Finite Group Databasebeta

← #6795512

Subgroups — #6795512

Order 512.

1983 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#32278C214yesmaximal, complemented
5#31930C214yesmaximal, complemented
6#29920C212yesmaximal, complemented
7#28689C211yesmaximal, characteristic, complemented
8#28527C212yesmaximal, complemented
9#28279C211yesmaximal, characteristic, complemented
10#27727C212yesmaximal, complemented
11#27677C212yesmaximal, complemented
12#25703C212yesmaximal, complemented
13#24650C212yesmaximal, complemented
14#24116C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21982C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2212yes—
21C2×#529—21no—
22C2×#500C2212yes—
23C2×#474C2212yes—
24C2×#474C2212yes—
25C2×#459C2211yescharacteristic
26C2×C4×D8C2212yes—
27C2×#431C2212yes—
28#2868C2211yescharacteristic
29#2856C2212yes—
30#2553—22no—
31#2552C2211yescharacteristic
32#2547C2211yescharacteristic
33#2542—21no—
34#2542—21no—
35#2542C2212yes—
36#2542C2211yescharacteristic
37#2537C2211yescharacteristic
38#2501—22no—
39#2501—22no—
40#2501C2214yes—
41#2496C2212yes—
42#2489—21no—
43#2427—21no—
44#2422C2212yes—
45#2395—22no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×C22⋊Q8—21no—
49C4×#121C2212yes—
50#2285C2212yes—
51C2×#384C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2099C2214yes—
55#2093C2214yes—
56#2093C2214yes—
57#2083C2214yes—
58#2071C2214yes—
59#2067C2214yes—
60#2033C2214yes—
61#2033C2214yes—
62#2033C2214yes—
63#1999C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1982C2212yes—
67#1982C2211yescharacteristic
68#1970C2214yes—
69#1954C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1934C2214yes—
77#1925C2212yes—
78#1923C2214yes—
79#1879C2212yes—
80#1874C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1852C2214yes—
84#1845C2214yes—
85#1842C2211yescharacteristic
86#1834C2214yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×D8C2312yes—
97#547—44no—
98#547—44no—
99#547—48no—
100#547—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106#529—48no—
107C2×#126—22no—
108C2×#125—21no—
109C2×#124—21no—
110C2×#123—22no—
111C2×#123—24no—
112C2×#123—24no—
113C2×#123—24no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×#121—21no—
120C2×#121—22no—
121C2×#121—22no—
122C2×#121C2312yes—
123C2×#121C2314yes—
124C2×C4×Q8—21no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2312yes—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×#117—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4C2314yes—
139C22×C42—21no—
140C22×C42C2312yes—
141#500—44no—
142#474—44no—
143#474—44no—
144#459—24no—
145C4×D8—44no—
146#431—28no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#107C2312yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2311yescharacteristic
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C4×C8C2314yes—
167C2×C4×C8C2312yes—
168#400—22no—
169#399—22no—
170#399—24no—
171#399—24no—
172#399—24no—
173#398—22no—
174#397—22no—
175#397—22no—
176#397—22no—
177#394—22no—
178#394—22no—
179#391—21no—
180#391—21no—
181#390—21no—
182#390—21no—
183#390—22no—
184#390—22no—
185#390C2312yes—
186#390C2314yes—
187#390C2312yes—
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#387—22no—
194#386—22no—
195#385—22no—
196#384—21no—
197#384—21no—
198#384—22no—
199#384—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#384C2312yes—
205#384C2314yes—
206#384C2312yes—
207#382—22no—
208#380—22no—
209#380—21no—
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C2×#41—48no—
240C2×#41—48no—
241C2×#41—48no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×D4—24no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2412yes—
250C22×D4C2411yescharacteristic
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×D8—24no—
260C2×D8—24no—
261C2×D8—42no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#127—22no—
272#127—22no—
273#126—48no—
274#125—44no—
275#124—44no—
276#123—416no—
277#123—416no—
278#123—48no—
