Finite Group Databasebeta

← #6541821

Subgroups — #6541821

Order 512.

2093 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
1#36134C211yesmaximal, characteristic, complemented
2#34310C211yesmaximal, characteristic
3#30113C212yesmaximal, complemented
4#29798C211yesmaximal, characteristic, complemented
5#29254C211yesmaximal, characteristic, complemented
6#28438C211yesmaximal, characteristic, complemented
7#28311C212yesmaximal, complemented
8#26945C214yesmaximal, complemented
9#26236C212yesmaximal, complemented
10#25753C212yesmaximal
11#25176C214yesmaximal, complemented
12#23933C212yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14#23574C212yesmaximal, complemented
15#23190C212yesmaximal, complemented
16#23100C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×D42C2211yescharacteristic
19C2×#533C2212yescomplemented
20C2×#518—21no—
21C2×#492C2212yes—
22C2×#484C2214yescomplemented
23C2×#461C2212yes—
24C2×#461C2214yescomplemented
25C2×#450C2214yes—
26C2×#450C2214yes—
27C2×#440C2214yescomplemented
28#2868C2211yescharacteristic, complemented
29#2857C2212yescomplemented
30#2552—21no—
31#2542C2212yes—
32#2538—21no—
33#2441—22no—
34#2429C2214yescomplemented
35#2423—22no—
36#2366C2211yescharacteristic
37#2365C2212yescomplemented
38#2363—21no—
39#2361C2212yes—
40#2358—22no—
41#2354—21no—
42#2354C2211yescharacteristic
43Q8×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45D4×C22⋊C4—21no—
46D4×C22⋊C4C2212yes—
47#2324—21no—
48#2321—22no—
49C4×C4⋊Q8C2211yescharacteristic, complemented
50C4×#127C2211yescharacteristic
51C4×#121C2212yes—
52#2293C2211yescharacteristic
53C2×#390C2212yes—
54C42×D4C2211yescharacteristic
55#2099C2211yescharacteristic, complemented
56#2093C2211yescharacteristic
57#2093C2212yescomplemented
58#2086C2212yescomplemented
59#2067C2212yes—
60#2057C2211yescharacteristic, complemented
61#2043C2212yescomplemented
62#2043C2212yescomplemented
63#2036C2214yescomplemented
64#2036C2214yescomplemented
65#2023C2211yescharacteristic
66#2023C2212yescomplemented
67#2013C2212yes—
68#2013C2212yes—
69#2013C2212yescomplemented
70#2013C2212yescomplemented
71#2000C2214yes—
72#2000C2214yes—
73#1982C2212yes—
74#1981C2212yescomplemented
75#1973C2214yes—
76#1973C2214yescomplemented
77#1956C2212yescomplemented
78#1954C2212yes—
79#1947C2212yescomplemented
80#1928C2214yescomplemented
81#1924C2214yescomplemented
82#1880C2214yes—
83#1880C2214yescomplemented
84#1877C2214yescomplemented
85#1872C2214yescomplemented
86#1862C2214yescomplemented
87#1831C2211yescharacteristic, complemented
88#1817C2212yescomplemented
89C4×#107C2212yescomplemented
90#1770C2212yescomplemented
91C4×C4⋊C8C2211yescharacteristic, complemented
92#1763C2214yescomplemented
93C4×#102C2212yes—
94#1749C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×SD16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100D42—24no—
101D42—24no—
102#533—48no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2314yes—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×#120—22no—
115#518—48no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2314yes—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×#117—22no—
132C2×#117—21no—
133C2×#117—21no—
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2312yes—
139#492—28no—
140#484—48no—
141#461—28no—
142#461—48no—
143#450—48no—
144#450—48no—
145#440—48no—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#107C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162C2×#97C2314yes—
163C2×C4×C8C2312yes—
164#399—24no—
165#398—24no—
166#396—24no—
167#391—21no—
168#391C2314yes—
169#390—22no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—22no—
175#390C2312yes—
176#390C2311yescharacteristic
177#390C2312yes—
178#390C2311yescharacteristic
179#390C2312yes—
180#389—21no—
181#389—22no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—24no—
186#389C2312yes—
187#389C2311yescharacteristic
188#389C2311yescharacteristic
189#388—24no—
190#388—21no—
191#388—22no—
192#388—21no—
193#388—22no—
194#388—22no—
195#387—24no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#380—22no—
211#380—22no—
212#380—21no—
213#380C2312yes—
214#379—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C43C2311yescharacteristic
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2312yes—
237#336C2314yes—
238#336C2312yes—
239C25—22no—
240C22×Q8—21no—
241C22×Q8C2414yes—
242C22×D4—28no—
243C22×D4—22no—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—44no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×SD16—44no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280#127—22no—
281#127#4114yes—
282#125—48no—
283C22⋊Q8—28no—
284C22⋊Q8—44no—
285C22⋊Q8—48no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288#121—28no—
289#121—24no—
290#121—24no—
291#120—216no—
292C4×Q8—28no—
293C4×Q8—44no—
294C4×Q8—48no—
295C4×Q8—44no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—48no—
304C4×D4—44no—
305C4×D4—48no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—82no—
309C4×D4#4114yes—
310#117—44no—
