Finite Group Databasebeta

← #6467729

Subgroups — #6467729

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3172C212yesmaximal, complemented
2#36631C211yesmaximal, characteristic
3C4×#533C211yesmaximal, characteristic, complemented
4#28753C211yesmaximal, characteristic, complemented
5#28723C211yesmaximal, characteristic, complemented
6#28692C211yesmaximal, characteristic, complemented
7#28659C212yesmaximal, complemented
8#28619C212yesmaximal, complemented
9#28581C212yesmaximal, complemented
10#28580C212yesmaximal
11#28447C211yesmaximal, characteristic, complemented
12#24757C214yesmaximal, complemented
13#23679C212yesmaximal, complemented
14#23678C212yesmaximal, complemented
15C4×#464C214yesmaximal, complemented
16#21403C212yesmaximal, complemented
17C4×#423C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2212yes—
20#3172—216no—
21C2×#501C2212yes—
22C2×#497C2212yes—
23C2×#491C2212yes—
24C2×#489C2212yes—
25C2×#470C2214yes—
26C2×#468C2214yes—
27C2×#464C2214yes—
28C2×C4×Q16C2214yes—
29C2×C4×SD16C2214yes—
30C2×#423C2212yes—
31C2×C4×M16C2212yes—
32#2861C2212yes—
33#2860—21no—
34#2860—21no—
35#2859C2211yescharacteristic
36#2858C2211yescharacteristic
37#2857C2211yescharacteristic
38#2856C2211yescharacteristic
39#2854—21no—
40#2854—21no—
41#2850C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#125—21no—
49C4×#125—21no—
50C4×#124C2212yes—
51C4×C22⋊Q8—21no—
52C4×C22⋊Q8—21no—
53C4×C22⋊Q8C2212yes—
54C4×C22⋊Q8C2212yes—
55C4×#121—21no—
56C4×#121—21no—
57#2296—21no—
58#2295—21no—
59#2295C2212yes—
60#2293—21no—
61#2293—21no—
62#2285C2211yescharacteristic
63C42×Q8—21no—
64C42×Q8C2212yes—
65C4×#117—21no—
66C2×C43C2211yescharacteristic
67#2084C2212yes—
68#2083C2212yes—
69#2072C2212yes—
70#2072C2212yes—
71#2068C2214yes—
72#2065C2214yes—
73#2056C2214yes—
74#2055C2214yes—
75#2043C2212yes—
76#2041C2212yes—
77#2016C2212yes—
78#2014C2212yes—
79#1992C2212yes—
80#1992C2212yes—
81#1983C2214yes—
82#1959C2212yes—
83#1959C2212yes—
84#1949C2214yes—
85#1927C2214yes—
86#1925C2214yes—
87#1895C2214yes—
88#1894C2214yes—
89#1880C2214yes—
90#1878C2214yes—
91#1805C2214yes—
92#1767C2211yescharacteristic
93#1767C2211yescharacteristic
94#1767C2212yes—
95C4×C4⋊C8C2212yes—
96C4×C4⋊C8C2212yes—
97C4×#103C2214yes—
98C4×#102C2214yes—
99#1728C2212yes—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102C22×M16C2314yes—
103#533—24no—
104#533D418yescomplemented
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109#529—28no—
110#529—28no—
111C2×#126—22no—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#124C2312yes—
115C2×#124C2312yes—
116C2×#123—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×D4—22no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—22no—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#501—28no—
145#497—28no—
146#491—28no—
147#489—28no—
148#470—216no—
149#468—216no—
150#464—216no—
151C4×Q16—216no—
152C4×SD16—216no—
153C2×#107C2314yes—
154C2×#106C2314yes—
155#423—28no—
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C4×M16—28no—
168C2×#97C2312yes—
169C2×#97C2312yes—
170C2×C4×C8C2314yes—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389C2312yes—
202#389C2312yes—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#386—22no—
216#386—22no—
217#385—22no—
218#385—22no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#382—21no—
224#382C2312yes—
225#380—21no—
226#380—21no—
227#380—21no—
228#380C2312yes—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—22no—
251#376—22no—
252C43—22no—
253C43D411yescharacteristic
254C43D411yescharacteristic
255C43D411yescharacteristic
256C43D411yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2314yes—
268C22×Q8—21no—
269C22×Q8C2412yes—
270C22×Q8C2412yes—
271C22×D4—22no—
272C22×D4—21no—
273C22×D4—21no—
274C22×D4C2412yes—
275C23×C4—22no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×Q16—28no—
284C2×Q16—28no—
285C2×Q16—44no—
286C2×SD16—28no—
287C2×SD16—28no—
288C2×SD16—44no—
289C2×M16—28no—
290C2×M16—28no—
291C2×M16—44no—
292C22×C8C2414yes—
293C22×C8C2414yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—42no—
297C4⋊Q8C2×D414yes—
298C4⋊Q8C2×D414yes—
299#127—42no—
300#126—48no—
301#125—28no—
302#125—24no—
303#125—24no—
304#124—24no—
305#124—24no—
306#124—28no—
307#123—48no—