279#123—416no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C22⋊Q8—48no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285#121—24no—
286#121—42no—
287#121—44no—
288#121—44no—
289#121—48no—
290#121—44no—
291#121—48no—
292C4×Q8—44no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—41no—
304C4×D4—44no—
305C4×D4—48no—
306C4×D4—44no—
307C4×D4D818yescomplemented
308#117—48no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4—48no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—42no—
378#107—44no—
379#107—44no—
380#107—44no—
381#107—44no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384C4⋊C8—22no—
385C4⋊C8—42no—
386C4⋊C8#4114yescomplemented
387#102—24no—
388#102—42no—
389#102—44no—
390#102—42no—
391#102—42no—
392#102—44no—
393#102—44no—
394C22⋊C8—28no—
395C22⋊C8—44no—
396C4×C8—28no—
397C4×C8—24no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410C24—21no—
411C24—22no—
412C24C22×D412yes—
413#41—832no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—48no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—82no—
445C2×D4C2×D814yes—
446C2×D42−1+412yes—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4C2×D818yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2511yescharacteristic
491D8—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8C2×#4114yes—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C4⋊C4C2×D814yes—
536C4⋊C4AGL(1,ℤ/8)14yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—44no—
554C22⋊C4C2×D818yes—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—21no—
562C42—21no—
563C42—41no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—44no—
568C42C2×D814yes—
569C42C2×#4112yes—
570C42H(5,𝔽2)12yes—
571C42H(5,𝔽2)12yes—
572C422−1+412yes—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23C22×D814yes—
594C23C23×D411yescharacteristic
595Q8—84no—
596Q8—88no—
597D4—28no—
598D4—22no—
599D4—22no—
600D4—41no—
601D4—42no—
602D4—42no—
603D4—41no—
604D4—42no—
605D4—42no—
606D4—84no—
607D4—88no—
608D4—816no—
609D4—84no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—216no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—84no—
654C2×C4—816no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—88no—
658C2×C4#53512yes—
659C2×C4#53512yes—
660C2×C4#54712yes—
661C2×C4#54712yes—
662C2×C4C22×D812yes—
663C2×C4C22×D814yes—
664C2×C4C22×D812yes—
665C2×C4C22×D812yes—
666C2×C4C2×AGL(1,ℤ/8)14yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4#57812yes—
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C2×C4#58511yescharacteristic
675C8—24no—
676C8—44no—
677C8—42no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22C2×#53511yescharacteristic
700C22C2×#54711yescharacteristic
701C22#348911yescharacteristic
702C22C23×D811yescharacteristic
703C22C22×AGL(1,ℤ/8)11yescharacteristic
704C22C2×#57811yescharacteristic
705C22#358611yescharacteristic
706C4—24no—
707C4—22no—
708C4—24no—
709C4—22no—
710C4—22no—
711C4—42no—
712C4—41no—
713C4—81no—
714C4—81no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4—81no—
719C4—82no—
720C4#320514yes—
721C4#333414yes—
722C4#351711yescharacteristic
723C4#352011yescharacteristic
724C4#354711yescharacteristic
725C4#355211yescharacteristic
726C2—44no—
727C2—82no—
728C2—162no—
729C2#3892011yescharacteristic
730C2C2×#320511yescharacteristic
731C2#6083711yescharacteristic
732C2C2×#333411yescharacteristic
733C2#6125211yescharacteristic
734C2#6209511yescharacteristic
735C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36179C2semidirect
2. #34932C2semidirect
3. #33195C2semidirect
4. #32278C2semidirect
5. #31930C2semidirect
6. #29920C2semidirect
7. #28689C2semidirect
8. #28527C2semidirect
9. #28279C2semidirect
10. #27727C2semidirect
11. #27677C2semidirect
12. #25703C2semidirect
13. #24650C2semidirect
14. #24116C2semidirect
15. #22795C2semidirect
16. #21982C2semidirect
17. C4×#459C2semidirect
307. C4×D4D8semidirect
386. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#361792
#349322
#331952
#322782
#319302
#299202
#286892
#285272
#282792
#277272
#276772
#257032
#246502
#241162
#227952
#219822
C4×#4592