Finite Group Databasebeta

← #781064

Subgroups — #781064

Order 512.

3646 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C212yesmaximal, complemented
2C2×D4×C4⋊C4C212yesmaximal, complemented
3#16358C212yesmaximal, complemented
4#15951C212yesmaximal, complemented
5#15779C212yesmaximal, complemented
6#15462C214yesmaximal, complemented
7#15181C212yesmaximal, complemented
8#15125C211yesmaximal, characteristic, complemented
9#14723C214yesmaximal, complemented
10#14171C212yesmaximal
11#14130C211yesmaximal, characteristic, complemented
12#14048C212yesmaximal, complemented
13#13993C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2214yes—
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23#2440—216no—
24C2×#398C2214yes—
25C2×#395C2214yes—
26C2×#393C2212yes—
27C2×#392C2211yescharacteristic
28D4×C4⋊C4—216no—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#386C2214yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#384C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2212yes—
47#1444C2211yescharacteristic
48#1444C2212yes—
49#1443C2214yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2211yescharacteristic
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2211yescharacteristic
64#1438C2212yes—
65#1438C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×Q8C2312yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2314yes—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—42no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—48no—
88C2×C22⋊Q8—48no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—216no—
93C2×C4×D4—28no—
94C2×C4×D4—42no—
95C2×C4×D4—48no—
96C2×C4×D4—48no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131#398—216no—
132#395—216no—
133#393—44no—
134#392Q818yescomplemented
135#389—216no—
136#389—28no—
137#389—28no—
138#386—48no—
139#385—216no—
140#385—216no—
141#385—216no—
142#385—44no—
143#384D4116yescomplemented
144#380—24no—
145#380—28no—
146#380—28no—
147#380—28no—
148#380—28no—
149C4×C4⋊C4—28no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—42no—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C25C2412yes—
176C25C2411yescharacteristic
177C22×Q8—28no—
178C22×Q8—24no—
179C22×Q8—44no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4C2×Q818yes—
193C22×D4C2×Q814yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4#4114yes—
217C23×C4#4118yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C4⋊Q8—416no—
232C22⋊Q8—464no—
233C4×D4—264no—
234C4×D4—416no—
235C4×D4—464no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D4116yes—
273C2×C4⋊C4C2×D4116yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×Q818yes—
309C2×C22⋊C4C2×Q814yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C2×D418yes—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—41no—
333#95—44no—
334#95—42no—
335#95—41no—
336#95—41no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95C2×Q814yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—22no—
363C24C22×Q814yes—
364C24C22×Q812yes—
365C24C2×#4112yes—
366C24C2×#4114yes—
367C24H(5,𝔽2)14yes—
368C24H(5,𝔽2)12yes—
369C24C2511yescharacteristic
370C2×Q8—416no—
371C2×Q8—48no—
372C2×Q8—88no—
373C2×Q8—84no—
374C2×Q8—84no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—216no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—44no—
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—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4C22×D414yes—
475C22×C4C22×D418yes—
476C22×C4C22×D418yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q812yes—
481C22×C4C22×Q814yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4114yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+414yes—
489C22×C42−1+412yes—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—44no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C22⋊C4—216no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—88no—
515C42—216no—
516C42—416no—
517C42—44no—
518C42—44no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—41no—
573C23—42no—
574C23—42no—
575C23#55418yes—
576C23#55414yes—
577C23C23×D412yes—
578C23C23×Q811yescharacteristic
579C23C22×#4111yescharacteristic
580C23C22×#4112yes—
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)12yes—
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×2−1+411yescharacteristic
586C23C2×2−1+412yes—
587C23#58511yescharacteristic
588Q8—816no—
589D4—264no—
590D4—416no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—416no—
646C2×C4—432no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—44no—
666C2×C4—816no—
667C2×C4—832no—
668C2×C4#52518yes—
669C2×C4#53914yes—
670C2×C4D4214yes—
671C2×C4#54818yes—
672C2×C4D4×Q818yes—
673C2×C4#55114yes—
674C22—28no—
675C22—28no—
676C22—28no—
677C22—28no—
678C22—24no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—42no—
698C22—44no—
699C22—44no—
700C22C2×#52512yes—
701C22C2×#53911yescharacteristic
702C22C2×D4211yescharacteristic
703C22C2×#54812yes—
704C22C2×D4×Q812yes—
705C22C2×#55111yescharacteristic
706C22C2×#55412yes—
707C22C2×#55411yescharacteristic
708C22#348412yes—
709C22#348412yes—
710C22#349212yes—
711C22#349212yes—
712C22#349412yes—
713C22#349712yes—
714C22#350111yescharacteristic
715C22#350311yescharacteristic
716C22#350611yescharacteristic
717C22#351112yes—
718C22#353711yescharacteristic
719C22#353711yescharacteristic
720C22#354412yes—
721C22#355011yescharacteristic
722C22#355911yescharacteristic
723C4—28no—
724C4—48no—
725C4—48no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—44no—
731C4—42no—
732C4—81no—
733C4—84no—
734C4—84no—
735C4#246318yes—
736C2—48no—
737C2—44no—
738C2C2×#246311yescharacteristic
739C2#3568212yes—
740C2#3654612yes—
741C2#3963111yescharacteristic
742C2#4250611yescharacteristic
743C2#4487812yes—
744C2#4533011yescharacteristic
745C2#4535511yescharacteristic
746C2#5913612yes—
747C2#5919911yescharacteristic
748C2#5981711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×D4×C4⋊C4C2semidirect
3. #16358C2semidirect
4. #15951C2semidirect
5. #15779C2semidirect
6. #15462C2semidirect
7. #15181C2semidirect
8. #15125C2semidirect
9. #14723C2semidirect
11. #14130C2semidirect
12. #14048C2semidirect
13. #13993C2semidirect
14. #13872C2semidirect
134. #392Q8semidirect
143. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×D4×C4⋊C42
#163582
#159512
#157792
#154622
#151812
#151252
#147232
#141712
#141302
#140482
#139932
#138722
C4×#3922