308C22⋊Q8—28no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—24no—
312C22⋊Q8C2×D418yes—
313#121—24no—
314#121—24no—
315#121—44no—
316#121—44no—
317C4×Q8—28no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—28no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8C2×D418yes—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—44no—
329C4×D4—48no—
330C4×D4—44no—
331#117—28no—
332#117—44no—
333#117—48no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C42—22no—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—42no—
385C2×C42—42no—
386C2×C42—42no—
387C2×C42—42no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D414yes—
410C2×C42C2×D411yescharacteristic
411C2×C42#4114yes—
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416C2×C42C2411yescharacteristic
417C2×C42C2411yescharacteristic
418#107—28no—
419#106—28no—
420C4⋊C8—28no—
421C4⋊C8—24no—
422C4⋊C8#4118yes—
423#103—28no—
424#103—28no—
425#103—28no—
426#103—28no—
427#103—44no—
428#102—28no—
429#102—28no—
430#102—44no—
431#97—24no—
432#97—24no—
433C4×C8—28no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450C24—22no—
451C24C22×D411yescharacteristic
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—22no—
458C2×Q8—24no—
459C2×Q8—24no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8C22×D414yes—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—42no—
475C2×D4—42no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4#12114yes—
505C22×C4#12114yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C2×#4112yes—
521C22×C4C2511yescharacteristic
522Q16—416no—
523SD16—416no—
524M16—416no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8C2×#4118yes—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4C22×D414yes—
562C22⋊C4—28no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C42—28no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—41no—
591C42—41no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—44no—
599C42—44no—
600C42—44no—
601C42—44no—
602C42—44no—
603C42—44no—
604C42#12114yes—
605C42#12114yes—
606C42#13714yes—
607C42C22×D412yes—
608C42C22×D412yes—
609C42C2×#4112yes—
610C42C2×#4112yes—
611C42C2×#4114yes—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23C2×#12112yes—
626C23C23×D411yescharacteristic
627Q8—28no—
628Q8—48no—
629Q8—44no—
630Q8—44no—
631Q8—44no—
632D4—44no—
633D4—44no—
634D4—84no—
635D4—84no—
636D4—88no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—28no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—88no—
693C2×C4C2×#12112yes—
694C2×C4C2×#12112yes—
695C2×C4C2×#12112yes—
696C2×C4C2×#12112yes—
697C2×C4C2×#12112yes—
698C2×C4#53212yes—
699C2×C4#53212yes—
700C2×C4C2×#13712yes—
701C2×C4C2×#13712yes—
702C2×C4C2×#13712yes—
703C2×C4#57512yes—
704C2×C4#57512yes—
705C2×C4#57512yes—
706C2×C4C23×D411yescharacteristic
707C2×C4C22×#4112yes—
708C2×C4C22×#4111yescharacteristic
709C2×C4#58512yes—
710C8—28no—
711C8—44no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22C22×#12111yescharacteristic
727C22C2×#53211yescharacteristic
728C22#344911yescharacteristic
729C22C22×#13711yescharacteristic
730C22C2×#57511yescharacteristic
731C22C2×#57511yescharacteristic
732C22C2×#57511yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—22no—
740C4—22no—
741C4—22no—
742C4—22no—
743C4—41no—
744C4—41no—
745C4—44no—
746C4—44no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#303714yes—
753C4D4×#4112yes—
754C4#348012yes—
755C2—42no—
756C2—162no—
757C2#3493711yescharacteristic
758C2C2×#303711yescharacteristic
759C2#6041511yescharacteristic
760C2#6067611yescharacteristic
761C2#6070111yescharacteristic
762C2#6095011yescharacteristic
763C2#6095411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3172C2semidirect
3. C4×#533C2semidirect
4. #28753C2semidirect
5. #28723C2semidirect
6. #28692C2semidirect
7. #28659C2semidirect
8. #28619C2semidirect
9. #28581C2semidirect
11. #28447C2semidirect
12. #24757C2semidirect
13. #23679C2semidirect
14. #23678C2semidirect
15. C4×#464C2semidirect
16. #21403C2semidirect
17. C4×#423C2semidirect
104. #533D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31722
#366312
C4×#5332
#287532
#287232
#286922
#286592
#286192
#285812
#285802
#284472
#247572
#236792
#236782
C4×#4642
#214032
C4×#4232