Finite Group Databasebeta

← #1409256

Subgroups — #1409256

Order 512.

2444 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2496C211yesmaximal, characteristic
3C2×#2664C212yesmaximal
4#16683C212yesmaximal, complemented
5#16673C212yesmaximal
6#16529C214yesmaximal, complemented
7#16314C214yesmaximal
8#15156C211yesmaximal, characteristic, complemented
9#15150C211yesmaximal, characteristic
10#15149C212yesmaximal, complemented
11#15145C211yesmaximal, characteristic, complemented
12#15131C212yesmaximal
13#14818C212yesmaximal, complemented
14#13910C212yesmaximal, complemented
15#13876C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic, complemented
18C22×#126C2212yes—
19C22×#125C2212yes—
20#2664—216no—
21#2552—28no—
22#2496—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#397C2212yes—
31C2×#397C2212yes—
32C2×#397C2212yes—
33C2×#396C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2214yes—
41C2×#388C2214yes—
42C2×#387C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2211yescharacteristic
57#1445C2211yescharacteristic
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1444C2214yes—
62#1444C2211yescharacteristic
63#1444C2214yes—
64#1444C2214yes—
65#1443C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2214yes—
73#1439C2214yes—
74#1439C2211yescharacteristic
75#1439C2214yes—
76#1439C2212yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2211yescharacteristic
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#126—28no—
88C2×#126—28no—
89C2×#126—44no—
90C2×#125—28no—
91C2×#125—28no—
92C2×#125—44no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128#400—28no—
129#400—28no—
130#399D418yes—
131#398—216no—
132#398—28no—
133#398—216no—
134#398—44no—
135#397—28no—
136#397—28no—
137#397—44no—
138#396—216no—
139#389—216no—
140#389—24no—
141#389—28no—
142#389—24no—
143#389—28no—
144#389—28no—
145#389—48no—
146#388—216no—
147#387—28no—
148#384—24no—
149#384—28no—
150#384—24no—
151#384—44no—
152#380—24no—
153#380—24no—
154#380—44no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—44no—
159C4×C4⋊C4D418yes—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—28no—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C25C2411yescharacteristic
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213#126—432no—
214#125—432no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D418yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42#4118yes—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—22no—
335#95—24no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95C2×D414yes—
343#95C2×D418yes—
344C24—21no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24C22×D412yes—
355C24C2511yescharacteristic
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+414yes—
454C22×C42−1+414yes—
455C22×C42−1+414yes—
456C22×C42−1+412yes—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—88no—
476C4⋊C4—84no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C22⋊C4—28no—
480C22⋊C4—216no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—88no—
486C42—28no—
487C42—216no—
488C42—216no—
489C42—44no—
490C42—48no—
491C42—88no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×2−1+412yes—
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+412yes—
524C23C2×2−1+412yes—
525C23#58511yescharacteristic
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4#53214yes—
617C2×C4#53614yes—
618C2×C4#53714yes—
619C2×C4#53918yes—
620C2×C4#54118yes—
621C2×C4D4214yes—
622C2×C4#54718yes—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—82no—
633C22—82no—
634C22C2×#53211yescharacteristic
635C22C2×#53611yescharacteristic
636C22C2×#53711yescharacteristic
637C22C2×#53912yes—
638C22C2×#54112yes—
639C22C2×D4211yescharacteristic
640C22C2×#54712yes—
641C22#348611yescharacteristic
642C22#348811yescharacteristic
643C22#349212yes—
644C22#349512yes—
645C22#349711yescharacteristic
646C22#349811yescharacteristic
647C22#349911yescharacteristic
648C22#349912yes—
649C22#351112yes—
650C22#354412yes—
651C22#354611yescharacteristic
652C22#354811yescharacteristic
653C22#355212yes—
654C22#355212yes—
655C22#355512yes—
656C22#355612yes—
657C4—24no—
658C4—28no—
659C4—44no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—44no—
667C4—81no—
668C4—82no—
669C4—82no—
670C4—84no—
671C4—84no—
672C4#266418yescomplemented
673C2—82no—
674C2C2×#266411yescharacteristic
675C2#3592611yescharacteristic
676C2#3612711yescharacteristic
677C2#3933711yescharacteristic
678C2#3934111yescharacteristic
679C2#3934812yes—
680C2#3935312yes—
681C2#4245512yes—
682C2#4279111yescharacteristic
683C2#4291811yescharacteristic
684C2#4292412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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
4. #16683C2semidirect
6. #16529C2semidirect
8. #15156C2semidirect
10. #15149C2semidirect
11. #15145C2semidirect
13. #14818C2semidirect
14. #13910C2semidirect
17. #13821C2semidirect
672. C4#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24962
C2×#26642
#166832
#166732
#165292
#163142
#151562
#151502
#151492
#151452
#151312
#148182
#139102
#138762
C4×#3992
#138212