Finite Group Databasebeta

← #764891

Subgroups — #764891

Order 512.

4006 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
1C22×#539C211yesmaximal, characteristic
2C2×#2591C211yesmaximal, characteristic
3C2×#2583C211yesmaximal, characteristic, complemented
4C22×#398C211yesmaximal, characteristic, complemented
5#15912C214yesmaximal, complemented
6#15847C214yesmaximal, complemented
7#15279C214yesmaximal, complemented
8#15271C214yesmaximal, complemented
9#14912C212yesmaximal, complemented
10#14912C212yesmaximal, complemented
11#14139C211yesmaximal, characteristic, complemented
12#14137C211yesmaximal, characteristic
13#14016C212yesmaximal, complemented
14#14015C212yesmaximal, complemented
15#13776C211yesmaximal, characteristic, complemented
16C2×#539—28no—
17C2×#539—28no—
18C2×#539—28no—
19C22×#126C2212yes—
20C22×#125C2212yes—
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C22×C4×D4C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31#2591—28no—
32#2583—28no—
33C2×#400C2212yes—
34C2×#398—28no—
35C2×#398C2212yes—
36C2×#398C2214yes—
37C2×#398C2214yes—
38C2×#395C2212yes—
39C2×#394C2212yes—
40C2×#391C2212yes—
41C2×#390C2212yes—
42C2×#389C2214yes—
43C2×#388C2214yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#386C2212yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#385C2214yes—
51C2×#381C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C22×#95C2211yescharacteristic
55C22×#95C2211yescharacteristic
56C22×#95C2211yescharacteristic
57#1444C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92#539—2128no—
93C2×#126—28no—
94C2×#126—28no—
95C2×#126—28no—
96C2×#125—28no—
97C2×#125—28no—
98C2×#125—28no—
99C2×#123—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—28no—
104C2×#123—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8—28no—
110C2×C22⋊Q8D418yes—
111C2×#121—24no—
112C2×#121—24no—
113C2×#121D418yes—
114C2×C4×D4—28no—
115C2×C4×D4—28no—
116C2×C4×D4—28no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—44no—
125C22×C4⋊C4—44no—
126C22×C4⋊C4D418yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2314yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2312yes—
149C22×C22⋊C4C2312yes—
150C22×C22⋊C4C2312yes—
151C22×C22⋊C4C2314yes—
152C22×C22⋊C4C2314yes—
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156#400—28no—
157#398—28no—
158#398—216no—
159#398—216no—
160#395—28no—
161#394—28no—
162#391—44no—
163#390—44no—
164#389—216no—
165#388—216no—
166#386—28no—
167#386—28no—
168#386—28no—
169#386—28no—
170#385—28no—
171#385—28no—
172#385—48no—
173#381—28no—
174C4×C22⋊C4—24no—
175C4×C22⋊C4—24no—
176C2×#95—24no—
177C2×#95—24no—
178C2×#95—24no—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C25C2412yes—
199C25C2411yescharacteristic
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8C2×D414yes—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4C2×D418yes—
216C22×D4C2×D414yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—24no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2×D414yes—
253C23×C4C2×D414yes—
254C23×C4C2×D414yes—
255C23×C4C2414yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2414yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268#126—264no—
269#125—264no—
270#123—264no—
271#123—264no—
272C22⋊Q8—264no—
273C22⋊Q8—232no—
274#121—232no—
275C4×D4—264no—
276C2×C4⋊C4—232no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—416no—
316C2×C4⋊C4—416no—
317C2×C4⋊C4—416no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D418yes—
321C2×C4⋊C4C2×D418yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D418yes—
324C2×C4⋊C4C2×D414yes—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
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—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—28no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4C2×D418yes—
356C2×C22⋊C4C2×D418yes—
357C2×C22⋊C4#4118yes—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—42no—
364C2×C42—42no—
365C2×C42—42no—
366C2×C42—42no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—24no—
371#95—24no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—28no—
379#95—28no—
