Finite Group Databasebeta

← #6448142

Subgroups — #6448142

Order 512.

2304 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3036C211yesmaximal, characteristic
2C2×Q8×M16C211yesmaximal, characteristic, complemented
3#34955C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic, complemented
5#25308C214yesmaximal, complemented
6#25304C214yesmaximal, complemented
7#25298C214yesmaximal, complemented
8#25296C214yesmaximal, complemented
9#23617C212yesmaximal, complemented
10#23546C212yesmaximal, complemented
11#21236C212yesmaximal, complemented
12#21201C211yesmaximal, characteristic, complemented
13#21172C211yesmaximal, characteristic, complemented
14#21093C212yesmaximal, complemented
15C4×#418C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3036—28no—
21C2×#463C2214yes—
22C2×#461C2212yes—
23C2×#459C2212yes—
24Q8×M16—28no—
25C2×#446C2212yescomplemented
26C2×#446C2214yescomplemented
27C2×C8×Q8C2212yescomplemented
28C2×#428C2212yes—
29C2×#423C2212yes—
30C2×#423C2211yescharacteristic
31C2×#417C2212yes—
32C2×#417C2212yes—
33C2×#417C2212yes—
34C2×#418C2211yescharacteristic
35C2×#418C2211yescharacteristic
36C2×C4×M16C2212yes—
37C2×C4×M16C2211yescharacteristic
38#2592C2212yescomplemented
39#2592C2212yescomplemented
40#2591—22no—
41#2580—21no—
42#2580—21no—
43#2580—22no—
44#2580C2212yes—
45#2580C2212yes—
46#2580C2214yes—
47#2568C2212yescomplemented
48#2548C2212yescomplemented
49#2504—21no—
50#2436—21no—
51#2398—21no—
52#2398—21no—
53C2×#399—21no—
54C2×#393—21no—
55C4×#124—22no—
56#2297C2212yes—
57#2297C2212yes—
58C2×#390C2212yes—
59C2×#389C2212yes—
60C2×#385—22no—
61C2×#382C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2068C2214yescomplemented
64#2067C2214yescomplemented
65#2066C2214yescomplemented
66#2065C2214yescomplemented
67#2055C2214yes—
68#2055C2214yes—
69#2054C2214yes—
70#2054C2214yes—
71#1973C2214yes—
72#1972C2214yes—
73#1970C2214yes—
74#1968C2214yes—
75#1961C2212yescomplemented
76#1960C2212yescomplemented
77#1958C2214yescomplemented
78#1949C2214yescomplemented
79#1949C2214yescomplemented
80#1936C2214yescomplemented
81#1935C2214yescomplemented
82#1934C2214yescomplemented
83#1933C2214yescomplemented
84#1762C2214yes—
85C4×#102C2214yes—
86#1736C2211yescharacteristic
87#1736C2211yescharacteristic
88#1736C2212yes—
89C22×#41C2312yes—
90C23×Q8D411yescharacteristic
91C23×D4C2311yescharacteristic
92C24×C4—21no—
93C2×AGL(1,ℤ/8)—28no—
94C2×AGL(1,ℤ/8)—28no—
95C2×AGL(1,ℤ/8)—44no—
96C22×SD16C2314yes—
97C22×D8C2314yes—
98C22×M16C2311yescharacteristic
99C22×M16C2312yes—
100C22×M16C2311yescharacteristic
101#532—24no—
102#532—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#124—22no—
106C2×#124C2312yes—
107C2×#121—21no—
108C2×#121—21no—
109C2×C4×Q8—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D412yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—24no—
122C2×C4×D4—24no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—22no—
128C22×C4⋊C4D412yes—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#463—216no—
138#461—28no—
139#459—28no—
140#446—216no—
141#446—28no—
142C8×Q8—28no—
143#428—44no—
144C2×#107C2314yes—
145C2×#106C2314yes—
146#423—28no—
147#423D418yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152#418—24no—
153#418—24no—
154#417—28no—
155#417—28no—
156#417—44no—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C4×M16—28no—
166C4×M16—24no—
167C2×#97C2314yes—
168C2×#97C2312yes—
169C2×C4×C8C2314yes—
170C2×C4×C8C2312yes—
171#400—22no—
172#400—22no—
173#399—24no—
174#399—22no—
175#399—22no—
176#397—22no—
177#397—22no—
178#394—22no—
179#394—22no—
180#393—22no—
181#393—24no—
182#393—22no—
183#393—22no—
184#393—22no—
185#391—22no—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#390—22no—
191#390—22no—
192#390—22no—
193#390C2312yes—
194#390C2312yes—
195#390C2314yes—
196#390C2314yes—
197#390C2312yes—
198#390C2312yes—
199#389—22no—
200#389—22no—
201#389C2314yes—
202#389C2314yes—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#386—22no—
208#386—22no—
209#386—22no—
210#386—22no—
211#385—24no—
212#385—24no—
213#385—24no—
214#385—24no—
215#382—22no—
216#382—21no—
217#382—21no—
218#382C2312yes—
219#382C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228#376—21no—
229C2×#95—21no—
230C2×#95—22no—
231C2×#95—21no—
232C43—21no—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—24no—
244C22×Q8—24no—
245C22×Q8—44no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×D412yes—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2412yes—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4#4114yes—
260C22×D4C2412yes—
261C22×D4C2412yes—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—44no—
268C23×C4—84no—
269C23×C4C2×D412yes—
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274AGL(1,ℤ/8)—432no—
275C2×SD16—28no—
276C2×SD16—28no—
277C2×SD16—44no—
278C2×D8—28no—
279C2×D8—28no—
280C2×D8—44no—
281C2×M16—22no—
282C2×M16—24no—
