Finite Group Databasebeta

← #495160

Subgroups — #495160

Order 512.

1409 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34935C211yesmaximal, characteristic
2C2×#2120C214yesmaximal, complemented
3#20637C211yesmaximal, characteristic, complemented
4C4×#410C211yesmaximal, characteristic, complemented
5#13222C214yesmaximal, complemented
6#13221C212yesmaximal, complemented
7#13220C212yesmaximal, complemented
8C2×#525C2212yes—
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C411yescharacteristic, complemented
11C2×#423—21no—
12C2×#413C2212yes—
13C2×#410C2212yes—
14#2538—21no—
15#2537—21no—
16#2536—21no—
17#2535—21no—
18#2534C412yescomplemented
19#2534C2212yes—
20#2486C411yescharacteristic, complemented
21#2486C2211yescharacteristic
22#2408C411yescharacteristic, complemented
23#2408C2211yescharacteristic
24#2392—21no—
25#2392—21no—
26C4×#121—21no—
27C4×#121—21no—
28#2285C411yescharacteristic, complemented
29#2280C412yescomplemented
30C4×#117—21no—
31#2120—216no—
32C2×#352C2218yes—
33C2×AΣL(1,𝔽16)C2218yes—
34#1801—21no—
35#1801—21no—
36#1800C2212yes—
37#1800C2212yes—
38C4×#99—21no—
39C4×#99—21no—
40C4×#99C2212yes—
41C4×#99C2212yes—
42#1742—21no—
43#1740—21no—
44C22×#41C2311yescharacteristic
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C22×M16—22no—
48#525—28no—
49#525—28no—
50C2×#123C2×C412yes—
51C2×#123C2312yes—
52C2×C22⋊Q8—22no—
53C2×#121—22no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#120C2×C412yes—
57C2×#120C2312yes—
58C4×#41—41no—
59C4×#41—41no—
60C4×#41—41no—
61C4×#41—41no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—41no—
71C2×C4×D4—41no—
72C2×C4×D4—41no—
73C2×C4×D4—41no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—22no—
81C2×#117C2×C412yes—
82C2×#117C2×C411yescharacteristic
83C2×#117D411yescharacteristic
84C2×#117D411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C412yes—
87C22×C42C2×C411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90#423—22no—
91#423—22no—
92#423—41no—
93#423—41no—
94C2×C4⋊C8—41no—
95C2×C4⋊C8—41no—
96#413—28no—
97C2×#101—22no—
98C2×#100C2314yes—
99#410—24no—
100#410D418yescomplemented
101C2×#99—22no—
102C2×#99C2314yes—
103#398—21no—
104#398—21no—
105#397—21no—
106#397—21no—
107#396—22no—
108#396—22no—
109#394—21no—
110#394—21no—
111#394—22no—
112#394—22no—
113#392—21no—
114#392—21no—
115#390—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#388—22no—
120#388—22no—
121#386—22no—
122#386—22no—
123#384—21no—
124#382—21no—
125#381C2×C412yes—
126#381C2×C412yes—
127#380—21no—
128#380—21no—
129#380—22no—
130#380—22no—
131#380C2×C411yescharacteristic
132#380C2×C411yescharacteristic
133#380C2×C412yes—
134#379C2×C412yes—
135#379C2×C412yes—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—41no—
139C4×C4⋊C4—41no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144#376—41no—
145C43—41no—
146#352—216no—
147AΣL(1,𝔽16)—216no—
148#343—22no—
149#343—22no—
150#342—22no—
151#342—22no—
152C25C22×C412yes—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—42no—
156C2×#41C2×D414yes—
157C2×#41#4114yes—
158C22×Q8C2×D411yescharacteristic
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—41no—
164C22×D4—41no—
165C22×D4—44no—
166C22×D4C22×C411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C22×D4C2411yescharacteristic
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—28no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C2×M16—24no—
186C2×M16—42no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8—42no—
190#123—28no—
191#123—24no—
192#123—24no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195#121—44no—
196#121—44no—
197#121—48no—
198#120—28no—
199#120—24no—
200#120—24no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—48no—