311#117—48no—
312#117—44no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—24no—
380#107—24no—
381#107—44no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#103—28no—
385#103—28no—
386#103—44no—
387#103—44no—
388#103—44no—
389#102—24no—
390#102—28no—
391#102—24no—
392#102—44no—
393C22⋊C8—28no—
394C22⋊C8—44no—
395#97—44no—
396C4×C8—24no—
397#95—21no—
398#95—22no—
399#95—21no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—22no—
405#95—24no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—28no—
410C24—21no—
411C24—22no—
412C24—44no—
413C24—44no—
414C24—82no—
415C24—82no—
416C24—82no—
417C24—82no—
418C24C22×D412yes—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×D4—28no—
429C2×D4—28no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—432no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—81no—
446C2×D4—81no—
447C2×D4—82no—
448C2×D4—82no—
449C2×D4—88no—
450C2×D4—88no—
451C2×D4#13714yes—
452C2×D4C2×#4112yes—
453C2×D4C2×#4114yes—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2511yescharacteristic
493SD16—416no—
494SD16—88no—
495SD16—88no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8H(5,𝔽2)14yes—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—82no—
529C4⋊C4—81no—
530C4⋊C4—84no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4C2×#4112yes—
534C4⋊C4H(5,𝔽2)14yes—
535C22⋊C4—28no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—81no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—84no—
555C42—22no—
556C42—22no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—42no—
561C42—42no—
562C42—41no—
563C42—48no—
564C42—44no—
565C42—42no—
566C42—44no—
567C42AGL(1,ℤ/8)14yes—
568C42C2×#4112yes—
569C42C2×#4112yes—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—416no—
579C23—41no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—42no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—41no—
588C23—42no—
589C23—41no—
590C23—42no—
591C23—88no—
592C23—88no—
593C23—164no—
594C23—164no—
595C23—164no—
596C23—164no—
597C23C2×#13714yes—
598C23C23×D411yescharacteristic
599Q8—48no—
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604D4—28no—
605D4—24no—
606D4—28no—
607D4—416no—
608D4—48no—
609D4—44no—
610D4—42no—
611D4—84no—
612D4—88no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—48no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—84no—
659C2×C4—164no—
660C2×C4—164no—
661C2×C4—164no—
662C2×C4#52512yes—
663C2×C4#52512yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×#13712yes—
668C2×C4#57712yes—
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)12yes—
672C2×C4C2×H(5,𝔽2)12yes—
673C8—24no—
674C8—44no—
675C8—42no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—82no—
690C22—81no—
691C22—82no—
692C22—81no—
693C22—82no—
694C22—81no—
695C22—82no—
696C22—81no—
697C22—1616no—
698C22C2×#52511yescharacteristic
699C22#348411yescharacteristic
700C22#349811yescharacteristic
701C22C22×AGL(1,ℤ/8)11yescharacteristic
702C22C22×#13711yescharacteristic
703C22C2×#57711yescharacteristic
704C22#358611yescharacteristic
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—42no—
710C4—42no—
711C4—44no—
712C4—42no—
713C4—42no—
714C4—41no—
715C4—81no—
716C4—84no—
717C4—82no—
718C4—82no—
719C4—81no—
720C4#311014yes—
721C4#350812yes—
722C4#351912yes—
723C2—44no—
724C2—84no—
725C2—82no—
726C2#3574111yescharacteristic
727C2C2×#311011yescharacteristic
728C2#6059111yescharacteristic
729C2#6158511yescharacteristic
730C2#6159311yescharacteristic
731C2#6242111yescharacteristic
732C2#6244811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
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. #36134C2semidirect
3. #30113C2semidirect
4. #29798C2semidirect
5. #29254C2semidirect
6. #28438C2semidirect
7. #28311C2semidirect
8. #26945C2semidirect
9. #26236C2semidirect
11. #25176C2semidirect
12. #23933C2semidirect
13. #23774C2semidirect
14. #23574C2semidirect
15. #23190C2semidirect
16. #23100C2semidirect
17. #22839C2semidirect
19. C2×#533C22semidirect
22. C2×#484C22semidirect
24. C2×#461C22semidirect
27. C2×#440C22semidirect
28. #2868C22semidirect
29. #2857C22semidirect
34. #2429C22semidirect
37. #2365C22semidirect
49. C4×C4⋊Q8C22semidirect
55. #2099C22semidirect
57. #2093C22semidirect
58. #2086C22semidirect
60. #2057C22semidirect
61. #2043C22semidirect
62. #2043C22semidirect
63. #2036C22semidirect
64. #2036C22semidirect
66. #2023C22semidirect
69. #2013C22semidirect
70. #2013C22semidirect
74. #1981C22semidirect
76. #1973C22semidirect
77. #1956C22semidirect
79. #1947C22semidirect
80. #1928C22semidirect
81. #1924C22semidirect
83. #1880C22semidirect
84. #1877C22semidirect
85. #1872C22semidirect
86. #1862C22semidirect
87. #1831C22semidirect
88. #1817C22semidirect
89. C4×#107C22semidirect
90. #1770C22semidirect
91. C4×C4⋊C8C22semidirect
92. #1763C22semidirect
94. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#361342
#343102
#301132
#297982
#292542
#284382
#283112
#269452
#262362
#257532
#251762
#239332
#237742
#235742
#231902
#231002
#228392