Finite Group Databasebeta

← #4312846

Subgroups — #4312846

Order 512.

4574 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#536C211yesmaximal, characteristic
2C2×#2734C212yesmaximal, complemented
3C2×#2432C211yesmaximal, characteristic
4#16681C214yesmaximal, complemented
5#16264C212yesmaximal, complemented
6#16254C212yesmaximal, complemented
7#15983C212yesmaximal, complemented
8#15271C214yesmaximal, complemented
9#15262C212yesmaximal, complemented
10#15114C212yesmaximal, complemented
11#15001C212yesmaximal, complemented
12#14874C212yesmaximal, complemented
13#14763C212yesmaximal, complemented
14#14450C212yesmaximal, complemented
15#13937C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18C2×#536—28no—
19C2×#536—216no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×#124C2212yes—
23C22×#124C2211yescharacteristic
24C22×#123C2212yes—
25C22×#123C2212yes—
26C22×C22⋊Q8C2212yes—
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2211yescharacteristic
29C22×#117C2211yescharacteristic
30#2734—216no—
31#2432—28no—
32C2×#398C2214yes—
33C2×#395C2214yes—
34C2×#395C2214yes—
35C2×#395C2212yes—
36C2×#395C2212yes—
37C2×#394C2212yes—
38C2×#394C2212yes—
39C2×#391C2212yes—
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2212yes—
43C2×#389C2214yes—
44C2×#387C2214yes—
45C2×#387C2214yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#385C2214yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1446C2214yes—
53#1446C2212yes—
54#1446C2212yes—
55#1444C2212yes—
56#1442C2212yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1438C2212yes—
73#1436C2214yes—
74#1436C2214yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2214yes—
78C22⋊#95C2212yes—
79C22×#41—28no—
80C22×#41—42no—
81C22×#41—42no—
82C22×#41—44no—
83C22×#41D418yes—
84C23×Q8—28no—
85C23×Q8—24no—
86C23×Q8—44no—
87C23×Q8—42no—
88C23×Q8—42no—
89C23×Q8C2312yes—
90C23×Q8C2311yescharacteristic
91C23×Q8C2312yes—
92C23×Q8C2311yescharacteristic
93C23×D4C2311yescharacteristic
94C23×D4C2311yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2312yes—
98C24×C4C2311yescharacteristic
99C24×C4C2311yescharacteristic
100#536—2128no—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—28no—
105C2×C4⋊Q8D4116yes—
106C2×#124—28no—
107C2×#124—28no—
108C2×#124—24no—
109C2×#124—28no—
110C2×#124—44no—
111C2×#123—28no—
112C2×#123—28no—
113C2×#123—216no—
114C2×#123—216no—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—216no—
119C2×C22⋊Q8—216no—
120C2×C22⋊Q8—42no—
121C2×C22⋊Q8—42no—
122C2×#117—24no—
123C2×#117—28no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151C22×C42C2312yes—
152#398—216no—
153#395—216no—
154#395—216no—
155#395—44no—
156#395—44no—
157#394—28no—
158#394—28no—
159#391—24no—
160#391—28no—
161#390—24no—
162#390—44no—
163#389—216no—
164#387—216no—
165#387—216no—
166#385—28no—
167#385—28no—
168#385—48no—
169#380—42no—
170C4×C22⋊C4—24no—
171C4×C22⋊C4—42no—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C25C2411yescharacteristic
193C25C2411yescharacteristic
194C25C2411yescharacteristic
195C2×#41—232no—
196C2×#41—416no—
197C2×#41—416no—
198C2×#41—432no—
199C2×#41—432no—
200C22×Q8—232no—
201C22×Q8—28no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—44no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—41no—
215C22×Q8—42no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8—41no—
219C22×Q8—42no—
220C22×Q8—432no—
221C22×Q8—432no—
222C22×Q8—416no—
223C22×Q8—416no—
224C22×Q8C2×D418yes—
225C22×Q8C2×D414yes—
226C22×Q8C2×D418yes—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—41no—
233C22×D4—41no—
234C22×D4—41no—
235C22×D4—41no—
236C22×D4—41no—
237C22×D4—41no—
238C22×D4—42no—
239C22×D4C2×D414yes—
240C22×D4C2×D414yes—
241C22×D4C2×D414yes—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—28no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—28no—
255C23×C4—42no—
256C23×C4—42no—
257C23×C4—42no—
258C23×C4—42no—
259C23×C4C2×D414yes—
260C23×C4C2×D414yes—
261C23×C4C2×D414yes—
262C23×C4C2412yes—
263C23×C4C2414yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2414yes—
267C23×C4C2414yes—
268C23×C4C2412yes—
269C23×C4C2414yes—
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C4⋊Q8—264no—
275C4⋊Q8—416no—
276#124—264no—
277#124—416no—
278#123—264no—
279#123—264no—
280C22⋊Q8—264no—
281C22⋊Q8—264no—
282C22⋊Q8—416no—
283#117—232no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—41no—
