Finite Group Databasebeta

← #6519833

Subgroups — #6519833

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic, complemented
2C2×#2988C211yesmaximal, characteristic
3#35280C211yesmaximal, characteristic
4#34947C211yesmaximal, characteristic
5#28553C214yesmaximal, complemented
6#25471C212yesmaximal, complemented
7#25386C212yesmaximal, complemented
8#25340C212yesmaximal, complemented
9#25294C212yesmaximal, complemented
10#24837C212yesmaximal, complemented
11#24781C214yesmaximal, complemented
12#24752C212yesmaximal, complemented
13#24702C212yesmaximal, complemented
14#24652C212yesmaximal, complemented
15#21860C211yesmaximal, characteristic, complemented
16#21385C211yesmaximal, characteristic, complemented
17C2×#1947C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3249—28no—
22C2×#477C2212yes—
23C2×#477C2212yes—
24C2×#467C2214yes—
25C2×#467C2214yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28#2988—28no—
29C2×C8×Q8C2212yes—
30C2×C8×D4C2212yes—
31C2×C8×D4C2214yes—
32C2×#431C2212yes—
33C2×#431C2211yescharacteristic
34C22×#107C2212yes—
35C22×C4⋊C8C2212yes—
36C22×C4⋊C8C2211yescharacteristic
37#2857C2212yes—
38#2857C2212yes—
39#2629—22no—
40#2582—21no—
41#2582C2212yes—
42#2568C2211yescharacteristic
43#2567C2211yescharacteristic
44#2566—21no—
45#2565—21no—
46#2564—22no—
47#2563C2211yescharacteristic
48#2542C2211yescharacteristic
49#2538—21no—
50#2538C2212yes—
51#2417—22no—
52#2417C2212yes—
53#2417C2212yes—
54#2417C2214yes—
55#2396—22no—
56#2316—21no—
57#2311—21no—
58C4×C22⋊Q8C2212yes—
59C4×C22⋊Q8C2212yes—
60#2297—21no—
61#2285—21no—
62C2×#384C2211yescharacteristic
63C2×C4×C4⋊C4C2211yescharacteristic
64#2087C2214yes—
65#2076C2214yes—
66#2068C2214yes—
67#2068C2214yes—
68#2056C2214yes—
69#2036C2214yes—
70#2036C2214yes—
71#2001C2214yes—
72#1990C2212yes—
73#1987C2211yescharacteristic
74#1987C2211yescharacteristic
75#1981C2211yescharacteristic
76#1981C2211yescharacteristic
77#1969C2214yes—
78#1956C2212yes—
79#1947C2212yes—
80#1947C2212yes—
81#1947C2212yes—
82#1947C2212yes—
83#1942C2214yes—
84#1933C2214yes—
85#1933C2214yes—
86#1892C2214yes—
87#1892C2214yes—
88#1872C2214yes—
89#1872C2212yes—
90#1872C2212yes—
91#1808C2214yes—
92#1803C2212yes—
93#1803C2212yes—
94C4×#103C2212yes—
95C4×#103C2212yes—
96C2×#340C2212yes—
97C22×#41D411yescharacteristic
98C22×Q16C2314yes—
99C23×C8C2314yes—
100C23×C8C2312yes—
101#533—24no—
102#533—24no—
103#533—24no—
104#533—24no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125—21no—
110C2×#125—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2314yes—
119C4×#41—24no—
120C4×#41—24no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4D412yes—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117—21no—
133C2×#117D411yescharacteristic
134C2×#117D412yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C42—21no—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143#477—28no—
144#477—28no—
145#467—216no—
146#467—48no—
147#462—28no—
148#462—28no—
149C8×Q8—28no—
150C8×D4—216no—
151C8×D4—28no—
152#431—24no—
153#431—28no—
154C2×#107—44no—
155C2×#107C2314yes—
156C2×#107C2314yes—
157C2×#107C2314yes—
158C2×C4⋊C8—28no—
159C2×C4⋊C8D418yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2312yes—
173C2×C22⋊C8C2314yes—
174C2×C22⋊C8C2312yes—
175C2×C4×C8C2314yes—
176C2×C4×C8C2312yes—
177#400—22no—
178#399—22no—
179#398—22no—
180#398—22no—
181#398—24no—
182#398—24no—
183#396—22no—
184#396—22no—
185#396—24no—
186#396—24no—
187#395—22no—
188#393—22no—
189#391—21no—
190#391—21no—
191#391—22no—
192#391—22no—
193#391C2312yes—
194#391C2314yes—
195#391C2312yes—
196#390—21no—
197#390—21no—
198#390—22no—
199#389—22no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389C2312yes—
205#389C2312yes—
206#389C2314yes—
207#387—22no—
208#387—22no—
209#386—22no—
210#386—22no—
211#386—24no—
212#385—22no—
213#385—22no—
214#384—21no—
215#384—21no—
216#384—21no—
217#384—21no—
218#384C2312yes—
219#384C2312yes—
220#382—21no—
221#382—21no—
222#380—22no—
223#380—22no—
224#380—22no—
225#380—22no—
226#380—22no—
227#380—24no—
228#380C2312yes—
229#380C2312yes—
230#380C2314yes—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4C2311yescharacteristic
236C4×C4⋊C4C2311yescharacteristic
237C4×C4⋊C4C2311yescharacteristic
238C4×C4⋊C4C2311yescharacteristic
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241C2×#95—21no—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41Q1618yescomplemented
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8C2×D411yescharacteristic
256C22×Q8C2412yes—
257C22×Q8C2412yes—
258C22×D4C2×D412yes—
259C22×D4C2×D411yescharacteristic
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C2×Q16—28no—
270C2×Q16—44no—
271C2×Q16—44no—
272C22×C8—28no—
