Finite Group Databasebeta

← #948612

Subgroups — #948612

Order 512.

2948 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C212yesmaximal, complemented
2C2×#2628C212yesmaximal, complemented
3#17104C214yesmaximal, complemented
4#17072C212yesmaximal, complemented
5#16877C212yesmaximal, complemented
6#16339C212yesmaximal, complemented
7#16254C212yesmaximal, complemented
8#15325C214yesmaximal
9#15220C212yesmaximal, complemented
10#15143C212yesmaximal, complemented
11#15131C212yesmaximal, complemented
12#14006C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15#13839C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2628—216no—
22#2440—216no—
23C2×#400C2214yes—
24C2×#399C2214yes—
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#393C2214yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1441C2211yescharacteristic
54#1441C2211yescharacteristic
55#1441C2214yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2212yes—
61#1439C2211yescharacteristic
62#1439C2211yescharacteristic
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2314yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—44no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—48no—
87C2×C22⋊Q8—48no—
88C2×C4×D4—42no—
89C2×C4×D4—42no—
90C2×C4×D4Q818yescomplemented
91C2×#117—28no—
92C2×#117—42no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#400—216no—
127#399—216no—
128#398—216no—
129#396—48no—
130#395—216no—
131#395—216no—
132#395—48no—
133#395—48no—
134#393—216no—
135#390—44no—
136#389—28no—
137#389—28no—
138#389—44no—
139#389D4116yescomplemented
140#386—44no—
141#385—216no—
142#384—28no—
143#384—42no—
144#382—28no—
145#380—28no—
146#380—28no—
147#380—28no—
148#380—42no—
149C4×C4⋊C4—42no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—42no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4C2×Q814yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—42no—
204C23×C4C2×Q814yes—
205C23×C4C2×Q814yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—432no—
220C22⋊Q8—464no—
221C4×D4—416no—
222#117—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D4116yes—
270C2×C4⋊C4C2×Q814yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×Q814yes—
300C2×C22⋊C4C2×Q814yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C2×D418yes—
312C2×C42C2×Q814yes—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—41no—
329#95—41no—
330#95—42no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C2×D4116yes—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—21no—
353C24—21no—
354C24C22×Q812yes—
355C24C22×Q812yes—
356C24H(5,𝔽2)12yes—
357C242−1+412yes—
358C24C2511yescharacteristic
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—88no—
362C2×Q8—88no—
363C2×Q8—88no—
364C2×Q8—88no—
365C2×Q8—88no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—88no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4C22×D418yes—
455C22×C4C22×D414yes—
456C22×C4C22×D418yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+414yes—
469C22×C42−1+412yes—
470C22×C42−1+414yes—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—416no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4—84no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—84no—
498C42—28no—
499C42—216no—
500C42—48no—
501C42—84no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23#54414yes—
548C23C23×D412yes—
549C23C23×Q811yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×H(5,𝔽2)12yes—
555C23C2×2−1+412yes—
556C23C2×2−1+411yescharacteristic
557C23C2×2−1+411yescharacteristic
558C23C2×2−1+412yes—
559Q8—832no—
560D4—416no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—816no—
641C2×C4#53718yes—
642C2×C4#54314yes—
643C2×C4#54414yes—
644C2×C4D4214yes—
645C2×C4D4×Q818yes—
646C2×C4#56314yes—
647C22—28no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—81no—
664C22—81no—
665C22—82no—
666C22—81no—
667C22—82no—
668C22—81no—
669C22C2×#53712yes—
670C22C2×#54311yescharacteristic
671C22C2×#54411yescharacteristic
672C22C2×#54411yescharacteristic
673C22C2×D4211yescharacteristic
674C22C2×D4×Q812yes—
675C22#348412yes—
676C22#348612yes—
677C22#349812yes—
678C22#349912yes—
679C22#350312yes—
680C22C2×#56311yescharacteristic
681C22#353611yescharacteristic
682C22#353612yes—
683C22#353611yescharacteristic
684C22#353711yescharacteristic
685C22#353811yescharacteristic
686C22#354412yes—
687C22#354412yes—
688C22#354911yescharacteristic
689C22#355011yescharacteristic
690C22#355212yes—
691C22#355212yes—
692C4—24no—
693C4—44no—
694C4—44no—
695C4—48no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—48no—
701C4—44no—
702C4—84no—
703C4—84no—
704C4#268318yes—
705C2—44no—
706C2—82no—
707C2#3730611yescharacteristic
708C2#3747711yescharacteristic
709C2#3768612yes—
710C2#3936912yes—
711C2#4242812yes—
712C2#4279412yes—
713C2#4487011yescharacteristic
714C2#4532111yescharacteristic
715C2C2×#268311yescharacteristic
716C2#5296811yescharacteristic
717C2#5300911yescharacteristic
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×#2440C2semidirect
2. C2×#2628C2semidirect
3. #17104C2semidirect
4. #17072C2semidirect
5. #16877C2semidirect
6. #16339C2semidirect
7. #16254C2semidirect
9. #15220C2semidirect
10. #15143C2semidirect
11. #15131C2semidirect
12. #14006C2semidirect
90. C2×C4×D4Q8semidirect
139. #389D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#26282
#171042
#170722
#168772
#163392
#162542
#153252
#152202
#151432
#151312
#140062
#139062
#138722
#138392