Finite Group Databasebeta

← #6564395

Subgroups — #6564395

Order 512.

2195 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3330C212yesmaximal, complemented
2#36179C211yesmaximal, characteristic, complemented
3#34316C211yesmaximal, characteristic, complemented
4#29864C211yesmaximal, characteristic, complemented
5#29716C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28580C212yesmaximal, complemented
8#28157C211yesmaximal, characteristic, complemented
9#27648C212yesmaximal, complemented
10#25320C212yesmaximal, complemented
11#25217C214yesmaximal, complemented
12#24676C212yesmaximal, complemented
13#24542C214yesmaximal, complemented
14#23676C211yesmaximal, characteristic, complemented
15#23630C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22500C212yesmaximal, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2211yescharacteristic
21C2×#547C2212yes—
22#3330—216no—
23C2×#487C2212yes—
24C2×#484C2214yes—
25C2×#474C2212yes—
26C2×#468C2214yes—
27C2×#461C2212yes—
28C2×#449C2214yes—
29C2×C4×D8C2212yes—
30C2×#436C2212yes—
31C2×#417C2214yes—
32#2868C2211yescharacteristic
33#2856C2212yes—
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2501—22no—
38#2501—22no—
39#2501C2214yes—
40#2374—22no—
41#2365C2211yescharacteristic
42#2363—21no—
43#2362C2212yes—
44#2354C2211yescharacteristic
45#2354C2211yescharacteristic
46#2345—21no—
47#2341C2212yes—
48#2335—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2297C2212yes—
54#2293C2211yescharacteristic
55C2×#384C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2083C2211yescharacteristic
59#2083C2212yes—
60#2083C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2212yes—
64#2072C2214yes—
65#2071C2212yes—
66#2071C2211yescharacteristic
67#2071C2214yes—
68#2071C2212yes—
69#2067C2212yes—
70#2055C2214yes—
71#2054C2214yes—
72#2054C2212yes—
73#2041C2211yescharacteristic
74#2014C2212yes—
75#2011C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1947C2214yes—
79#1939C2212yes—
80#1925C2214yes—
81#1895C2214yes—
82#1880C2214yes—
83#1880C2214yes—
84#1875C2214yes—
85#1851C2214yes—
86#1831C2211yescharacteristic
87#1824C2214yes—
88#1803C2214yes—
89#1803C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×#41—22no—
97C22×#41C2312yes—
98C22×SD16C2314yes—
99C22×D8C2312yes—
100C22×M16C2314yes—
101#547—28no—
102#547—24no—
103#547—28no—
104#547—48no—
105#547—48no—
106#547D418yescomplemented
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×#127C2311yescharacteristic
112C2×#123—22no—
113C2×#123—24no—
114C2×#123—24no—
115C2×#123—24no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—22no—
122C2×#121—22no—
123C2×#121C2314yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2312yes—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C2×#117—21no—
135C2×#117C2312yes—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2314yes—
140C22×C42C2312yes—
141#487—28no—
142#484—216no—
143#474—28no—
144#468—216no—
145#461—28no—
146#449—216no—
147C4×D8—28no—
148#436—28no—
149C2×#107C2312yes—
150C2×#107C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2312yes—
154#417—216no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×C22⋊C8C2314yes—
165C2×C4×C8C2312yes—
166#399—24no—
167#399—24no—
168#399—24no—
169#391C2312yes—
170#391C2312yes—
171#390—21no—
172#390—22no—
173#390—22no—
174#390—21no—
175#390—21no—
176#390—22no—
177#390—22no—
178#390C2312yes—
179#390C2311yescharacteristic
180#390C2312yes—
181#390C2314yes—
182#390C2311yescharacteristic
183#389—21no—
184#389—21no—
185#389—21no—
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#388—22no—
189#388—22no—
190#388—22no—
191#388—21no—
192#388—21no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#386—22no—
198#386—22no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384—22no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2314yes—
208#384C2312yes—
209#380—21no—
210#380—22no—
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231C43C2311yescharacteristic
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2311yescharacteristic
242#336C2314yes—
243#336C2312yes—
244C2×#41—24no—
245C2×#41—44no—
246C2×#41—44no—
247C2×#41—48no—
248C2×#41—48no—
249C2×#41—48no—
250C2×#41C2×D418yes—
251C22×Q8—22no—
252C22×Q8C2412yes—
253C22×D4—22no—
254C22×D4—24no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4C2×D411yescharacteristic
258C22×D4C2412yes—
259C22×D4C2411yescharacteristic