273C22×C8—42no—
274C22×C8—42no—
275C22×C8—44no—
276C22×C8—44no—
277C22×C8C2×D418yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289#125—44no—
290#125—44no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—42no—
298C22⋊Q8—42no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—84no—
304C4×Q8—24no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×D4—28no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—22no—
318C4×D4—22no—
319#117—24no—
320#117—28no—
321#117—44no—
322#117—44no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—82no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2411yescharacteristic
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—44no—
385C2×C42C2×D412yes—
386C2×C42C2×D412yes—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D414yes—
390C2×C42#4114yes—
391C2×C42C2412yes—
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#107—28no—
395#107—28no—
396#107—44no—
397C4⋊C8—28no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8C2×D418yes—
401C4⋊C8C2×D418yes—
402#103—28no—
403#103—28no—
404#103—24no—
405#103—24no—
406#103—42no—
407#103—42no—
408#103—44no—
409#103—44no—
410C22⋊C8—24no—
411C22⋊C8—28no—
412C4×C8—28no—
413C4×C8—24no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430C24C22×D412yes—
431C24C22×D411yescharacteristic
432#41—232no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—81no—
452C2×Q8—81no—
453C2×Q8C2×Q1614yes—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4C2×Q1618yes—
459C2×D4C2×Q1614yes—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4#12114yes—
496C22×C4#12114yes—
497C22×C4C2×Q1618yes—
498C22×C4C2×Q1614yes—
499C22×C4#13514yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C2×#4112yes—
510C22×C4C2511yescharacteristic
511Q16—88no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—48no—
521C2×C8—88no—
522C2×C8C22×D418yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—24no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—84no—
561C4⋊C4—84no—
562C4⋊C4H(5,𝔽2)12yes—
563C4⋊C42−1+412yes—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—84no—
574C22⋊C4—84no—
575C42—24no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—44no—
580C42—44no—
581C42—42no—
582C42—42no—
583C42—42no—
584C42—42no—
585C42#12114yes—
586C42#12114yes—
587C42C22×D412yes—
588C42C22×D412yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23C2×#12112yes—
602C23C22×Q1614yes—
603C23C22×Q1612yes—
604C23C2×#13512yes—
605C23C23×D411yescharacteristic
606Q8—28no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—42no—
613Q8—42no—
614Q8—42no—
615Q8—84no—
616Q8—84no—
617D4—216no—
618D4—28no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—216no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—44no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—164no—
674C2×C4#467116yescomplemented
675C2×C4C2×#12112yes—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#12112yes—
679C2×C4C2×#12112yes—
680C2×C4C22×Q1614yes—
681C2×C4C22×Q1612yes—
682C2×C4C22×Q1612yes—
683C2×C4C2×#13512yes—
684C2×C4C2×#13512yes—
685C2×C4C23×D411yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)11yescharacteristic
688C2×C4C2×2−1+411yescharacteristic
689C2×C4#58511yescharacteristic
690C2×C4#58511yescharacteristic
691C8—28no—
692C8—42no—
693C8—42no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22C2×#46714yes—
708C22C22×#12111yescharacteristic
709C22#348811yescharacteristic
710C22#348911yescharacteristic
711C22C23×Q1611yescharacteristic
712C22C22×#13511yescharacteristic
713C22#358511yescharacteristic
714C22#358611yescharacteristic
715C4—24no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—82no—
724C4—82no—
725C4—81no—
726C4—82no—
727C4—82no—
728C4—81no—
729C4—84no—
730C4C2×#46714yes—
731C4C2×#46714yes—
732C4#349611yescharacteristic
733C4#349711yescharacteristic
734C4#351111yescharacteristic
735C4#351211yescharacteristic
736C2—24no—
737C2—44no—
738C2#3556411yescharacteristic
739C2C22×#46711yescharacteristic
740C2#6045611yescharacteristic
741C2#6194911yescharacteristic
742C2#6195211yescharacteristic
743C2#6209711yescharacteristic
744C2#6210011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3249C2semidirect
5. #28553C2semidirect
6. #25471C2semidirect
7. #25386C2semidirect
8. #25340C2semidirect
9. #25294C2semidirect
10. #24837C2semidirect
11. #24781C2semidirect
12. #24752C2semidirect
13. #24702C2semidirect
14. #24652C2semidirect
15. #21860C2semidirect
16. #21385C2semidirect
17. C2×#1947C2semidirect
252. C2×#41Q16semidirect
674. C2×C4#467semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
C2×#29882
#352802
#349472
#285532
#254712
#253862
#253402
#252942
#248372
#247812
#247522
#247022
#246522
#218602
#213852
C2×#19472