Finite Group Databasebeta

← #846786

Subgroups — #846786

Order 512.

4118 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2413C212yesmaximal, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2406C212yesmaximal
4C22×#384C211yesmaximal, characteristic, complemented
5#16522C212yesmaximal, complemented
6#16514C212yesmaximal, complemented
7#15643C212yesmaximal, complemented
8#15625C212yesmaximal, complemented
9#15392C214yesmaximal, complemented
10#15073C212yesmaximal, complemented
11#15069C212yesmaximal, complemented
12#15069C212yesmaximal, complemented
13#14696C212yesmaximal, complemented
14#14405C214yesmaximal, complemented
15#13870C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#123C2212yes—
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C42C2211yescharacteristic
26#2413—216no—
27#2406—216no—
28#2406—28no—
29C2×#398C2214yes—
30C2×#396C2214yes—
31C2×#395C2211yescharacteristic
32C2×#395C2214yes—
33C2×#395C2211yescharacteristic
34C2×#394C2211yescharacteristic
35C2×#394C2214yes—
36C2×#394C2211yescharacteristic
37C2×#389C2214yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2212yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#384—28no—
49C2×#384C2212yes—
50C2×#384C2212yes—
51C2×#384C2212yes—
52C2×#384C2212yes—
53C2×#381C2212yes—
54C2×#381C2211yescharacteristic
55C2×#381C2212yes—
56C2×#381C2211yescharacteristic
57C2×#381C2212yes—
58#1444C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1437C2214yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2212yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22×#41—28no—
80C22×#41—24no—
81C22×#41—44no—
82C22×#41—42no—
83C22×#41—42no—
84C23×Q8C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×#123—44no—
99C2×#123—44no—
100C2×#123—44no—
101C2×#123—44no—
102C2×#117—28no—
103C2×#117—42no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—42no—
110C22×C4⋊C4—42no—
111C22×C4⋊C4—44no—
112C22×C4⋊C4—44no—
113C22×C4⋊C4—42no—
114C22×C4⋊C4—42no—
115C22×C4⋊C4D418yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2314yes—
143C22×C42—24no—
144C22×C42—28no—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2312yes—
148#398—216no—
149#396—216no—
150#395—24no—
151#395—24no—
152#395—216no—
153#394—24no—
154#394—24no—
155#394—216no—
156#389—216no—
157#388—216no—
158#388—216no—
159#386—28no—
160#386—28no—
161#386D4116yes—
162#385—28no—
163#385—28no—
164#385—28no—
165#385—216no—
166#385—216no—
167#384—28no—
168#384—28no—
169#384—28no—
170#384—28no—
171#381—24no—
172#381—28no—
173#381—28no—
174#381—42no—
175#381—44no—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C25C2412yes—
197C25C2411yescharacteristic
198C2×#41—432no—
199C2×#41—416no—
200C2×#41—416no—
201C2×#41—432no—
202C2×#41—416no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—41no—
206C22×Q8—42no—
207C22×Q8—41no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—42no—
217C22×D4—41no—
218C22×D4—41no—
219C22×D4C2×D418yes—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—28no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2×D418yes—
253C23×C4C2×D414yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2414yes—
257C23×C4C2414yes—
258C23×C4C2414yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2414yes—
266C23×C4C2411yescharacteristic
267#123—432no—
268#123—432no—
269#123—432no—
270#117—416no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—416no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4—41no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—41no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—432no—
316C2×C4⋊C4—432no—
317C2×C4⋊C4—416no—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—42no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4C2×D418yes—
352C2×C22⋊C4C2×D4116yes—
353C2×C42—232no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—28no—
357C2×C42—24no—
358C2×C42—28no—
359C2×C42—42no—
360C2×C42#4118yescomplemented
361#95—28no—
362#95—24no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—42no—
377#95—42no—
378#95—44no—
379#95—44no—
380#95C2×D4116yes—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24C22×D414yes—
395C24C22×D412yes—
396C24C22×D414yes—
397C24H(5,𝔽2)12yes—
398C24H(5,𝔽2)12yes—
399C24H(5,𝔽2)14yes—
400C24C2511yescharacteristic
401#41—4128no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—216no—
410C2×D4—48no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—232no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—216no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—232no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—216no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4C22×D418yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D414yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D418yes—
524C22×C4C2×#4114yes—
525C22×C4C2×#4118yes—
526C22×C42−1+414yes—
527C22×C42−1+418yes—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—416no—
533C4⋊C4—416no—
534C4⋊C4—416no—
535C4⋊C4—416no—
536C4⋊C4—48no—
537C4⋊C4—416no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—416no—
543C4⋊C4—416no—
544C22⋊C4—216no—
545C22⋊C4—232no—
546C22⋊C4—416no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—416no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—416no—
555C42—216no—
556C42—216no—
557C42—48no—
558C23—22no—
559C23—22no—
560C23—28no—
561C23—24no—
562C23—28no—
563C23—24no—
564C23—24no—
565C23—28no—
566C23—24no—
567C23—24no—
568C23—28no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—28no—
573C23—28no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—42no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23#53414yes—
593C23#53518yes—
594C23D4214yes—
595C23D4218yes—
596C23#56014yes—
597C23C23×D411yescharacteristic
598C23C23×D412yes—
599C23C22×#4112yes—
600C23C2×H(5,𝔽2)11yescharacteristic
601C23C2×H(5,𝔽2)11yescharacteristic
602C23C2×H(5,𝔽2)12yes—
603C23C2×2−1+412yes—
604C23C2×2−1+414yes—
605Q8—416no—
606D4—416no—
607D4—432no—
608C2×C4—264no—
609C2×C4—232no—
610C2×C4—22no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—432no—
639C2×C4—416no—
640C2×C4—416no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—816no—
686C2×C4#52514yes—
687C2×C4#53614yes—
688C2×C4#547116yes—
689C22—28no—
690C22—24no—
691C22—28no—
692C22—28no—
693C22—216no—
694C22—24no—
695C22—28no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—48no—
700C22—48no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—84no—
705C22—82no—
706C22—84no—
707C22#266918yes—
708C22C2×#52511yescharacteristic
709C22C2×#53411yescharacteristic
710C22C2×#53512yes—
711C22C2×#53611yescharacteristic
712C22C2×D4212yes—
713C22C2×D4211yescharacteristic
714C22C2×#54714yes—
715C22#348612yes—
716C22#349214yes—
717C22#349212yes—
718C22#349212yes—
719C22#349914yes—
720C22C2×#56011yescharacteristic
721C22#354412yes—
722C22#354412yes—
723C22#354414yes—
724C4—216no—
725C4—24no—
726C4—24no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—44no—
732C4—48no—
733C4—82no—
734C4—84no—
735C4—82no—
736C4—84no—
737C2—28no—
738C2—84no—
739C2#3505711yescharacteristic
740C2#3610312yes—
741C2#3725811yescharacteristic
742C2#3786314yes—
743C2#3933712yes—
744C2#4632712yes—
745C2C2×#266911yescharacteristic
746C2#5931312yes—
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
1. C2×#2413C2semidirect
4. C22×#384C2semidirect
5. #16522C2semidirect
6. #16514C2semidirect
7. #15643C2semidirect
8. #15625C2semidirect
9. #15392C2semidirect
10. #15073C2semidirect
11. #15069C2semidirect
12. #15069C2semidirect
13. #14696C2semidirect
14. #14405C2semidirect
15. #13870C2semidirect
360. C2×C42#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24132
C2×#24062
C22×#3842
#165222
#165142
#156432
#156252
#153922
#150732
#150692
#146962
#144052
#138702