Finite Group Databasebeta

← #848404

Subgroups — #848404

Order 512.

3460 conjugacy classes of subgroups.

The subgroup lattice has 813 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×C4×#123C211yesmaximal, characteristic, complemented
3#16537C214yesmaximal, complemented
4#15635C212yesmaximal, complemented
5#15561C212yesmaximal, complemented
6#15313C212yesmaximal, complemented
7#15261C212yesmaximal, complemented
8#15127C212yesmaximal, complemented
9#15126C212yesmaximal, complemented
10#15126C212yesmaximal, complemented
11#14698C212yesmaximal, complemented
12#14568C214yesmaximal, complemented
13#14136C212yesmaximal, complemented
14#14041C212yesmaximal, complemented
15#13871C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#127C2212yes—
18C22×#123C2211yescharacteristic
19C22×#121C2212yes—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2541—28no—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#392C2212yes—
30C2×#392C2212yes—
31C4×#123—28no—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#386C2211yescharacteristic
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#385C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51#1445C2214yes—
52#1442C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2214yes—
65#1437C2212yes—
66#1437C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2314yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2314yes—
82C23×D4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#127—28no—
88C2×#127—24no—
89C2×#127—42no—
90C2×#127—42no—
91C2×#127—48no—
92C2×#123—24no—
93C2×#123—28no—
94C2×#121—28no—
95C2×#121—28no—
96C2×#121—28no—
97C2×#121—28no—
98C2×#121—44no—
99C2×#121—44no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4D418yes—
103C2×#117—44no—
104C2×#117—42no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C42—24no—
132C22×C42—42no—
133C22×C42—42no—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140#394—216no—
141#394—48no—
142#394—48no—
143#394—48no—
144#392—28no—
145#392—28no—
146#390—28no—
147#390—44no—
148#390—44no—
149#390—44no—
150#390—44no—
151#389—28no—
152#388—28no—
153#387—28no—
154#386—24no—
155#386—48no—
156#386—48no—
157#385—24no—
158#381—24no—
159#380—44no—
160#380—42no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—28no—
163C4×C22⋊C4—28no—
164C4×C22⋊C4—24no—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C25C2414yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×D4—24no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4C2×D414yes—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—42no—
235C23×C4—41no—
236C23×C4—41no—
237C23×C4—41no—
238C23×C4—41no—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253#127—416no—
254#127—432no—
255#123—232no—
256#121—432no—
257#121—432no—
258C4×D4—416no—
259#117—416no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4#4118yes—
280C2×C4⋊C4#4118yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42—42no—
346C2×C42—44no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—416no—
350C2×C42C2×D414yes—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—44no—
361#95—44no—
362#95—42no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24C22×D412yes—
399C24C22×D412yes—
400C24H(5,𝔽2)14yes—
401C24C2511yescharacteristic
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—416no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—416no—
415C2×D4—416no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—88no—
431C2×D4—88no—
432C2×D4—88no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—88no—
436C2×D4—88no—
437C2×D4—88no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—216no—
481C22×C4—216no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—44no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—44no—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C2×#4118yes—
524C22×C4C2×#4114yes—
525C22×C4C2×#4114yes—
526C22×C4C2×#4114yes—
527C22×C4C2×#4114yes—
528C22×C4H(5,𝔽2)14yes—
529C22×C4H(5,𝔽2)12yes—