224C4×D4—44no—
225C4×D4—44no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230#117—28no—
231#117—28no—
232#117—41no—
233#117—41no—
234#117—41no—
235#117—41no—
236#117—41no—
237#117—41no—
238#117—44no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22×C412yes—
272C2×C22⋊C4C22×C412yes—
273C2×C22⋊C4C22×C411yescharacteristic
274C2×C22⋊C4C22×C411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C4⋊C8—42no—
304C4⋊C8—42no—
305C4⋊C8—42no—
306C4⋊C8—42no—
307#101—44no—
308#100—28no—
309#99—28no—
310#99—24no—
311#99C2×D418yes—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—21no—
318#95—21no—
319#95—42no—
320#95—42no—
321C24—22no—
322C24—24no—
323C24—42no—
324C24—44no—
325C24—42no—
326C24—82no—
327C24C2×C22⋊C411yescharacteristic
328C24C23×C411yescharacteristic
329C24C22×D411yescharacteristic
330#41—44no—
331#41—44no—
332#41—44no—
333#41—44no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8#12412yes—
337C2×Q8#12712yes—
338C2×D4—24no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4C4×D414yes—
360C2×D4#12114yes—
361C2×D4C22×D412yes—
362C2×D4C2×#4112yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C4×D414yes—
418C22×C4#12114yes—
419C22×C4#12412yes—
420C22×C4#12712yes—
421C22×C4C22×D411yescharacteristic
422M16—44no—
423M16—44no—
424M16—84no—
425C2×C8—44no—
426C2×C8—84no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—82no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—28no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—48no—
485C42—44no—
486C42—84no—
487C42—84no—
488C42#9911yescharacteristic
489C42#9911yescharacteristic
490C42#9911yescharacteristic
491C42#9911yescharacteristic
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—84no—
514C23—88no—
515C23—88no—
516C23—84no—
517C23#39012yes—
518C23C22×C22⋊C411yescharacteristic
519C23C2×C4×D412yes—
520C23C2×#12112yes—
521Q8—42no—
522Q8—42no—
523D4—44no—
524D4—44no—
525D4—42no—
526D4—42no—
527D4—44no—
528D4—48no—
529D4—44no—
530D4—88no—
531D4—88no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—88no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#39012yes—
589C2×C4#39012yes—
590C2×C4#39012yes—
591C2×C4C2×#9911yescharacteristic
592C2×C4C2×#9911yescharacteristic
593C2×C4C2×#9911yescharacteristic
594C2×C4C2×#9911yescharacteristic
595C2×C4C2×#9911yescharacteristic
596C2×C4C2×#9911yescharacteristic
597C2×C4#41012yes—
598C2×C4C2×#12411yescharacteristic
599C2×C4C2×#12711yescharacteristic
600C8—84no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—44no—
619C22—81no—
620C22—81no—
621C22—82no—
622C22—82no—
623C22—88no—
624C22—88no—
625C22—82no—
626C22—82no—
627C22C2×#39011yescharacteristic
628C22C22×#9911yescharacteristic
629C22C2×#41011yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—21no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—82no—
644C4—82no—
645C4—88no—
646C4—82no—
647C4#196412yes—
648C4#196412yes—
649C4#212012yescomplemented
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2—42no—
655C2—84no—
656C2C2×#196411yescharacteristic
657C2C2×#212011yescharacteristic
658C2#3334711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radical#34935C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2120C2semidirect
3. #20637C2semidirect
4. C4×#410C2semidirect
5. #13222C2semidirect
6. #13221C2semidirect
7. #13220C2semidirect
10. C22×C4×D4C4semidirect
18. #2534C4semidirect
20. #2486C4semidirect
22. #2408C4semidirect
28. #2285C4semidirect
29. #2280C4semidirect
100. #410D4semidirect
649. C4#2120semidirect

Maximal subgroups

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

subgroupindex
#349352
C2×#21202
#206372
C4×#4102
#132222
#132212
#132202