Finite Group Databasebeta

← #6566098

Subgroups — #6566098

Order 512.

2057 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3354C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic, complemented
3#34308C211yesmaximal, characteristic, complemented
4#29868C211yesmaximal, characteristic, complemented
5#29755C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28578C212yesmaximal, complemented
8#28201C211yesmaximal, characteristic, complemented
9#27650C212yesmaximal, complemented
10#25452C212yesmaximal, complemented
11#25228C214yesmaximal, complemented
12#24812C212yesmaximal, complemented
13#24589C214yesmaximal, complemented
14#23676C211yesmaximal, characteristic, complemented
15#23657C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22503C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×#547C2211yescharacteristic
20C2×#520—21no—
21#3354—216no—
22C2×#498C2212yes—
23C2×#481C2214yes—
24C2×#482C2214yes—
25C2×#476C2212yes—
26C2×#474C2212yes—
27C2×#469C2214yes—
28C2×C4×D8C2212yes—
29C2×#436C2212yes—
30C2×#428C2214yes—
31#2868C2211yescharacteristic
32#2856C2212yes—
33#2563C2212yes—
34#2537—21no—
35#2537—21no—
36#2500C2214yes—
37#2491—22no—
38#2491—22no—
39#2365C2211yescharacteristic
40#2361C2212yes—
41#2360—21no—
42#2359—22no—
43#2354—21no—
44#2354—21no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47#2344C2212yes—
48#2339—21no—
49#2323—21no—
50#2322—22no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2211yescharacteristic
53C4×#121C2212yes—
54#2293C2211yescharacteristic
55C2×#389C2212yes—
56#2285C2212yes—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2212yes—
63#2084C2212yes—
64#2083C2212yes—
65#2083C2211yescharacteristic
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2212yes—
69#2067C2212yes—
70#2065C2214yes—
71#2065C2214yes—
72#2054C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2011C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1949C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1926C2214yes—
82#1925C2214yes—
83#1895C2214yes—
84#1879C2214yes—
85#1874C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1804C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×#41—21no—
97C22×D8C2312yes—
98C22×M16C2314yes—
99#555—28no—
100#555—28no—
101#547—24no—
102#547D418yescomplemented
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#124—24no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×#121C2312yes—
116#520—48no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C2×#117—22no—
132C2×#117—21no—
133C2×#117C2314yes—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C42C2312yes—
139#498—28no—
140#482—216no—
141#481—216no—
142#476—28no—
143#474—28no—
144#469—216no—
145C4×D8—28no—
146#436—28no—
147#428—216no—
148C2×#107C2314yes—
149C2×#107C2312yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8C2312yes—
165#394—24no—
166#394—24no—
167#393—24no—
168#391—21no—
169#391—21no—
170#390—21no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390C2312yes—
175#390C2314yes—
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2311yescharacteristic
186#389C2311yescharacteristic
187#388—21no—
188#388—22no—
189#388—21no—
190#388—22no—
191#388—24no—
192#388—24no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#386—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2312yes—
207#382C2314yes—
208#381—22no—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C2×#95—22no—
230C43C2311yescharacteristic
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2312yes—
241#336C2311yescharacteristic
242#336C2314yes—
243C2×#41—28no—
244C2×#41—44no—
245C2×#41—44no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×D4—22no—
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2412yes—
254C22×D4C2411yescharacteristic
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—42no—
266C2×M16—28no—
267C2×M16—28no—
268C2×M16—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—22no—
274C4⋊Q8—24no—
275C4⋊Q8—82no—
276C4⋊Q8—82no—
277C4⋊Q8C2×D414yes—
278#127—42no—
279#124—416no—
280#123—216no—
281C22⋊Q8—28no—
282#121—24no—
283#121—28no—
284#121—28no—
285#121—48no—
286#121—84no—
287#121—84no—
288#121C2×D418yes—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—48no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4C2×D414yes—
303C4×D4C2×D414yes—
304C4×D4#4114yes—
305#117—28no—
306#117—28no—
307#117—48no—
308#117—48no—
309#117—44no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D418yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—44no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—24no—
376#107—28no—
377#106—28no—
378#106—28no—
379#106—28no—
380C4⋊C8—24no—
381C4⋊C8—24no—
382C4⋊C8—44no—
383#102—24no—
384#102—28no—
385#102—24no—
386#102—24no—
387#102—28no—
388#102—42no—
389#102—44no—
390C22⋊C8—28no—
391C4×C8—24no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—21no—
407C24—22no—
408C24C22×D412yes—
409#41—432no—
410C2×Q8—24no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4#12114yes—
440C2×D4C22×D412yes—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D414yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4114yes—
482C22×C4C2511yescharacteristic
483D8—48no—
484M16—416no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C82−1+414yes—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4#12114yes—
523C4⋊C4C22×D412yes—
524C4⋊C4C22×D414yes—
525C22⋊C4—24no—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—81no—
539C22⋊C4—82no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4#12118yes—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—22no—
548C42—42no—
549C42—42no—
550C42—41no—
551C42—44no—
552C42—48no—
553C42—44no—
554C42—42no—
555C42AGL(1,ℤ/8)14yes—
556C42C22×D412yes—
557C42C2×#4112yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23C2×#12114yes—
579C23C23×D411yescharacteristic
580Q8—48no—
581Q8—164no—
582D4—24no—
583D4—28no—
584D4—416no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—84no—
591D4—84no—
592D4—168no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—416no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—88no—
640C2×C4—164no—
641C2×C4—164no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4#57512yes—
649C2×C4#57712yes—
650C2×C4C23×D411yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×2−1+412yes—
653C2×C4#58512yes—
654C8—24no—
655C8—42no—
656C8—44no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22C22×#12111yescharacteristic
679C22#348511yescharacteristic
680C22#349911yescharacteristic
681C22C22×AGL(1,ℤ/8)11yescharacteristic
682C22C2×#57511yescharacteristic
683C22C2×#57711yescharacteristic
684C22#358511yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—24no—
690C4—42no—
691C4—42no—
692C4—41no—
693C4—42no—
694C4—44no—
695C4—81no—
696C4—84no—
697C4—81no—
698C4—82no—
699C4—82no—
700C4#303614yes—
701C4#348912yes—
702C4#351512yes—
703C2—44no—
704C2—82no—
705C2—162no—
706C2#3602011yescharacteristic
707C2C2×#303611yescharacteristic
708C2#6043111yescharacteristic
709C2#6172011yescharacteristic
710C2#6173111yescharacteristic
711C2#6247711yescharacteristic
712C2#6251211yescharacteristic
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×#3354C2semidirect
2. #36187C2semidirect
3. #34308C2semidirect
4. #29868C2semidirect
5. #29755C2semidirect
6. #28687C2semidirect
7. #28578C2semidirect
8. #28201C2semidirect
9. #27650C2semidirect
10. #25452C2semidirect
11. #25228C2semidirect
12. #24812C2semidirect
13. #24589C2semidirect
14. #23676C2semidirect
15. #23657C2semidirect
16. #22700C2semidirect
17. #22503C2semidirect
102. #547D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33542
#361872
#343082
#298682
#297552
#286872
#285782
#282012
#276502
#254522
#252282
#248122
#245892
#236762
#236572
#227002
#225032