530C22×C4H(5,𝔽2)12yes—
531C22×C4H(5,𝔽2)14yes—
532C4⋊C4—216no—
533C4⋊C4—216no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—88no—
539C22⋊C4—28no—
540C22⋊C4—216no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—416no—
544C22⋊C4—416no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—44no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C22⋊C4—88no—
554C22⋊C4—88no—
555C22⋊C4—84no—
556C22⋊C4—88no—
557C22⋊C4—88no—
558C42—28no—
559C42—48no—
560C42—48no—
561C42—48no—
562C42—48no—
563C42—48no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—24no—
579C23—42no—
580C23—41no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—42no—
589C23—42no—
590C23—41no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—44no—
626C23—44no—
627C23—44no—
628C23—44no—
629C23—44no—
630C23—44no—
631C23—44no—
632C23—44no—
633C23—44no—
634C23—44no—
635C23—44no—
636C23—44no—
637C23—44no—
638C23—44no—
639C23—44no—
640C23#53514yes—
641C23C23×D411yescharacteristic
642C23C22×#4112yes—
643C23C22×#4112yes—
644C23C2×H(5,𝔽2)11yescharacteristic
645C23C2×H(5,𝔽2)11yescharacteristic
646C23C2×H(5,𝔽2)12yes—
647C23C2×H(5,𝔽2)12yes—
648C23C2×H(5,𝔽2)12yes—
649C23#58512yes—
650D4—416no—
651D4—816no—
652D4—832no—
653D4—832no—
654D4—816no—
655D4—816no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—28no—
666C2×C4—24no—
667C2×C4—24no—
668C2×C4—24no—
669C2×C4—24no—
670C2×C4—24no—
671C2×C4—24no—
672C2×C4—28no—
673C2×C4—24no—
674C2×C4—24no—
675C2×C4—28no—
676C2×C4—24no—
677C2×C4—24no—
678C2×C4—416no—
679C2×C4—416no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—41no—
697C2×C4—41no—
698C2×C4—41no—
699C2×C4—42no—
700C2×C4—41no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—416no—
706C2×C4—416no—
707C2×C4—42no—
708C2×C4—44no—
709C2×C4—42no—
710C2×C4—44no—
711C2×C4—42no—
712C2×C4—44no—
713C2×C4—44no—
714C2×C4—42no—
715C2×C4—44no—
716C2×C4—44no—
717C2×C4—44no—
718C2×C4—42no—
719C2×C4—44no—
720C2×C4—44no—
721C2×C4—44no—
722C2×C4—44no—
723C2×C4—42no—
724C2×C4—44no—
725C2×C4—44no—
726C2×C4—42no—
727C2×C4—44no—
728C2×C4#52514yes—
729C2×C4#53414yes—
730C2×C4#54018yes—
731C2×C4#54618yes—
732C2×C4#55914yes—
733C2×C4#56014yes—
734C22—24no—
735C22—28no—
736C22—44no—
737C22—44no—
738C22—44no—
739C22—42no—
740C22—42no—
741C22—44no—
742C22—44no—
743C22—42no—
744C22—44no—
745C22—44no—
746C22—42no—
747C22—42no—
748C22—82no—
749C22—82no—
750C22—82no—
751C22—82no—
752C22—84no—
753C22—84no—
754C22—84no—
755C22—84no—
756C22—84no—
757C22—84no—
758C22—84no—
759C22—84no—
760C22—84no—
761C22—84no—
762C22—84no—
763C22—84no—
764C22—84no—
765C22—84no—
766C22—84no—
767C22C2×#52511yescharacteristic
768C22C2×#53411yescharacteristic
769C22C2×#53511yescharacteristic
770C22C2×#54012yes—
771C22C2×#54612yes—
772C22C2×#55911yescharacteristic
773C22#348012yes—
774C22#348312yes—
775C22#348812yes—
776C22#350812yes—
777C22#351312yes—
778C22#351712yes—
779C22#351912yes—
780C22#351912yes—
781C22#351912yes—
782C22C2×#56011yescharacteristic
783C22#353711yescharacteristic
784C22#354011yescharacteristic
785C22#354212yes—
786C22#354312yes—
787C22#355812yes—
788C4—24no—
789C4—44no—
790C4—44no—
791C4—48no—
792C4—44no—
793C4—42no—
794C4—42no—
795C4—42no—
796C4—84no—
797C4—82no—
798C4—84no—
799C4#270918yes—
800C2—44no—
801C2—82no—
802C2—164no—
803C2#3507611yescharacteristic
804C2#3641012yes—
805C2#3726911yescharacteristic
806C2#3802612yes—
807C2#3923712yes—
808C2#4012612yes—
809C2#4560412yes—
810C2C2×#270911yescharacteristic
811C2#5963312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×C4×#123C2semidirect
3. #16537C2semidirect
4. #15635C2semidirect
5. #15561C2semidirect
6. #15313C2semidirect
7. #15261C2semidirect
8. #15127C2semidirect
9. #15126C2semidirect
10. #15126C2semidirect
11. #14698C2semidirect
12. #14568C2semidirect
13. #14136C2semidirect
14. #14041C2semidirect
15. #13871C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×C4×#1232
#165372
#156352
#155612
#153132
#152612
#151272
#151262
#146982
#145682
#141362
#140412
#138712