380#95—28no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95—44no—
385#95—44no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—24no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24—24no—
399C24—24no—
400C24C22×D414yes—
401C24C22×D412yes—
402C24C22×D412yes—
403C24C22×D412yes—
404C24C2×#4114yes—
405C242−1+414yes—
406C24C2511yescharacteristic
407C2×Q8—28no—
408C2×Q8—48no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—84no—
412C2×D4—216no—
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—28no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—84no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—216no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
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—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—416no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—84no—
514C22×C4—84no—
515C22×C4—88no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4—88no—
519C22×C4—84no—
520C22×C4—84no—
521C22×C4—88no—
522C22×C4#120116yes—
523C22×C4C22×D414yes—
524C22×C4C22×D414yes—
525C22×C4C22×D414yes—
526C22×C4C22×D414yes—
527C22×C4C22×D414yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C2×#4118yes—
535C22×C4H(5,𝔽2)14yes—
536C22×C42−1+414yes—
537C4⋊C4—216no—
538C4⋊C4—216no—
539C4⋊C4—216no—
540C4⋊C4—216no—
541C4⋊C4—216no—
542C4⋊C4—28no—
543C4⋊C4—216no—
544C4⋊C4—28no—
545C4⋊C4—416no—
546C4⋊C4—48no—
547C4⋊C4—48no—
548C4⋊C4—416no—
549C4⋊C4—48no—
550C4⋊C4—48no—
551C4⋊C4—88no—
552C22⋊C4—216no—
553C22⋊C4—216no—
554C22⋊C4—216no—
555C22⋊C4—216no—
556C22⋊C4—216no—
557C22⋊C4—216no—
558C22⋊C4—416no—
559C22⋊C4—416no—
560C22⋊C4—48no—
561C22⋊C4—88no—
562C22⋊C4—88no—
563C42—216no—
564C42—84no—
565C42—84no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—28no—
575C23—28no—
576C23—28no—
577C23—28no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—28no—
582C23—28no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23C2×#12014yes—
602C23#53214yes—
603C23#53614yes—
604C23#54718yes—
605C23#54818yes—
606C23C23×D411yescharacteristic
607C23C23×D411yescharacteristic
608C23C23×D411yescharacteristic
609C23C22×#4112yes—
610C23C2×H(5,𝔽2)12yes—
611C23C2×2−1+412yes—
612C23C2×2−1+412yes—
613C23#58512yes—
614C23#58512yes—
615Q8—88no—
616D4—432no—
617D4—88no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—416no—
648C2×C4—416no—
649C2×C4—416no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—416no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—88no—
696C2×C4—816no—
697C2×C4—88no—
698C2×C4—88no—
699C2×C4C2×#12014yes—
700C2×C4C2×#12014yes—
701C22—24no—
702C22—24no—
703C22—216no—
704C22—216no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—24no—
709C22—44no—
710C22—48no—
711C22—48no—
712C22—44no—
713C22—44no—
714C22—82no—
715C22—84no—
716C22—84no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22#243218yes—
721C22C22×#12011yescharacteristic
722C22C2×#53211yescharacteristic
723C22C2×#53611yescharacteristic
724C22C2×#54712yes—
725C22C2×#54812yes—
726C22#348411yescharacteristic
727C22#348511yescharacteristic
728C22#348611yescharacteristic
729C22#348711yescharacteristic
730C22#348912yes—
731C22#348912yes—
732C22#349612yes—
733C22#349612yes—
734C22#351314yes—
735C22#351514yes—
736C22#355214yes—
737C22#355314yes—
738C4—24no—
739C4—24no—
740C4—44no—
741C4—44no—
742C4—48no—
743C4—44no—
744C4—44no—
745C4—44no—
746C4—44no—
747C4—44no—
748C4—82no—
749C4—84no—
750C4—82no—
751C2—28no—
752C2—84no—
753C2C2×#243211yescharacteristic
754C2#3534711yescharacteristic
755C2#3539611yescharacteristic
756C2#3546811yescharacteristic
757C2#3565911yescharacteristic
758C2#3712011yescharacteristic
759C2#3723311yescharacteristic
760C2#4302714yes—
761C2#4302814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2583C2semidirect
4. C22×#398C2semidirect
5. #15912C2semidirect
6. #15847C2semidirect
7. #15279C2semidirect
8. #15271C2semidirect
9. #14912C2semidirect
10. #14912C2semidirect
11. #14139C2semidirect
13. #14016C2semidirect
14. #14015C2semidirect
15. #13776C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#25912
C2×#25832
C22×#3982
#159122
#158472
#152792
#152712
#149122
#141392
#141372
#140162
#140152
#137762