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4—22no—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C2×SD16—28no—
268C2×SD16—44no—
269C2×SD16—44no—
270C2×D8—24no—
271C2×D8—42no—
272C2×D8—42no—
273C2×M16—28no—
274C2×M16—44no—
275C2×M16—44no—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—24no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8C2×D414yes—
285#127—42no—
286#123—216no—
287#123—416no—
288#123—416no—
289#123—416no—
290C22⋊Q8—24no—
291C22⋊Q8—48no—
292C22⋊Q8—48no—
293C22⋊Q8—48no—
294C22⋊Q8C2×D418yes—
295#121—28no—
296#121—28no—
297#121—28no—
298#121—48no—
299#121—48no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4—24no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4C2×D414yes—
310C4×D4C2×D414yes—
311C4×D4#4114yes—
312#117—44no—
313#117—44no—
314#117—44no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—48no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—44no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—28no—
385C4⋊C8—24no—
386C4⋊C8—44no—
387C4⋊C8—42no—
388#103—28no—
389#103—28no—
390#103—28no—
391#102—24no—
392#102—28no—
393#102—24no—
394#102—42no—
395#102—44no—
396#102—42no—
397C22⋊C8—28no—
398C4×C8—24no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—21no—
408#95C2×D412yes—
409#95C2×D412yes—
410#95C2×D412yes—
411C24—21no—
412C24—22no—
413C24C22×D412yes—
414#41—416no—
415#41—88no—
416#41—832no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8C22×D414yes—
424C2×D4—28no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—48no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4#12114yes—
448C2×D4C22×D412yes—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D414yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4114yes—
487C22×C4C2511yescharacteristic
488SD16—88no—
489D8—48no—
490M16—88no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—81no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4#12114yes—
529C4⋊C4AGL(1,ℤ/8)14yes—
530C4⋊C4C22×D412yes—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—42no—
545C22⋊C4—81no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4#12118yes—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—42no—
557C42—41no—
558C42—44no—
559C42—42no—
560C42—82no—
561C42—82no—
562C42—82no—
563C42AGL(1,ℤ/8)14yes—
564C42C22×D412yes—
565C42C2×#4112yes—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23C2×#12114yes—
587C23C23×D411yescharacteristic
588Q8—28no—
589Q8—44no—
590Q8—88no—
591D4—24no—
592D4—28no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—84no—
599D4—84no—
600D4—88no—
601D4—816no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—48no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—88no—
647C2×C4—816no—
648C2×C4—88no—
649C2×C4C2×#12112yes—
650C2×C4C2×#12114yes—
651C2×C4C2×#12112yes—
652C2×C4C2×AGL(1,ℤ/8)14yes—
653C2×C4C2×AGL(1,ℤ/8)12yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4#57812yes—
658C2×C4C23×D411yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4#58512yes—
661C2×C4#58512yes—
662C8—42no—
663C8—44no—
664C8—42no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22C22×#12111yescharacteristic
687C22#348511yescharacteristic
688C22#350011yescharacteristic
689C22C22×AGL(1,ℤ/8)11yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C2×#57811yescharacteristic
692C22#358611yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—24no—
697C4—22no—
698C4—24no—
699C4—42no—
700C4—41no—
701C4—44no—
702C4—42no—
703C4—81no—
704C4—81no—
705C4—82no—
706C4—82no—
707C4—84no—
708C4#303814yes—
709C4#348512yes—
710C4#352112yes—
711C2—44no—
712C2—82no—
713C2—162no—
714C2#3600911yescharacteristic
715C2C2×#303811yescharacteristic
716C2#6043211yescharacteristic
717C2#6171711yescharacteristic
718C2#6173711yescharacteristic
719C2#6253811yescharacteristic
720C2#6259511yescharacteristic
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. C2×#3330C2semidirect
2. #36179C2semidirect
3. #34316C2semidirect
4. #29864C2semidirect
5. #29716C2semidirect
6. #28717C2semidirect
7. #28580C2semidirect
8. #28157C2semidirect
9. #27648C2semidirect
10. #25320C2semidirect
11. #25217C2semidirect
12. #24676C2semidirect
13. #24542C2semidirect
14. #23676C2semidirect
15. #23630C2semidirect
16. #22700C2semidirect
17. #22500C2semidirect
106. #547D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33302
#361792
#343162
#298642
#297162
#287172
#285802
#281572
#276482
#253202
#252172
#246762
#245422
#236762
#236302
#227002
#225002