305C2×C4⋊C4—41no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D4116yes—
313C2×C4⋊C4C2×D4116yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—42no—
347C2×C22⋊C4—44no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42C2×D418yes—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—24no—
369#95—24no—
370#95—28no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95—44no—
376#95—44no—
377#95—44no—
378C24—22no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—21no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—24no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24C22×D412yes—
399C24C22×D412yes—
400C24C22×D412yes—
401C24H(5,𝔽2)14yes—
402C24C2511yescharacteristic
403#41—4128no—
404C2×Q8—216no—
405C2×Q8—216no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—416no—
413C2×Q8—416no—
414C2×Q8—4128no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—48no—
424C2×Q8—84no—
425C2×Q8—84no—
426C2×Q8#120116yes—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—28no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—48no—
442C22×C4—216no—
443C22×C4—216no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—216no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
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—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—24no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—416no—
516C22×C4—416no—
517C22×C4—48no—
518C22×C4—48no—
519C22×C4C22×D418yes—
520C22×C4C22×D418yes—
521C22×C4C22×D412yes—
522C22×C4C22×D414yes—
523C22×C4C22×D414yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D414yes—
527C22×C4C22×D412yes—
528C22×C4H(5,𝔽2)12yes—
529C22×C4H(5,𝔽2)12yes—
530C22×C42−1+418yes—
531C22×C42−1+414yes—
532C4⋊C4—232no—
533C4⋊C4—216no—
534C4⋊C4—232no—
535C4⋊C4—232no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—84no—
541C4⋊C4—88no—
542C4⋊C4—88no—
543C22⋊C4—216no—
544C22⋊C4—232no—
545C22⋊C4—232no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—416no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—88no—
555C22⋊C4—84no—
556C42—216no—
557C42—44no—
558C42—44no—
559C42—84no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—28no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—44no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—42no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—42no—
601C23—41no—
602C23—41no—
603C23—42no—
604C23—42no—
605C23—41no—
606C23—41no—
607C23#53414yes—
608C23C23×D411yescharacteristic
609C23C23×D412yes—
610C23C2×H(5,𝔽2)12yes—
611C23C2×H(5,𝔽2)11yescharacteristic
612C23C2×H(5,𝔽2)11yescharacteristic
613C23C2×2−1+412yes—
614C23C2×2−1+414yes—
615C23#58512yes—
616Q8—416no—
617Q8—432no—
618Q8—432no—
619Q8—88no—
620D4—416no—
621D4—416no—
622D4—416no—
623C2×C4—232no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—416no—
652C2×C4—416no—
653C2×C4—432no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—42no—
693C2×C4C2×#12014yes—
694C2×C4C2×#12014yes—
695C2×C4C2×#12014yes—
696C2×C4#53414yes—
697C22—28no—
698C22—24no—
699C22—42no—
700C22—42no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—48no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—42no—
714C22—42no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—84no—
720C22—84no—
721C22C22×#12011yescharacteristic
722C22C2×#53411yescharacteristic
723C22C2×#53411yescharacteristic
724C22#348412yes—
725C22#348412yes—
726C22#348614yes—
727C22#348912yes—
728C22#349714yes—
729C22#349912yes—
730C22#353514yes—
731C22#353812yes—
732C22#354414yes—
733C22#354712yes—
734C22#355011yescharacteristic
735C22#355012yes—
736C22#355111yescharacteristic
737C4—24no—
738C4—24no—
739C4—24no—
740C4—24no—
741C4—48no—
742C4—44no—
743C4—48no—
744C4—48no—
745C4—82no—
746C4—82no—
747C4—84no—
748C4—82no—
749C2—44no—
750C2—82no—
751C2—82no—
752C2#3498011yescharacteristic
753C2#3568712yes—
754C2#3737611yescharacteristic
755C2#3738611yescharacteristic
756C2#4229512yes—
757C2#4779514yes—
758C2#5207212yes—
759C2#5207512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2734C2semidirect
4. #16681C2semidirect
5. #16264C2semidirect
6. #16254C2semidirect
7. #15983C2semidirect
8. #15271C2semidirect
9. #15262C2semidirect
10. #15114C2semidirect
11. #15001C2semidirect
12. #14874C2semidirect
13. #14763C2semidirect
14. #14450C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5362
C2×#27342
C2×#24322
#166812
#162642
#162542
#159832
#152712
#152622
#151142
#150012
#148742
#147632
#144502
#139372