Finite Group Databasebeta

← #6793022

Subgroups — #6793022

Order 512.

1763 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#31617C214yesmaximal, complemented
5#31559C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28279C211yesmaximal, characteristic, complemented
9#28141C212yesmaximal, complemented
10#26111C212yesmaximal, complemented
11#25338C212yesmaximal, complemented
12#24837C212yesmaximal, complemented
13#24816C212yesmaximal, complemented
14#24156C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21888C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#477C2212yes—
21C2×#474C2212yes—
22C2×#460C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2573—22no—
29#2571C2211yescharacteristic
30#2568C2211yescharacteristic
31#2547—21no—
32#2547C2211yescharacteristic
33#2538C2211yescharacteristic
34#2537—21no—
35#2537C2212yes—
36#2514—21no—
37#2494C2212yes—
38#2447—21no—
39#2432—22no—
40#2422—22no—
41#2422C2214yes—
42#2421C2212yes—
43#2401—22no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#124—21no—
49C4×C22⋊Q8C2212yes—
50#2295—21no—
51#2283C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2100C2214yes—
56#2087C2214yes—
57#2085C2214yes—
58#2077C2214yes—
59#2075C2214yes—
60#2068C2214yes—
61#2067C2214yes—
62#2033C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1982C2212yes—
69#1982C2211yescharacteristic
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1926C2212yes—
76#1893C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1880C2212yes—
80#1877C2214yes—
81#1862C2214yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1808C2214yes—
86#1804C2214yes—
87#1804C2214yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97C23×C8C2314yes—
98#547—48no—
99#532—28no—
100#532—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#124—21no—
107C2×#124—22no—
108C2×#123—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121C2314yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C22×C4⋊C4—22no—
127C22×C42C2312yes—
128#477—44no—
129#474—44no—
130#460—24no—
131C4×SD16—44no—
132C8×D4—44no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2312yes—
136C2×C4⋊C8—44no—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#401—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#397—24no—
160#396—22no—
161#394—24no—
162#394—24no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—24no—
187#385—24no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#383—22no—
195#382—21no—
196#382C2312yes—
197#382C2312yes—
198#381—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376—22no—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—48no—
238C2×#41—48no—
239C2×#41—84no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8C2411yescharacteristic
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8—44no—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267C4⋊Q8—82no—
268#127—22no—
269#127#4114yes—
270#124—48no—
271#124—44no—
272#123—416no—
273C22⋊Q8—24no—
274C22⋊Q8—42no—
275C22⋊Q8—48no—
276#121—28no—
277#121—44no—
278#121—48no—
279#121—48no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—41no—
286C4×Q8—44no—
287C4×D4—24no—
288C4×D4—28no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—24no—
294C4×D4—44no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—44no—
359#107—44no—
360#106—42no—
361C4⋊C8—22no—
362C4⋊C8—22no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366C4⋊C8—42no—
367#103—42no—
368#103—44no—
369#102—24no—
370#102—42no—
371#102—44no—
372#102—42no—
373#102—44no—
374C22⋊C8—44no—
375C4×C8—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—22no—
397C24C22×D412yes—
398#41—832no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q82−1+412yes—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—84no—
427C2×D4AGL(1,ℤ/8)14yes—
428C2×D4C2×#4114yes—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—84no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465SD16—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—88no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—41no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—81no—
539C42#13514yes—
540C42C2×#4112yes—
541C42H(5,𝔽2)12yes—
542C42H(5,𝔽2)12yes—
543C422−1+412yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23C2×AGL(1,ℤ/8)14yes—
558C23C23×D411yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563Q8—88no—
564Q8—84no—
565D4—28no—
566D4—24no—
567D4—24no—
568D4—48no—
569D4—42no—
570D4—42no—
571D4—48no—
572D4—816no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—816no—
622C2×C4—88no—
623C2×C4#53712yes—
624C2×C4#53712yes—
625C2×C4#54612yes—
626C2×C4#54612yes—
627C2×C4C2×#13512yes—
628C2×C4C2×#13512yes—
629C2×C4C2×#13512yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57812yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4#58511yescharacteristic
638C8—42no—
639C8—44no—
640C8—42no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22C2×#53711yescharacteristic
656C22C2×#54611yescharacteristic
657C22#348911yescharacteristic
658C22C22×#13511yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C2×#57811yescharacteristic
661C22#358511yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—82no—
675C4—81no—
676C4—84no—
677C4#324714yes—
678C4#330214yes—
679C4#351511yescharacteristic
680C4#352011yescharacteristic
681C4#354611yescharacteristic
682C4#355211yescharacteristic
683C2—44no—
684C2—84no—
685C2#3890811yescharacteristic
686C2C2×#324711yescharacteristic
687C2#6093011yescharacteristic
688C2C2×#330211yescharacteristic
689C2#6115511yescharacteristic
690C2#6210211yescharacteristic
691C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
2. #34945C2semidirect
3. #33198C2semidirect
4. #31617C2semidirect
5. #31559C2semidirect
6. #29942C2semidirect
7. #29218C2semidirect
8. #28279C2semidirect
9. #28141C2semidirect
10. #26111C2semidirect
11. #25338C2semidirect
12. #24837C2semidirect
13. #24816C2semidirect
14. #24156C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21888C2semidirect
17. C4×#460C2semidirect

Maximal subgroups

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

subgroupindex
#361272
#349452
#331982
#316172
#315592
#299422
#292182
#282792
#281412
#261112
#253382
#248372
#248162
#241562
D4×C4⋊C82
#218882
C4×#4602