Finite Group Databasebeta

← #6514127

Subgroups — #6514127

Order 512.

1902 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic
3#34953C211yesmaximal, characteristic
4#34214C211yesmaximal, characteristic
5#28284C212yesmaximal, complemented
6#28255C212yesmaximal, complemented
7#28206C212yesmaximal, complemented
8#28161C212yesmaximal, complemented
9#25206C214yesmaximal
10#25082C214yesmaximal
11#24817C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#24682C212yesmaximal, complemented
14#24632C212yesmaximal, complemented
15#21972C211yesmaximal, characteristic
16#21860C211yesmaximal, characteristic
17C2×#1803C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3249—28no—
22#3104—28no—
23C2×#484C2212yes—
24C2×#477C2211yescharacteristic
25C2×#477C2212yes—
26C2×#477C2212yes—
27C2×#477C2211yescharacteristic
28C2×#467C2214yes—
29C2×#462C2211yescharacteristic
30C2×#462C2212yes—
31C2×#462C2212yes—
32C2×#462C2211yescharacteristic
33C2×#462C2214yes—
34C2×#451C2212yes—
35C2×C4×Q16C2212yes—
36C2×C4×Q16C2212yes—
37C2×#431C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C22×#103C2212yes—
40#2584—21no—
41#2584—21no—
42#2583—22no—
43#2567C2211yescharacteristic
44#2567C2211yescharacteristic
45#2546C2211yescharacteristic
46#2546C2211yescharacteristic
47#2502C2212yes—
48#2434C2212yes—
49#2399—22no—
50#2360—22no—
51#2360C2214yes—
52#2355—21no—
53#2355C2212yes—
54Q8×C4⋊C4—21no—
55Q8×C4⋊C4C2212yes—
56#2330—22no—
57#2317—21no—
58C4×C4⋊Q8C2212yes—
59C4×C4⋊Q8C2212yes—
60C4×C4⋊Q8C2212yes—
61C4×C4⋊Q8C2212yes—
62#2296—21no—
63#2296—21no—
64C2×#384C2211yescharacteristic
65#2274—21no—
66C4×#117C2211yescharacteristic
67#2083C2214yes—
68#2072C2214yes—
69#2068C2214yes—
70#2068C2214yes—
71#2056C2214yes—
72#2056C2214yes—
73#2043C2214yes—
74#2016C2214yes—
75#1990C2212yes—
76#1981C2214yes—
77#1956C2212yes—
78#1947C2214yes—
79#1939C2212yes—
80#1939C2212yes—
81#1928C2214yes—
82#1928C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1872C2214yes—
86#1850C2214yes—
87#1842C2212yes—
88#1831C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1803C2212yes—
93#1803C2212yes—
94#1803C2214yes—
95#1803C2212yes—
96C2×#340C2212yes—
97C23×Q8C2311yescharacteristic
98C22×Q16C2312yes—
99C22×Q16C2312yes—
100C22×Q16C2314yes—
101C23×C8C2312yes—
102#533—24no—
103#533—24no—
104#533—24no—
105#533D418yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2314yes—
132C2×#117—21no—
133C2×#117—21no—
134C2×#117D411yescharacteristic
135C2×#117D411yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#484—28no—
144#477—24no—
145#477—24no—
146#477—28no—
147#477—28no—
148#467—216no—
149#462—24no—
150#462—24no—
151#462—28no—
152#462—28no—
153#462—48no—
154#451—28no—
155C4×Q16—28no—
156C4×Q16—28no—
157#431—24no—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×C4⋊C8—24no—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2314yes—
167C2×C4⋊C8C2314yes—
168C2×#103—44no—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2314yes—
175C2×#103C2312yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×C22⋊C8C2312yes—
180C2×C4×C8C2312yes—
181#398—22no—
182#398—22no—
183#398—22no—
184#398—22no—
185#395—22no—
186#395—22no—
187#395—22no—
188#395—22no—
189#391—22no—
190#391C2312yes—
191#391C2312yes—
192#391C2314yes—
193#391C2312yes—
194#391C2312yes—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—22no—
204#389—22no—
205#389—22no—
206#389—22no—
207#389—22no—
208#389—22no—
209#389—22no—
210#389—24no—
211#389—24no—
212#389C2314yes—
213#387—24no—
214#387—24no—
215#385—24no—
216#385—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#384—22no—
221#384—21no—
222#384—21no—
223#384C2312yes—
224#384C2312yes—
225#384C2312yes—
226#384C2312yes—
227#382—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4C2314yes—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2311yescharacteristic
242C4×C22⋊C4—24no—
243C4×C22⋊C4D412yes—
244C4×C22⋊C4C2312yes—
245#376D412yes—
246C43C2312yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—41no—
261C22×Q8—41no—
262C22×Q8C2412yes—
263C22×Q8C2412yes—
264C22×Q8C2411yescharacteristic
265C22×Q8C2411yescharacteristic
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—28no—
277C2×Q16—42no—
278C2×Q16—42no—
279C2×Q16—44no—
280C2×Q16—44no—
281C22×C8—24no—
282C22×C8—42no—
283C22×C8—42no—
284C22×C8C2414yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297C4⋊Q8C2×D414yes—
298C4⋊Q8C2×D414yes—
299#125—28no—
300#125—44no—
301C22⋊Q8—28no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—44no—
306C22⋊Q8C2×D418yes—
307C4×Q8—28no—
308C4×Q8—24no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—24no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—84no—
