
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—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4C2×#12014yes—
659C2×C4C2×#12018yes—
660C22—44no—
661C22—48no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—44no—
666C22—48no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—48no—
671C22—48no—
672C22—864no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22C22×#12011yescharacteristic
686C22#348411yescharacteristic
687C22#348511yescharacteristic
688C22#348814yes—
689C22#348812yes—
690C22#349612yes—
691C22#349612yes—
692C22#350012yes—
693C22#350014yes—
694C22#352911yescharacteristic
695C22#354012yes—
696C22#354014yes—
697C22#354512yes—
698C22#354911yescharacteristic
699C22#355812yes—
700C22#356814yes—
701C4—28no—
702C4—24no—
703C4—42no—
704C4—42no—
705C4—44no—
706C4—48no—
707C4—44no—
708C4—82no—
709C4—84no—
710C4—82no—
711C4—84no—
712C2—84no—
713C2—88no—
714C2#3534711yescharacteristic
715C2#3608212yes—
716C2#4622511yescharacteristic
717C2#4623712yes—
718C2#5107312yes—
719C2#5142414yes—
720C2#5174911yescharacteristic
721C2#5178312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×H(3,𝔽4)C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2731C2semidirect
3. C2×#2545C2semidirect
4. #16559C2semidirect
5. #16505C2semidirect
6. #15281C2semidirect
7. #15268C2semidirect
8. #15268C2semidirect
9. #15259C2semidirect
10. #15070C2semidirect
11. #14932C2semidirect
12. #14492C2semidirect
13. #14026C2semidirect
16. C22×C4⋊Q8C22semidirect
24. C22×C4×Q8C22semidirect
27. C2×#396C22semidirect
28. C2×#396C22semidirect
29. C2×#395C22semidirect
33. C2×#393C22semidirect
34. C2×#391C22semidirect
35. C2×#391C22semidirect
39. C2×#388C22semidirect
40. C2×#388C22semidirect
41. C2×#388C22semidirect
42. C2×#387C22semidirect
43. C2×#379C22semidirect
44. C2×#379C22semidirect
45. C2×#379C22semidirect
46. C2×C4×C4⋊C4C22semidirect
47. #1446C22semidirect
48. #1446C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. #1437C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect
70. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×H(3,𝔽4)2
C2×#27312
C2×#25452
#165592
#165052
#152812
#152682
#152592
#150702
#149322
#144922
#140262