Finite Group Databasebeta

← #737748

Subgroups — #737748

Order 512.

2386 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2357C211yesmaximal, characteristic
3#16698C212yesmaximal, complemented
4#16364C212yesmaximal, complemented
5#15710C214yesmaximal, complemented
6#15648C214yesmaximal, complemented
7#15138C212yesmaximal
8#15138C212yesmaximal
9#15061C212yesmaximal, complemented
10#15001C212yesmaximal, complemented
11#14348C211yesmaximal, characteristic, complemented
12#14133C212yesmaximal, complemented
13#14132C212yesmaximal, complemented
14#13925C211yesmaximal, characteristic, complemented
15#13910C211yesmaximal, characteristic, complemented
16#13850C211yesmaximal, characteristic, complemented
17#13785C211yesmaximal, characteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2538—28no—
22C2×#397C2214yes—
23C2×#397C2214yes—
24C2×#396C2214yes—
25#2357—28no—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#387C2214yes—
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×#382C2211yescharacteristic
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2211yescharacteristic
47#1446C2211yescharacteristic
48#1446C2211yescharacteristic
49#1446C2212yes—
50#1445C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2214yes—
53#1445C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1443C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—48no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#397—48no—
128#397—48no—
129#396—216no—
130#390—24no—
131#390—24no—
132#390—28no—
133#390—28no—
134#389—28no—
135#389—28no—
136#389—28no—
137#388—28no—
138#388—28no—
139#387—48no—
140#384—24no—
141#384—44no—
142#382—24no—
143#381—28no—
144#380—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C22⋊C4—28no—
150#376—24no—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—42no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4C2×D414yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2×D418yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C22⋊Q8—432no—
215C4×Q8—416no—
216C4×D4—232no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4#4118yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C2×D418yes—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C2×D414yes—
299C2×C42#4114yes—
300C2×C42#4114yes—
301#95—22no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24C22×D414yes—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—48no—
353C2×D4—48no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—216no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C2×#4118yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C22×C42−1+414yes—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—84no—
466C4⋊C4—88no—
467C22⋊C4—216no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—88no—
476C42—48no—
477C42—48no—
478C42—44no—
479C42—44no—
480C42—48no—
481C42—84no—
482C42—84no—
483C42#123116yes—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23C23×D411yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4112yes—
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23C2×2−1+412yes—
514C23C2×2−1+411yescharacteristic
515C23#58511yescharacteristic
516C23#58511yescharacteristic
517C23#58512yes—
518Q8—416no—
519D4—416no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—816no—
608C2×C4C2×#12314yes—
609C2×C4C2×#12314yes—
610C2×C4C2×#12314yes—
611C2×C4#53214yes—
612C2×C4#53614yes—
613C2×C4#55218yes—
614C2×C4#55618yes—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22C22×#12311yescharacteristic
627C22C2×#53211yescharacteristic
628C22C2×#53611yescharacteristic
629C22#346011yescharacteristic
630C22#346111yescharacteristic
631C22C2×#55212yes—
632C22C2×#55612yes—
633C22#348411yescharacteristic
634C22#348511yescharacteristic
635C22#348811yescharacteristic
636C22#348911yescharacteristic
637C22#349412yes—
638C22#349412yes—
639C22#351111yescharacteristic
640C22#351211yescharacteristic
641C22#351612yes—
642C22#351712yes—
643C22#351912yes—
644C22#352112yes—
645C22#355012yes—
646C22#355412yes—
647C22#355412yes—
648C22#355712yes—
649C4—24no—
650C4—24no—
651C4—42no—
652C4—48no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—81no—
657C4—81no—
658C4—84no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#249918yes—
665C2—84no—
666C2C2×#249911yescharacteristic
667C2#3525511yescharacteristic
668C2#3554011yescharacteristic
669C2#3585611yescharacteristic
670C2#3594111yescharacteristic
671C2#3822612yes—
672C2#3823112yes—
673C2#4248111yescharacteristic
674C2#4267811yescharacteristic
675C2#4503712yes—
676C2#4519712yes—
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
3. #16698C2semidirect
4. #16364C2semidirect
5. #15710C2semidirect
6. #15648C2semidirect
9. #15061C2semidirect
10. #15001C2semidirect
11. #14348C2semidirect
12. #14133C2semidirect
13. #14132C2semidirect
14. #13925C2semidirect
15. #13910C2semidirect
16. #13850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#23572
#166982
#163642
#157102
#156482
#151382
#150612
#150012
#143482
#141332
#141322
#139252
#139102
#138502
#137852