Finite Group Databasebeta

← #6578222

Subgroups — #6578222

Order 512.

2123 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×#3104C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3C4×D4×Q8C211yesmaximal, characteristic
4#33163C211yesmaximal, characteristic, complemented
5#32906C212yesmaximal, complemented
6#28440C211yesmaximal, characteristic, complemented
7#28284C211yesmaximal, characteristic, complemented
8#26621C214yesmaximal, complemented
9#26236C212yesmaximal
10#25760C212yesmaximal
11#25161C214yesmaximal, complemented
12#24652C212yesmaximal, complemented
13#24155C212yesmaximal, complemented
14#23184C212yesmaximal, complemented
15#22998C212yesmaximal, complemented
16#22795C211yesmaximal, characteristic
17C4×#477C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#533C2212yes—
20C22×C4×Q8C2212yes—
21C2×#494C2212yes—
22#3104—216no—
23C2×#484C2214yes—
24C2×#477C2212yes—
25C2×#477C2211yescharacteristic
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#462C2212yes—
29C2×#462C2212yes—
30C2×#462C2212yes—
31C2×#451C2214yes—
32C2×#451C2214yes—
33C2×C4×Q16C2214yes—
34C2×C4×Q16C2214yes—
35C2×#431C2212yes—
36#2868C2211yescharacteristic
37#2857C2212yes—
38#2552—21no—
39#2542C2212yes—
40#2538—21no—
41#2441—22no—
42#2429C2214yes—
43#2423—22no—
44#2365—21no—
45#2365C2212yes—
46#2365C2211yescharacteristic
47#2360—22no—
48#2360C2212yes—
49#2354—21no—
50Q8×C4⋊C4—21no—
51Q8×C4⋊C4C2211yescharacteristic
52D4×C4⋊C4—21no—
53D4×C4⋊C4C2211yescharacteristic
54Q8×C22⋊C4C2212yes—
55C4×C4⋊Q8—21no—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C22⋊Q8—22no—
59C4×C22⋊Q8C2212yes—
60C42×Q8C2211yescharacteristic
61C42×D4—21no—
62C42×D4C2211yescharacteristic
63#2068C2214yes—
64#2036C2214yes—
65#2036C2214yes—
66#2016C2214yes—
67#2001C2214yes—
68#2001C2214yes—
69#1987C2211yescharacteristic
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1969C2214yes—
75#1956C2212yes—
76#1954C2212yes—
77#1947C2211yescharacteristic
78#1947C2212yes—
79#1933C2214yes—
80#1923C2214yes—
81#1921C2212yes—
82#1878C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1872C2214yes—
86#1863C2212yes—
87#1863C2214yes—
88#1842C2211yescharacteristic
89#1803C2212yes—
90#1803C2211yescharacteristic
91#1797C2212yes—
92C4×#107C2212yes—
93C8×C4⋊C4C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100D4×Q8—24no—
101D4×Q8—24no—
102#533—28no—
103#533—44no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2314yes—
115C2×C22⋊Q8C2314yes—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8—28no—
121C2×C4×Q8—44no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4D411yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×#117—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2312yes—
137C22×C42—22no—
138C22×C42C2312yes—
139#494—28no—
140#484—216no—
141#477—28no—
142#477D418yescomplemented
143#462—28no—
144#462—28no—
145#462—28no—
146#462—44no—
147#462—44no—
148#451—216no—
149#451—48no—
150C4×Q16—216no—
151C4×Q16—216no—
152#431—28no—
153C2×#107C2314yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2314yes—
172C2×C4×C8C2312yes—
173C2×C4×C8C2314yes—
174#399—24no—
175#398—24no—
176#396—24no—
177#391—21no—
178#391—22no—
179#391—22no—
180#391C2312yes—
181#391C2312yes—
182#391C2314yes—
183#390—22no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—24no—
194#389—22no—
195#389C2312yes—
196#389C2312yes—
197#388—24no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—24no—
202#385—22no—
203#385—22no—
204#385—22no—
205#384—21no—
206#384—22no—
207#384—21no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#380—22no—
214#380—22no—
215#380—22no—
216#380C2312yes—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4D412yes—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4C2312yes—
240C43—21no—
241C43C2311yescharacteristic
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2312yes—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8—42no—
256C22×Q8—42no—
257C22×Q8C2414yes—
258C22×Q8C2412yes—
259C22×Q8C2411yescharacteristic
260C22×D4C2×D411yescharacteristic
261C23×C4—22no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—28no—
269C2×Q16—28no—
270C2×Q16—28no—
271C2×Q16—28no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×Q16—44no—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C22×C8C2412yes—
278C4⋊Q8—28no—
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—24no—
285C4⋊Q8—42no—
286C4⋊Q8—42no—
287C4⋊Q8C2×D414yes—
288C4⋊Q8#4114yes—
289#125—48no—
290C22⋊Q8—216no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—28no—
294C22⋊Q8—44no—
295C22⋊Q8—48no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—22no—
