Finite Group Databasebeta

← #6795537

Subgroups — #6795537

Order 512.

1615 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32301C214yesmaximal, complemented
5#31950C214yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#28689C211yesmaximal, characteristic, complemented
8#28540C212yesmaximal, complemented
9#28283C211yesmaximal, characteristic, complemented
10#27737C212yesmaximal, complemented
11#27679C212yesmaximal, complemented
12#25705C212yesmaximal, complemented
13#24659C212yesmaximal, complemented
14#24119C212yesmaximal
15#22799C211yesmaximal, characteristic, complemented
16#21988C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#529—21no—
20C2×#500C2212yes—
21C2×#475C2212yes—
22C2×#474C2212yes—
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#432C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2568C2212yes—
29#2553—22no—
30#2552C2211yescharacteristic
31#2547—21no—
32#2547—21no—
33#2547C2211yescharacteristic
34#2542C2211yescharacteristic
35#2537C2211yescharacteristic
36#2499C2214yes—
37#2496—22no—
38#2496—22no—
39#2496C2212yes—
40#2489—21no—
41#2427—21no—
42#2422C2212yes—
43#2395—22no—
44#2365—21no—
45#2365—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8—21no—
49C4×#121C2212yes—
50#2285C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2095C2214yes—
56#2092C2214yes—
57#2083C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2036C2214yes—
61#2036C2214yes—
62#2036C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1981C2212yes—
67#1981C2211yescharacteristic
68#1973C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1935C2214yes—
77#1927C2212yes—
78#1924C2214yes—
79#1877C2212yes—
80#1875C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1836C2214yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#556—44no—
97#556—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103#529—48no—
104C2×#126—22no—
105C2×#126—24no—
106C2×#125—21no—
107C2×#124—21no—
108C2×#124—22no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2314yes—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117C2314yes—
127C22×C42—21no—
128C22×C42C2312yes—
129#500—44no—
130#475—44no—
131#474—44no—
132#461—24no—
133C4×SD16—44no—
134#432—28no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—24no—
157#400—22no—
158#399—22no—
159#398—22no—
160#397—22no—
161#397—22no—
162#397—24no—
163#397—24no—
164#397—22no—
165#394—22no—
166#394—22no—
167#391—22no—
168#391—22no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2314yes—
173#390C2312yes—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#387—24no—
184#387—24no—
185#387—22no—
186#386—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382C2314yes—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—42no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263#126—48no—
264#126—416no—
265#125—44no—
266#124—44no—
267#124—48no—
268#123—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273#121—24no—
274#121—42no—
275#121—44no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—48no—
279C4×Q8—44no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—42no—
289C4×D4—41no—
290C4×D4—48no—
291#117—44no—
292#117—44no—
293#117—48no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#107—44no—
359#107—44no—
360#106—42no—
361#106—44no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—42no—
366C4⋊C8#4114yes—
367#103—24no—
368#103—42no—
369#103—44no—
370#103—42no—
371#103—44no—
372#102—42no—
373#102—44no—
374C22⋊C8—28no—
375C22⋊C8—44no—
376#97—28no—
377C4×C8—24no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—22no—
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—21no—
397C24C22×D412yes—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—82no—
421C2×D42−1+412yes—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461SD16—48no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8C2×#4114yes—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—41no—
531C42—48no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42AGL(1,ℤ/8)14yes—
538C42C2×#4112yes—
539C42H(5,𝔽2)12yes—
540C42H(5,𝔽2)12yes—
541C422−1+412yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C23×D411yescharacteristic
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—88no—
562Q8—84no—
563D4—22no—
564D4—22no—
565D4—44no—
566D4—41no—
567D4—41no—
568D4—88no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4#53512yes—
621C2×C4#53512yes—
622C2×C4#54712yes—
623C2×C4#54712yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4#57512yes—
628C2×C4#57812yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C2×C4#58511yescharacteristic
635C8—24no—
636C8—44no—
637C8—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22C2×#53511yescharacteristic
653C22C2×#54711yescharacteristic
654C22#348911yescharacteristic
655C22C22×AGL(1,ℤ/8)11yescharacteristic
656C22C2×#57511yescharacteristic
657C22C2×#57811yescharacteristic
658C22#358611yescharacteristic
659C4—22no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—82no—
667C4—81no—
668C4—81no—
669C4—84no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4#321414yes—
675C4#335314yes—
676C4#351711yescharacteristic
677C4#352011yescharacteristic
678C4#354711yescharacteristic
679C4#355211yescharacteristic
680C2—82no—
681C2—82no—
682C2#3892011yescharacteristic
683C2C2×#321411yescharacteristic
684C2#6083711yescharacteristic
685C2C2×#335311yescharacteristic
686C2#6125711yescharacteristic
687C2#6210911yescharacteristic
688C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #36188C2semidirect
3. #33195C2semidirect
4. #32301C2semidirect
5. #31950C2semidirect
6. #29923C2semidirect
7. #28689C2semidirect
8. #28540C2semidirect
9. #28283C2semidirect
10. #27737C2semidirect
11. #27679C2semidirect
12. #25705C2semidirect
13. #24659C2semidirect
15. #22799C2semidirect
16. #21988C2semidirect
17. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349322
#331952
#323012
#319502
#299232
#286892
#285402
#282832
#277372
#276792
#257052
#246592
#241192
#227992
#219882
C4×#4612