Finite Group Databasebeta

← #4456912

Subgroups — #4456912

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2319C211yesmaximal, characteristic
3#17066C212yesmaximal, complemented
4#17064C212yesmaximal, complemented
5#16982C212yesmaximal, complemented
6#16978C212yesmaximal, complemented
7#16304C214yesmaximal, complemented
8#16300C214yesmaximal, complemented
9#15229C212yesmaximal
10#15218C212yesmaximal, complemented
11#15148C212yesmaximal
12#15139C212yesmaximal, complemented
13#14277C211yesmaximal, characteristic, complemented
14#13955C211yesmaximal, characteristic, complemented
15#13941C211yesmaximal, characteristic
16#13906C211yesmaximal, characteristic
17#13848C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2552—28no—
21C2×#400C2214yes—
22C2×#399C2214yes—
23C2×#398C2214yes—
24#2319—28no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×#382C2211yescharacteristic
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×#376C2211yescharacteristic
47#1444C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1443C2214yes—
54#1442C2211yescharacteristic
55#1442C2211yescharacteristic
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#125—28no—
88C2×#125—44no—
89C2×#125—44no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4Q818yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#400—48no—
127#399—48no—
128#398—216no—
129#390—24no—
130#390—24no—
131#390—44no—
132#390—44no—
133#389—24no—
134#389—24no—
135#389—28no—
136#389—28no—
137#389—44no—
138#388—28no—
139#388—28no—
140#384—28no—
141#384—24no—
142#382—24no—
143#381—28no—
144#380—28no—
145#380—48no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—44no—
149C4×C4⋊C4D418yes—
150#376—24no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C25C2412yes—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4C2×Q814yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×Q818yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210#125—432no—
211C4×D4—232no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×Q814yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
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⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×Q818yes—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42C2×D418yes—
297C2×C42C2×D414yes—
298C2×C42C2×Q814yes—
299C2×C42#4114yes—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×D418yes—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24C22×Q814yes—
341C24C2511yescharacteristic
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—216no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—416no—
428C22×C4—88no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q812yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+412yes—
447C22×C42−1+412yes—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—88no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C22⋊C4—216no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—48no—
476C42—44no—
477C42—48no—
478C42—44no—
479C42—48no—
480C42C22⋊Q8116yes—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×H(5,𝔽2)12yes—
510C23C2×2−1+411yescharacteristic
511C23C2×2−1+411yescharacteristic
512C23#58512yes—
513C23#58512yes—
514C23#58512yes—
515D4—416no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—816no—
602C2×C4C2×C22⋊Q814yes—
603C2×C4C2×C22⋊Q814yes—
604C2×C4C2×C22⋊Q814yes—
605C2×C4#53514yes—
606C2×C4#53614yes—
607C2×C4#54214yes—
608C2×C4#54214yes—
609C2×C4#54414yes—
610C2×C4#54414yes—
611C22—48no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—48no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22C22×C22⋊Q811yescharacteristic
623C22C2×#53511yescharacteristic
624C22C2×#53611yescharacteristic
625C22C2×#54211yescharacteristic
626C22C2×#54211yescharacteristic
627C22C2×#54411yescharacteristic
628C22C2×#54411yescharacteristic
629C22#348511yescharacteristic
630C22#348711yescharacteristic
631C22#349412yes—
632C22#349811yescharacteristic
633C22#350011yescharacteristic
634C22#350112yes—
635C22#350412yes—
636C22#351212yes—
637C22#353912yes—
638C22#354212yes—
639C22#354612yes—
640C22#355412yes—
641C22#355712yes—
642C22#355912yes—
643C22#356212yes—
644C22#356512yes—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4—84no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—84no—
659C4—82no—
660C4#262818yes—
661C2—84no—
662C2C2×#262811yescharacteristic
663C2#3552911yescharacteristic
664C2#3563911yescharacteristic
665C2#3736812yes—
666C2#3737312yes—
667C2#4249512yes—
668C2#4249812yes—
669C2#4265111yescharacteristic
670C2#4265211yescharacteristic
671C2#4287511yescharacteristic
672C2#4287611yescharacteristic
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
1. C2×#2552C2semidirect
3. #17066C2semidirect
4. #17064C2semidirect
5. #16982C2semidirect
6. #16978C2semidirect
7. #16304C2semidirect
8. #16300C2semidirect
10. #15218C2semidirect
12. #15139C2semidirect
13. #14277C2semidirect
14. #13955C2semidirect
17. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23192
#170662
#170642
#169822
#169782
#163042
#163002
#152292
#152182
#151482
#151392
#142772
#139552
#139412
#139062
#138482