302C4×Q8—24no—
303C4×Q8—28no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—48no—
307C4×Q8—42no—
308C4×Q8—42no—
309C4×Q8—41no—
310C4×Q8—44no—
311C4×Q8—48no—
312C4×Q8—48no—
313C4×Q8—44no—
314C4×Q8C2×D414yes—
315C4×Q8#4114yes—
316C4×D4—28no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—42no—
320C4×D4—44no—
321C4×D4Q1618yescomplemented
322#117—48no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—48no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—24no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—48no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#107—42no—
394#107C2×D418yes—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8—42no—
399C4⋊C8C2×D414yes—
400#103—24no—
401#103—28no—
402#103—28no—
403#103—28no—
404#103—28no—
405#103—44no—
406#103—44no—
407#103—44no—
408#103C2×D418yes—
409C22⋊C8—28no—
410C22⋊C8—44no—
411C4×C8—28no—
412C4×C8—24no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—24no—
426#95C2×D412yes—
427#95C2×D412yes—
428#95C2×D412yes—
429C24C22×D412yes—
430C2×Q8—28no—
431C2×Q8—24no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—48no—
451C2×Q8—48no—
452C2×Q8—48no—
453C2×Q8—48no—
454C2×Q8C22×D412yes—
455C2×Q8C2×#4114yes—
456C2×Q8C2×#4112yes—
457C2×Q8H(5,𝔽2)12yes—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4C2×Q1614yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—82no—
487C22×C4C2×Q1618yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498Q16—416no—
499Q16—416no—
500Q16—88no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—22no—
509C2×C8C22×D414yes—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—81no—
543C4⋊C4—82no—
544C4⋊C4—84no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4C2×Q1614yes—
548C4⋊C4C22×D412yes—
549C4⋊C4C22×D414yes—
550C4⋊C4C2×#4112yes—
551C22⋊C4—28no—
552C22⋊C4—24no—
553C22⋊C4—42no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—84no—
557C22⋊C4C2×Q1618yes—
558C42—24no—
559C42—24no—
560C42—24no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—22no—
567C42—21no—
568C42—41no—
569C42—44no—
570C42—44no—
571C42—48no—
572C42—42no—
573C42—44no—
574C42—42no—
575C42—44no—
576C42—42no—
577C42—81no—
578C42C2×Q1614yes—
579C42C22×D412yes—
580C42C2×#4112yes—
581C42C2×#4112yes—
582C42H(5,𝔽2)12yes—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23C22×Q1614yes—
590C23C23×D411yescharacteristic
591Q8—22no—
592Q8—22no—
593Q8—22no—
594Q8—24no—
595Q8—24no—
596Q8—48no—
597Q8—416no—
598Q8—42no—
599Q8—42no—
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604Q8#55014yes—
605D4—28no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—216no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—416no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—84no—
657C2×C4—88no—
658C2×C4#52512yes—
659C2×C4#52512yes—
660C2×C4D4212yes—
661C2×C4D4212yes—
662C2×C4C22×Q1612yes—
663C2×C4C22×Q1614yes—
664C2×C4C22×Q1612yes—
665C2×C4C22×Q1612yes—
666C2×C4#57512yes—
667C2×C4#57812yes—
668C2×C4C23×D411yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4#58511yescharacteristic
674C8—24no—
675C8—44no—
676C8—42no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22C2×#52511yescharacteristic
685C22C2×D4211yescharacteristic
686C22#348811yescharacteristic
687C22C23×Q1611yescharacteristic
688C22C2×#57511yescharacteristic
689C22C2×#57811yescharacteristic
690C22#358611yescharacteristic
691C4—22no—
692C4—24no—
693C4—24no—
694C4—22no—
695C4—22no—
696C4—42no—
697C4—44no—
698C4—41no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—84no—
704C4—81no—
705C4—82no—
706C4—82no—
707C4#310414yescomplemented
708C4D4×Q1614yes—
709C4C2×#55011yescharacteristic
710C4D4×#4111yescharacteristic
711C4#350611yescharacteristic
712C4#351911yescharacteristic
713C2—44no—
714C2#3610111yescharacteristic
715C2C2×#310411yescharacteristic
716C2#6059411yescharacteristic
717C2C2×D4×Q1611yescharacteristic
718C2#6106711yescharacteristic
719C2#6195211yescharacteristic
720C2#6199811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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×#3104C2semidirect
4. #33163C2semidirect
5. #32906C2semidirect
6. #28440C2semidirect
7. #28284C2semidirect
8. #26621C2semidirect
11. #25161C2semidirect
12. #24652C2semidirect
13. #24155C2semidirect
14. #23184C2semidirect
15. #22998C2semidirect
17. C4×#477C2semidirect
142. #477D4semidirect
321. C4×D4Q16semidirect
707. C4#3104semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#361342
C4×D4×Q82
#331632
#329062
#284402
#282842
#266212
#262362
#257602
#251612
#246522
#241552
#231842
#229982
#227952
C4×#4772