Finite Group Databasebeta

← #2737526

Subgroups — #2737526

Order 512.

3570 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2435C212yesmaximal
2#17193C212yesmaximal, complemented
3#17186C214yesmaximal, complemented
4#17118C212yesmaximal, complemented
5#16391C212yesmaximal, complemented
6#16391C214yesmaximal, complemented
7#15243C212yesmaximal, complemented
8#15226C212yesmaximal, complemented
9#15217C212yesmaximal, complemented
10#15179C212yesmaximal, complemented
11#15178C211yesmaximal, characteristic, complemented
12#15138C212yesmaximal, complemented
13#14673C212yesmaximal, complemented
14#13902C211yesmaximal, characteristic
15C4×#397C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic, complemented
18C22×#124C2212yes—
19C22×#124C2212yes—
20C22×C22⋊Q8C2212yes—
21#2435—216no—
22C2×#397C2211yescharacteristic
23C2×#397C2214yescomplemented
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#391C2212yescomplemented
27C2×#391C2212yescomplemented
28C2×#391C2212yescomplemented
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yescomplemented
34C2×#380C2211yescharacteristic
35C2×#379C2214yes—
36C2×#379C2212yes—
37C2×#379C2214yes—
38C2×#379C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic, complemented
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42#1445C2211yescharacteristic, complemented
43#1444C2212yescomplemented
44#1444C2212yescomplemented
45#1443C2212yescomplemented
46#1442C2214yescomplemented
47#1442C2211yescharacteristic, complemented
48#1441C2214yescomplemented
49#1440C2212yes—
50#1440C2214yes—
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1439C2212yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2211yescharacteristic
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1439C2214yescomplemented
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1439C2214yescomplemented
70#1438C2212yes—
71#1438C2212yes—
72C4⋊#95C2211yescharacteristic, complemented
73C4⋊#95C2212yescomplemented
74C4×#95C2212yescomplemented
75C4×#95C2211yescharacteristic, complemented
76C26C2312yes—
77C23×Q8C2312yes—
78C23×D4—44no—
79C23×D4—44no—
80C23×D4—42no—
81C23×D4—42no—
82C23×D4D418yes—
83C23×D4C2312yes—
84C23×D4C2312yes—
85C24×C4C2311yescharacteristic
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—44no—
88C2×#124—28no—
89C2×#124—28no—
90C2×#124—44no—
91C2×#124—44no—
92C2×#124—44no—
93C2×#124—44no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#397—24no—
128#397—48no—
129#394—216no—
130#394—216no—
131#391—44no—
132#391—44no—
133#391—44no—
134#390—28no—
135#390—28no—
136#390—44no—
137#390—44no—
138#389—44no—
139#380—42no—
140#379—216no—
141#379—216no—
142#379—28no—
143#379—44no—
144C4×C4⋊C4—42no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—42no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C25—24no—
172C25—24no—
173C25—44no—
174C25—44no—
175C25—42no—
176C25—42no—
177C25—44no—
178C25—44no—
179C25—42no—
180C25—42no—
181C25C2×D418yes—
182C25C2414yes—
183C25C2411yescharacteristic
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—432no—
198C22×D4—416no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—416no—
204C22×D4—432no—
205C22×D4—416no—
206C22×D4C2×D418yes—
207C22×D4C2×D418yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4C2×D414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—416no—
227#124—432no—
228#124—432no—
229C22⋊Q8—432no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—44no—
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—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4#4114yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—41no—
325#95—42no—
326#95—44no—
327#95—41no—
328#95—41no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—21no—
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—216no—
362C24—416no—
363C24—48no—
364C24—416no—
365C24—48no—
366C24—48no—
367C24—416no—
368C24—48no—
369C24—84no—
370C24—84no—
371C24—88no—
372C24—88no—
373C24—84no—
374C24—88no—
375C24—88no—
376C24—84no—