283C2×M16—24no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16C2×D418yes—
289C2×M16#4114yes—
290C22×C8C2412yes—
291C22×C8C2414yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295#127—22no—
296#127—22no—
297#124—24no—
298#124—24no—
299#124—48no—
300#121—44no—
301#121—44no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—28no—
305C4×Q8—22no—
306C4×Q8—24no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—44no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—48no—
366C2×C22⋊C4—44no—
367C2×C22⋊C4—84no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D414yes—
379C2×C42#4114yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—44no—
385#106—44no—
386C4⋊C8—28no—
387C4⋊C8—28no—
388C4⋊C8C2×D418yes—
389C4⋊C8C2×D418yes—
390#103—28no—
391#103—28no—
392#103—44no—
393#102—28no—
394#102—28no—
395#102—28no—
396#102—28no—
397#102—44no—
398#97—28no—
399#97—24no—
400C4×C8—28no—
401C4×C8—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418C24—24no—
419C24—22no—
420C24—22no—
421C24—44no—
422C24—44no—
423C24—42no—
424C24—42no—
425C24C22×D411yescharacteristic
426#41—48no—
427#41—48no—
428#41—48no—
429#41—48no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—416no—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—82no—
469C2×D4C2×#4114yes—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4—816no—
496C22×C4#12114yes—
497C22×C4#12114yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C42−1+412yes—
513C22×C4C2511yescharacteristic
514SD16—416no—
515D8—416no—
516M16—216no—
517M16—28no—
518M16—44no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8C22×D418yes—
527C2×C8C2×#4114yes—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—48no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—84no—
566C22⋊C4—84no—
567C42—24no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—44no—
574C42—42no—
575C42—42no—
576C42—82no—
577C42#12114yes—
578C42#12114yes—
579C42C22×D412yes—
580C42C22×D412yes—
581C42C2×#4112yes—
582C42C2×#4112yes—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—416no—
587C23—44no—
588C23—42no—
589C23—48no—
590C23—42no—
591C23—44no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—84no—
600C23—84no—
601C23—88no—
602C23C2×#12112yes—
603C23#54812yes—
604C23#55012yes—
605C23C23×D411yescharacteristic
606Q8—28no—
607Q8—48no—
608Q8—44no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—42no—
613D4—24no—
614D4—24no—
615D4—42no—
616D4—48no—
617D4—42no—
618D4—42no—
619D4—42no—
620D4—42no—
621D4—42no—
622D4—84no—
623D4—84no—
624D4—88no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—48no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—88no—
658C2×C4—88no—
659C2×C4—84no—
660C2×C4—88no—
661C2×C4—84no—
662C2×C4—88no—
663C2×C4C2×#12112yes—
664C2×C4C2×#12112yes—
665C2×C4C2×#12112yes—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4#52914yes—
669C2×C4#54214yes—
670C2×C4#54614yes—
671C2×C4#54812yes—
672C2×C4#54814yes—
673C2×C4#55012yes—
674C2×C4#57514yes—
675C2×C4C23×D411yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×2−1+411yescharacteristic
681C8—28no—
682C8—24no—
683C8—42no—
684C22—22no—
685C22—22no—
686C22—24no—
687C22—24no—
688C22—48no—
689C22—44no—
690C22—816no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22#242614yes—
702C22C22×#12111yescharacteristic
703C22C2×#54811yescharacteristic
704C22C2×#55011yescharacteristic
705C22C2×#57512yes—
706C22#358512yes—
707C4—22no—
708C4—22no—
709C4—24no—
710C4—22no—
711C4—22no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—84no—
720C4#242614yes—
721C4C2×#52911yescharacteristic
722C4C2×#54211yescharacteristic
723C4C2×#54611yescharacteristic
724C4C2×#54811yescharacteristic
725C2—24no—
726C2—84no—
727C2—162no—
728C2C2×#242611yescharacteristic
729C2#6044411yescharacteristic
730C2#6045411yescharacteristic
731C2#6131812yes—
732C2#6140512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3036C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×Q8×M16C2semidirect
3. #34955C2semidirect
4. C2×#2580C2semidirect
5. #25308C2semidirect
6. #25304C2semidirect
7. #25298C2semidirect
8. #25296C2semidirect
9. #23617C2semidirect
10. #23546C2semidirect
11. #21236C2semidirect
12. #21201C2semidirect
13. #21172C2semidirect
14. #21093C2semidirect
15. C4×#418C2semidirect
25. C2×#446C22semidirect
26. C2×#446C22semidirect
27. C2×C8×Q8C22semidirect
38. #2592C22semidirect
39. #2592C22semidirect
47. #2568C22semidirect
48. #2548C22semidirect
63. #2068C22semidirect
64. #2067C22semidirect
65. #2066C22semidirect
66. #2065C22semidirect
75. #1961C22semidirect
76. #1960C22semidirect
77. #1958C22semidirect
78. #1949C22semidirect
79. #1949C22semidirect
80. #1936C22semidirect
81. #1935C22semidirect
82. #1934C22semidirect
83. #1933C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30362
C2×Q8×M162
#349552
C2×#25802
#253082
#253042
#252982
#252962
#236172
#235462
#212362
#212012
#211722
#210932
C4×#4182