Finite Group Databasebeta

← #4344916

Subgroups — #4344916

Order 512.

3270 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#16895C212yesmaximal, complemented
4#16686C212yesmaximal, complemented
5#16556C212yesmaximal, complemented
6#16265C212yesmaximal, complemented
7#15506C214yesmaximal, complemented
8#15328C214yesmaximal, complemented
9#15119C212yesmaximal, complemented
10#14873C212yesmaximal, complemented
11#14865C212yesmaximal, complemented
12#14759C212yesmaximal, complemented
13#14513C212yesmaximal, complemented
14#14419C212yesmaximal
15#13960C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2409—28no—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C2×#393C2212yes—
26C2×#393C2212yes—
27C2×#393C2214yes—
28#2328—28no—
29C2×#391C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#382C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#376C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1445C2214yes—
50#1444C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68#1436C2214yes—
69#1434C2212yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2312yes—
92C2×#127—24no—
93C2×#127—24no—
94C2×#127Q818yescomplemented
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×C4×D4—24no—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4D418yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#396—48no—
128#395—28no—
129#393—28no—
130#393—44no—
131#393—44no—
132#393—48no—
133#391—28no—
134#390—24no—
135#390—24no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—48no—
143#386—48no—
144#385—28no—
145#382—24no—
146#380—24no—
147#380—42no—
148#376—24no—
149C2×#95—42no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—416no—
188C22×Q8—432no—
189C22×Q8—432no—
190C22×Q8—432no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—41no—
198C22×D4—41no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4C2×D414yes—
210C22×D4C2×Q818yes—
211C22×D4C2×Q814yes—
212C22×D4C2×Q818yescomplemented
213C22×D4C2×Q814yescomplemented
214C22×D4#4114yes—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—42no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—42no—
235C23×C4—44no—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250#127—232no—
251C22⋊Q8—432no—
252C4×D4—416no—
253C4×D4—416no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—41no—
317C2×C42—41no—
318C2×C42C2×D414yes—
319C2×C42C2×Q814yescomplemented
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344C24—24no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—24no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—24no—
354C24—24no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C22×Q814yes—
369C24C22×Q812yes—
370C24C22×Q812yes—
371C24C2×#4112yes—
372C24C2×#4112yes—
373C24C2511yescharacteristic
374C2×Q8—216no—
375C2×Q8—416no—
376C2×Q8—416no—
377C2×Q8—416no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—416no—
382C2×Q8—4128no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—28no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—84no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4C22⋊Q8116yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4—84no—
486C22×C4—88no—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×Q814yes—
493C22×C4C22×Q812yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)18yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C4⋊C4—44no—
499C4⋊C4—416no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—84no—
503C4⋊C4—88no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C42—28no—
520C42—28no—
521C42—44no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—28no—
541C23—28no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23C2×C22⋊Q814yes—
578C23C2×C22⋊Q814yes—
579C23C23×D411yescharacteristic
580C23C23×Q811yescharacteristic
581C23C22×#4111yescharacteristic
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×H(5,𝔽2)14yes—
585C23#58511yescharacteristic
586C23#58511yescharacteristic
587C23#58514yes—
588Q8—432no—
589Q8—432no—
590D4—416no—
591D4—416no—
592D4—816no—
593D4—816no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—416no—
615C2×C4—432no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—88no—
661C2×C4—816no—
662C2×C4—816no—
663C2×C4C2×C22⋊Q814yes—
664C2×C4#53414yes—
665C22—28no—
666C22—28no—
667C22—24no—
668C22—24no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—48no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—48no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22C22×C22⋊Q811yescharacteristic
695C22C2×#53411yescharacteristic
696C22#344911yescharacteristic
697C22#348412yes—
698C22#350012yes—
699C22#350111yescharacteristic
700C22#350114yes—
701C22#350111yescharacteristic
702C22#350511yescharacteristic
703C22#350511yescharacteristic
704C22#351414yes—
705C22#353212yes—
706C22#353212yes—
707C22#354012yes—
708C22#355114yes—
709C22#355112yes—
710C22#356214yes—
711C4—24no—
712C4—24no—
713C4—42no—
714C4—42no—
715C4—48no—
716C4—42no—
717C4—44no—
718C4—84no—
719C4—82no—
720C4—82no—
721C4—84no—
722C4—84no—
723C2—44no—
724C2—44no—
725C2—84no—
726C2#3501011yescharacteristic
727C2#3570212yes—
728C2#3670211yescharacteristic
729C2#3741911yescharacteristic
730C2#4215212yes—
731C2#4753912yes—
732C2#4781412yes—
733C2#5197112yes—
734C2#5203512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2409C2semidirect
3. #16895C2semidirect
4. #16686C2semidirect
5. #16556C2semidirect
6. #16265C2semidirect
7. #15506C2semidirect
8. #15328C2semidirect
9. #15119C2semidirect
10. #14873C2semidirect
11. #14865C2semidirect
12. #14759C2semidirect
13. #14513C2semidirect
15. #13960C2semidirect
94. C2×#127Q8semidirect
212. C22×D4C2×Q8semidirect
213. C22×D4C2×Q8semidirect
319. C2×C42C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23282
#168952
#166862
#165562
#162652
#155062
#153282
#151192
#148732
#148652
#147592
#145132
#144192
#139602