377C24—84no—
378C24—88no—
379C24—84no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24C22×D418yes—
385C24C22×D412yes—
386C24C2×#4112yes—
387C242−1+412yes—
388C24C2511yescharacteristic
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×D4—216no—
397C2×D4—216no—
398C2×D4—216no—
399C2×D4—416no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—416no—
403C2×D4—48no—
404C2×D4—4128no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—216no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—44no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)14yes—
493C4⋊C4—44no—
494C4⋊C4—416no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—88no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—84no—
518C22⋊C4—88no—
519C42—48no—
520C42—48no—
521C42—44no—
522C42—44no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—432no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
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—832no—
582C23—816no—
583C23—816no—
584C23—832no—
585C23—816no—
586C23—832no—
587C23—816no—
588C23—832no—
589C23—816no—
590C23—816no—
591C23#54714yes—
592C23C23×D411yescharacteristic
593C23C22×#4111yescharacteristic
594C23C2×H(5,𝔽2)11yescharacteristic
595C23C2×H(5,𝔽2)11yescharacteristic
596C23C2×H(5,𝔽2)12yes—
597C23C2×H(5,𝔽2)12yes—
598C23C2×H(5,𝔽2)11yescharacteristic
599C23C2×2−1+411yescharacteristic
600C23#58512yes—
601C23#58512yes—
602C23#58511yescharacteristic
603Q8—816no—
604D4—432no—
605D4—432no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—416no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4#53414yes—
682C2×C4#53418yes—
683C2×C4#56014yes—
684C2×C4#56018yes—
685C2×C4H(3,𝔽4)14yes—
686C22—28no—
687C22—24no—
688C22—42no—
689C22—42no—
690C22—44no—
691C22—42no—
692C22—42no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—48no—
698C22—48no—
699C22—48no—
700C22—48no—
701C22—48no—
702C22—48no—
703C22—48no—
704C22—864no—
705C22—84no—
706C22—84no—
707C22—84no—
708C22—84no—
709C22—84no—
710C22—84no—
711C22—84no—
712C22—84no—
713C22C2×#53412yes—
714C22C2×#53411yescharacteristic
715C22C2×#54711yescharacteristic
716C22#348512yes—
717C22#349611yescharacteristic
718C22#350511yescharacteristic
719C22#350811yescharacteristic
720C22#351412yes—
721C22#351412yes—
722C22C2×#56011yescharacteristic
723C22C2×#56012yes—
724C22C2×H(3,𝔽4)11yescharacteristic
725C22#353611yescharacteristic
726C22#354011yescharacteristic
727C22#354612yes—
728C22#355111yescharacteristic
729C22#355812yes—
730C22#356112yes—
731C22#356112yes—
732C22#356111yescharacteristic
733C22#356212yes—
734C22#356212yes—
735C22#356212yes—
736C4—42no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—82no—
741C4—81no—
742C4—84no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—84no—
747C4—84no—
748C4#281218yes—
749C2—44no—
750C2—88no—
751C2C2×#281211yescharacteristic
752C2#3891812yes—
753C2#3948611yescharacteristic
754C2#4243512yes—
755C2#4267311yescharacteristic
756C2#4569711yescharacteristic
757C2#4606112yes—
758C2#4607311yescharacteristic
759C2#5299011yescharacteristic
760C2#5301811yescharacteristic
761C2#5304312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17193C2semidirect
3. #17186C2semidirect
4. #17118C2semidirect
5. #16391C2semidirect
6. #16391C2semidirect
7. #15243C2semidirect
8. #15226C2semidirect
9. #15217C2semidirect
10. #15179C2semidirect
11. #15178C2semidirect
12. #15138C2semidirect
13. #14673C2semidirect
15. C4×#397C2semidirect
17. C22×C4⋊Q8C22semidirect
23. C2×#397C22semidirect
26. C2×#391C22semidirect
27. C2×#391C22semidirect
28. C2×#391C22semidirect
33. C2×#389C22semidirect
39. C2×C4×C4⋊C4C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1444C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
69. #1439C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
74. C4×#95C22semidirect
75. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24352
#171932
#171862
#171182
#163912
#152432
#152262
#152172
#151792
#151782
#151382
#146732
#139022
C4×#3972