Finite Group Databasebeta

← #636513

Subgroups — #636513

Order 512.

2820 conjugacy classes of subgroups.

The subgroup lattice has 780 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic, complemented
3#16575C214yesmaximal, complemented
4#16574C214yesmaximal, complemented
5#15225C212yesmaximal, complemented
6#15223C212yesmaximal, complemented
7#15187C212yesmaximal, complemented
8#15183C212yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14912C212yesmaximal
11#14697C211yesmaximal, characteristic, complemented
12#14342C212yesmaximal, complemented
13#14256C212yesmaximal
14#13904C211yesmaximal, characteristic, complemented
15#13881C211yesmaximal, characteristic, complemented
16#13872C211yesmaximal, characteristic
17C4×#386C211yesmaximal, characteristic
18C22×#123C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2486—28no—
25C2×#398C2212yes—
26C2×#397C2212yes—
27C2×#389—28no—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#386C2211yescharacteristic
35C2×#386C2211yescharacteristic
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#382C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2212yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2214yes—
54#1445C2214yes—
55#1445C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2212yes—
71#1436C2214yes—
72#1436C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—44no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4—44no—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C42—24no—
132C22×C42—44no—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137#398—28no—
138#397—28no—
139#389—24no—
140#389—24no—
141#389—28no—
142#389—28no—
143#389—216no—
144#389—44no—
145#386—24no—
146#386—24no—
147#386—44no—
148#385—44no—
149#382—44no—
150#381—28no—
151#381—44no—
152#381—44no—
153#380—24no—
154#380—24no—
155#380—24no—
156#380—24no—
157#380—44no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—44no—
160C2×#95—28no—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C25C2411yescharacteristic
191C25C2411yescharacteristic
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4#4114yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D418yes—
227C23×C4C2×D414yes—
228C23×C4#4114yes—
229C23×C4#4114yes—
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246#123—432no—
247C4×D4—416no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—432no—
284C2×C4⋊C4—416no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D418yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—416no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—44no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24C22×D412yes—
380C24C2×#4112yes—
381C24C2×#4112yes—
382C24C2×#4112yes—
383C24C2×#4112yes—
384C24C2×#4112yes—
385C24H(5,𝔽2)12yes—
386C24H(5,𝔽2)12yes—
387C24C2511yescharacteristic
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—88no—
505C22×C4—88no—
506C22×C4#123116yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4114yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4H(5,𝔽2)14yes—
518C22×C42−1+414yes—
519C22×C42−1+414yes—
520C4⋊C4—216no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—416no—
524C4⋊C4—416no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—88no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—88no—
545C42—48no—
546C42—44no—
547C42—48no—
548C42—48no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—24no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—42no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23C2×#12314yes—
589C23#52914yes—
590C23#54014yes—
591C23#54614yes—
592C23#54614yes—
593C23#55314yes—
594C23#55514yes—
595C23C23×D411yescharacteristic
596C23C22×#4111yescharacteristic
597C23C22×#4111yescharacteristic
598C23C22×#4111yescharacteristic
599C23C22×#4111yescharacteristic
600C23C2×H(5,𝔽2)12yes—
601C23C2×H(5,𝔽2)11yescharacteristic
602C23C2×H(5,𝔽2)11yescharacteristic
603C23C2×2−1+412yes—
604C23C2×2−1+412yes—
605C23#58512yes—
606D4—416no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—232no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—432no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—44no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—816no—
699C2×C4—88no—
700C2×C4C2×#12314yes—
701C2×C4C2×#12314yes—
702C2×C4#53514yes—
703C2×C4#53614yes—
704C2×C4#53918yes—
705C2×C4#53918yes—
706C22—24no—
707C22—28no—
708C22—24no—
709C22—24no—
710C22—28no—
711C22—24no—
712C22—28no—
713C22—28no—
714C22—24no—
715C22—24no—
716C22—44no—
717C22—42no—
718C22—44no—
719C22—42no—
720C22—42no—
721C22—81no—
722C22—81no—
723C22—81no—
724C22—82no—
725C22—81no—
726C22—82no—
727C22#242018yes—
728C22C22×#12311yescharacteristic
729C22C2×#52911yescharacteristic
730C22C2×#53511yescharacteristic
731C22C2×#53611yescharacteristic
732C22C2×#53912yes—
733C22C2×#53912yes—
734C22C2×#54011yescharacteristic
735C22#346112yes—
736C22C2×#54611yescharacteristic
737C22C2×#54611yescharacteristic
738C22C2×#55311yescharacteristic
739C22C2×#55511yescharacteristic
740C22#348611yescharacteristic
741C22#348711yescharacteristic
742C22#350512yes—
743C22#350712yes—
744C22#350912yes—
745C22#351512yes—
746C22#352012yes—
747C22#353612yes—
748C22#354212yes—
749C22#354412yes—
750C22#355912yes—
751C4—24no—
752C4—44no—
753C4—44no—
754C4—44no—
755C4—42no—
756C4—44no—
757C4—81no—
758C4—82no—
759C4—84no—
760C4—82no—
761C4—81no—
762C4—84no—
763C4—82no—
764C4—82no—
765C4#262318yes—
766C2—28no—
767C2—82no—
768C2C2×#262311yescharacteristic
769C2C2×#242011yescharacteristic
770C2#3502612yes—
771C2#3518211yescharacteristic
772C2#3750212yes—
773C2#3890511yescharacteristic
774C2#3890711yescharacteristic
775C2#3919012yes—
776C2#3946511yescharacteristic
777C2#3946711yescharacteristic
778C2#3950412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#389C2semidirect
3. #16575C2semidirect
4. #16574C2semidirect
5. #15225C2semidirect
6. #15223C2semidirect
7. #15187C2semidirect
8. #15183C2semidirect
9. #14919C2semidirect
11. #14697C2semidirect
12. #14342C2semidirect
14. #13904C2semidirect
15. #13881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C22×#3892
#165752
#165742
#152252
#152232
#151872
#151832
#149192
#149122
#146972
#143422
#142562
#139042
#138812
#138722
C4×#3862