Finite Group Databasebeta

← #642352

Subgroups — #642352

Order 512.

3132 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic, complemented
2C2×#2629C211yesmaximal, characteristic, complemented
3C2×#2619C211yesmaximal, characteristic
4C2×C4×#125C211yesmaximal, characteristic, complemented
5#16292C214yesmaximal, complemented
6#16289C214yesmaximal, complemented
7#15141C212yesmaximal, complemented
8#15126C212yesmaximal, complemented
9#14815C214yesmaximal, complemented
10#14780C214yesmaximal, complemented
11#14713C211yesmaximal, characteristic, complemented
12#14277C211yesmaximal, characteristic, complemented
13#14141C212yesmaximal, complemented
14#13998C212yesmaximal, complemented
15#13761C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#121C2212yes—
19C22×#117C2211yescharacteristic
20#2641—28no—
21#2629—28no—
22#2619—28no—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#392C2214yes—
31C4×#125—28no—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2214yes—
38C2×#390C2214yes—
39C2×#390C2214yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53C2×C43C2211yescharacteristic
54#1444C2214yes—
55#1442C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1437C2212yes—
68#1437C2214yes—
69#1437C2212yes—
70#1436C2212yes—
71#1436C2212yes—
72C4⋊#95C2212yes—
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2314yes—
78C24×C4C2311yescharacteristic
79C2×#127—24no—
80C2×#127—24no—
81C2×#127—24no—
82C2×#125—24no—
83C2×#125—24no—
84C2×#125D418yescomplemented
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—44no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—24no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#398—216no—
126#396—216no—
127#396—48no—
128#394—216no—
129#394—216no—
130#394—216no—
131#394—48no—
132#392—216no—
133#390—28no—
134#390—28no—
135#390—28no—
136#390—28no—
137#390—48no—
138#390—48no—
139#390—48no—
140#390—44no—
141#389—28no—
142#389—28no—
143#389—44no—
144#388—48no—
145#387—216no—
146#384—24no—
147#382D418yes—
148#380—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—216no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—24no—
153#376—24no—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C43—24no—
172C25C2414yes—
173C25C2411yescharacteristic
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4#4114yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215#127—416no—
216#125—232no—
217#121—432no—
218#117—416no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
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—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D418yes—
244C2×C4⋊C4C2×D418yes—
245C2×C4⋊C4C2×D4116yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
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⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—28no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—28no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—42no—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42#4114yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—42no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C2×D4116yes—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24C2×#4112yes—
362C24C2511yescharacteristic
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4C22×D418yes—
458C22×C4C22×D418yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C4⋊C4—232no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—28no—
500C42—48no—
501C42—416no—
502C42—48no—
503C42—44no—
504C42—44no—
505C42—44no—
506C42—44no—
507C42#121116yes—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23C23×D411yescharacteristic
567C23C23×D411yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4111yescharacteristic
570C23C22×#4111yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)12yes—
574C23C2×2−1+411yescharacteristic
575C23#58512yes—
576C23#58512yes—
577D4—832no—
578D4—832no—
579D4—816no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—416no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4C2×#12118yes—
656C2×C4C2×#12114yes—
657C2×C4#52914yes—
658C2×C4#54014yes—
659C2×C4#54214yes—
660C2×C4#54614yes—
661C2×C4#54814yes—
662C2×C4#54814yes—
663C2×C4#55014yes—
664C22—44no—
665C22—42no—
666C22—44no—
667C22—42no—
668C22—42no—
669C22—81no—
670C22—81no—
671C22—82no—
672C22—81no—
673C22—82no—
674C22—81no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22C22×#12111yescharacteristic
691C22C2×#52911yescharacteristic
692C22C2×#54011yescharacteristic
693C22C2×#54211yescharacteristic
694C22#345511yescharacteristic
695C22#346012yes—
696C22#346012yes—
697C22#346012yes—
698C22C2×#54611yescharacteristic
699C22C2×#54811yescharacteristic
700C22C2×#54811yescharacteristic
701C22C2×#55011yescharacteristic
702C22#348812yes—
703C22#348812yes—
704C22#349412yes—
705C22#349812yes—
706C22#350012yes—
707C22#350811yescharacteristic
708C22#351211yescharacteristic
709C22#354312yes—
710C22#354712yes—
711C22#355912yes—
712C22#356212yes—
713C4—24no—
714C4—28no—
715C4—42no—
716C4—44no—
717C4—48no—
718C4—42no—
719C4—48no—
720C4—82no—
721C4—82no—
722C4—81no—
723C4—81no—
724C4—84no—
725C4#242618yes—
726C2—82no—
727C2—164no—
728C2#3462111yescharacteristic
729C2C2×#242611yescharacteristic
730C2#3525612yes—
731C2#3528711yescharacteristic
732C2#3759112yes—
733C2#3770611yescharacteristic
734C2#3771011yescharacteristic
735C2#3775112yes—
736C2#3938211yescharacteristic
737C2#3938611yescharacteristic
738C2#3942612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2619C2
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×#2641C2semidirect
2. C2×#2629C2semidirect
4. C2×C4×#125C2semidirect
5. #16292C2semidirect
6. #16289C2semidirect
7. #15141C2semidirect
8. #15126C2semidirect
9. #14815C2semidirect
10. #14780C2semidirect
11. #14713C2semidirect
12. #14277C2semidirect
13. #14141C2semidirect
14. #13998C2semidirect
15. #13761C2semidirect
84. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
C2×#26292
C2×#26192
C2×C4×#1252
#162922
#162892
#151412
#151262
#148152
#147802
#147132
#142772
#141412
#139982
#137612