325C4×Q8C2×D418yes—
326#117—28no—
327#117—24no—
328#117—44no—
329#117Q1618yescomplemented
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
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—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4—82no—
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2414yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—24no—
387C2×C42—24no—
388C2×C42—44no—
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D412yes—
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D414yes—
395C2×C42C2×D412yes—
396C2×C42#4114yes—
397C2×C42C2412yes—
398C2×C42C2412yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—24no—
402#107—24no—
403#107—28no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—22no—
407C4⋊C8—22no—
408C4⋊C8—44no—
409C4⋊C8—44no—
410#103—24no—
411#103—24no—
412#103—28no—
413#103—28no—
414#103—24no—
415#103—24no—
416#103—28no—
417#103—44no—
418#103—44no—
419#103—44no—
420C22⋊C8—24no—
421C4×C8—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—24no—
434#95C2×D412yes—
435#95C2×D412yes—
436#95C2×D412yes—
437#95C2×D412yes—
438C24C22×D411yescharacteristic
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—28no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—81no—
472C2×Q8—81no—
473C2×Q8C22×D414yes—
474C2×Q8H(5,𝔽2)12yes—
475C2×Q82−1+412yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4#12114yes—
501C22×C4#12114yes—
502C22×C4C2×Q1614yes—
503C22×C4#13714yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2×#4112yes—
515C22×C4C2511yescharacteristic
516Q16—48no—
517Q16—48no—
518Q16—88no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—24no—
528C2×C8—84no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—28no—
533C4⋊C4—28no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—42no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—84no—
569C4⋊C4—84no—
570C4⋊C4C2×Q1614yes—
571C4⋊C4C2×Q1614yes—
572C4⋊C4C22×D414yes—
573C22⋊C4—28no—
574C22⋊C4—24no—
575C22⋊C4—44no—
576C22⋊C4C2×Q1618yes—
577C42—28no—
578C42—24no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—42no—
584C42—42no—
585C42—42no—
586C42—42no—
587C42—42no—
588C42—42no—
589C42—44no—
590C42—84no—
591C42—84no—
592C42#12114yes—
593C42#12114yes—
594C42C2×Q1618yes—
595C42C22×D412yes—
596C42C22×D412yes—
597C42C2×#4112yes—
598C42C2×#4112yes—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23C2×#12112yes—
604C23C22×Q1612yes—
605C23C2×#13712yes—
606C23C23×D411yescharacteristic
607Q8—22no—
608Q8—24no—
609Q8—22no—
610Q8—24no—
611Q8—22no—
612Q8—22no—
613Q8—48no—
614Q8—48no—
615Q8—41no—
616Q8—41no—
617Q8—42no—
618Q8—42no—
619Q8—41no—
620Q8—41no—
621Q8—42no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—84no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—44no—
651C2×C4—48no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—84no—
676C2×C4—164no—
677C2×C4#462116yescomplemented
678C2×C4C2×#12112yes—
679C2×C4C2×#12112yes—
680C2×C4C2×#12112yes—
681C2×C4C2×#12112yes—
682C2×C4C2×#12112yes—
683C2×C4C22×Q1614yes—
684C2×C4C22×Q1614yes—
685C2×C4C22×Q1612yes—
686C2×C4C22×Q1612yes—
687C2×C4C2×#13712yes—
688C2×C4C2×#13712yes—
689C2×C4C23×D411yescharacteristic
690C2×C4C22×#4111yescharacteristic
691C2×C4C2×H(5,𝔽2)11yescharacteristic
692C2×C4C2×2−1+411yescharacteristic
693C2×C4#58511yescharacteristic
694C2×C4#58511yescharacteristic
695C8—44no—
696C8—42no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22C2×#46214yes—
705C22C22×#12111yescharacteristic
706C22#348811yescharacteristic
707C22#348911yescharacteristic
708C22C23×Q1611yescharacteristic
709C22C22×#13711yescharacteristic
710C22#358511yescharacteristic
711C22#358611yescharacteristic
712C4—24no—
713C4—22no—
714C4—22no—
715C4—24no—
716C4—22no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—44no—
723C4—81no—
724C4—81no—
725C4—82no—
726C4—84no—
727C4—82no—
728C4C2×#46214yes—
729C4C2×#46214yes—
730C4#348811yescharacteristic
731C4#348911yescharacteristic
732C4#349211yescharacteristic
733C4#349311yescharacteristic
734C2—24no—
735C2#3551811yescharacteristic
736C2C22×#46211yescharacteristic
737C2#6042511yescharacteristic
738C2#6195211yescharacteristic
739C2#6196511yescharacteristic
740C2#6210011yescharacteristic
741C2#6211311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28284C2semidirect
6. #28255C2semidirect
7. #28206C2semidirect
8. #28161C2semidirect
11. #24817C2semidirect
12. #24732C2semidirect
13. #24682C2semidirect
14. #24632C2semidirect
329. #117Q16semidirect
677. C2×C4#462semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
C2×#31042
#349532
#342142
#282842
#282552
#282062
#281612
#252062
#250822
#248172
#247322
#246822
#246322
#219722
#218602
C2×#18032