Finite Group Databasebeta

← #6793353

Subgroups — #6793353

Order 512.

1919 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic, complemented
4#31288C214yesmaximal, complemented
5#30986C214yesmaximal, complemented
6#29922C212yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28400C212yesmaximal, complemented
9#28283C211yesmaximal, characteristic, complemented
10#27749C212yesmaximal, complemented
11#25884C212yesmaximal, complemented
12#24651C212yesmaximal, complemented
13#24239C212yesmaximal
14#24209C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21804C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C2×#529—21no—
21C2×#487C2212yes—
22C2×#475C2212yes—
23C2×#461C2211yescharacteristic
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×#461C2212yes—
27C2×#448C2214yes—
28C2×C4×SD16C2212yes—
29C2×#431C2212yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2553—22no—
33#2552C2211yescharacteristic
34#2551—21no—
35#2551—21no—
36#2547C2211yescharacteristic
37#2546C2212yes—
38#2542C2211yescharacteristic
39#2537C2211yescharacteristic
40#2496C2212yes—
41#2489—21no—
42#2440—22no—
43#2440—22no—
44#2434C2214yes—
45#2427—21no—
46#2422C2212yes—
47#2395—22no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8—21no—
53C4×#121C2212yes—
54#2285C2212yes—
55C42×D4C2211yescharacteristic
56C42×D4C2211yescharacteristic
57#2096C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2035C2214yes—
63#2035C2214yes—
64#2035C2214yes—
65#2013C2214yes—
66#2000C2214yes—
67#1981C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1968C2214yes—
75#1956C2214yes—
76#1956C2214yes—
77#1956C2212yes—
78#1954C2212yes—
79#1948C2212yes—
80#1948C2211yescharacteristic
81#1933C2214yes—
82#1927C2212yes—
83#1923C2214yes—
84#1877C2212yes—
85#1842C2211yescharacteristic
86#1803C2212yes—
87#1802C2211yescharacteristic
88#1797C2212yes—
89#1787C2214yes—
90#1787C2214yes—
91#1787C2214yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×Q8C2314yes—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98D4×Q8—44no—
99D4×Q8—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111#529—48no—
112C2×#126—22no—
113C2×#125—21no—
114C2×#124—21no—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8C2314yes—
120C2×#121—21no—
121C2×#121C2312yes—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—22no—
135C22×C42—21no—
136C22×C42C2312yes—
137#487—44no—
138#475—44no—
139#461—24no—
140#461—44no—
141#461—44no—
142#461—44no—
143#448—48no—
144C4×SD16—44no—
145#431—44no—
146C2×#106C2314yes—
147C2×#106C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2312yes—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2314yes—
168C2×C4×C8C2312yes—
169C2×C4×C8C2314yes—
170#400—22no—
171#399—22no—
172#398—22no—
173#398—24no—
174#397—22no—
175#397—22no—
176#397—22no—
177#395—24no—
178#395—24no—
179#394—22no—
180#394—22no—
181#391—21no—
182#391—21no—
183#391C2312yes—
184#391C2314yes—
185#390C2312yes—
186#389—22no—
187#389—22no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—22no—
193#389—22no—
194#389C2312yes—
195#389C2312yes—
196#388—22no—
197#387—22no—
198#386—22no—
199#385—24no—
200#385—24no—
201#385—22no—
202#384—21no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2312yes—
209#384C2312yes—
210#382—22no—
211#380—22no—
212#380—22no—
213#380—22no—
214#380—21no—
215#380C2312yes—
216#380C2312yes—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4D412yes—
235C4×C22⋊C4C2312yes—
236C43C2311yescharacteristic
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2314yes—
241#336C2312yes—
242C22×Q8—21no—
243C22×Q8—24no—
244C22×Q8—28no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8C2414yes—
251C22×Q8C2412yes—
252C22×D4—22no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2411yescharacteristic
255C23×C4—24no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—28no—
265C2×SD16—42no—
266C2×SD16—44no—
267C2×SD16—44no—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—24no—
279C4⋊Q8—48no—
280C4⋊Q8—48no—
281C4⋊Q8—44no—
282C4⋊Q8—42no—
283C4⋊Q8—42no—
284C4⋊Q8—48no—
285C4⋊Q8#4114yes—
286#126—48no—
287#125—44no—
288#124—44no—
289#123—48no—
290C22⋊Q8—416no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295#121—24no—
296#121—42no—
297#121—44no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—44no—
301C4×D4—24no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—41no—
310C4×D4—48no—
311C4×D4—48no—
312C4×D4SD1618yescomplemented
313#117—48no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
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—21no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—48no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#106—42no—
380#106—44no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C4⋊C8—42no—
386#103—24no—
387#103—42no—
388#103—44no—
389#103—42no—
390#103—44no—
391#103—44no—
392#103—44no—
393#103—44no—
394#103—44no—
395#103—44no—
396#103—44no—
397#103—44no—
398#102—42no—
399C22⋊C8—44no—
400C22⋊C8—44no—
401C4×C8—24no—
402C4×C8—44no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95C2×D412yes—
417#95C2×D412yes—
418#95C2×D412yes—
419C24—21no—
420C24C22×D412yes—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—84no—
438C2×Q8—88no—
439C2×Q8—88no—
440C2×Q8—88no—
441C2×Q8—88no—
442C2×Q8C2×#4114yes—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—82no—
453C2×D4C2×SD1614yes—
454C2×D4H(5,𝔽2)12yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—42no—
474C22×C4—48no—
475C22×C4—42no—
476C22×C4—48no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—84no—
486C22×C4C2×SD1618yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C2511yescharacteristic
497SD16—48no—
498SD16—88no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—81no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4C2×SD1614yes—
545C4⋊C4#13714yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—42no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4C2×SD1618yes—
559C42—24no—
560C42—24no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—41no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42—42no—
572C42—44no—
573C42C2×SD1614yes—
574C42C2×#4112yes—
575C42H(5,𝔽2)12yes—
576C42H(5,𝔽2)12yes—
577C42H(5,𝔽2)12yes—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23C22×SD1614yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—42no—
596Q8—42no—
597Q8—44no—
598Q8—44no—
599Q8—44no—
600Q8—44no—
601Q8—816no—
602Q8—84no—
603D4—28no—
604D4—22no—
605D4—22no—
606D4—41no—
607D4—41no—
608D4—88no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—216no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—84no—
656C2×C4—816no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4#53512yes—
660C2×C4#53512yes—
661C2×C4#54612yes—
662C2×C4#54612yes—
663C2×C4C22×SD1612yes—
664C2×C4C22×SD1614yes—
665C2×C4C22×SD1612yes—
666C2×C4C22×SD1612yes—
667C2×C4C2×#13714yes—
668C2×C4C2×#13712yes—
669C2×C4#57812yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4#58511yescharacteristic
676C8—42no—
677C8—44no—
678C8—42no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22C2×#53511yescharacteristic
694C22C2×#54611yescharacteristic
695C22#348811yescharacteristic
696C22C23×SD1611yescharacteristic
697C22C22×#13711yescharacteristic
698C22C2×#57811yescharacteristic
699C22#358611yescharacteristic
700C4—22no—
701C4—24no—
702C4—22no—
703C4—24no—
704C4—22no—
705C4—41no—
706C4—42no—
707C4—44no—
708C4—81no—
709C4—82no—
710C4—82no—
711C4—81no—
712C4—81no—
713C4—82no—
714C4—84no—
715C4#320614yes—
716C4#329814yes—
717C4#351411yescharacteristic
718C4#351911yescharacteristic
719C4#354311yescharacteristic
720C4#354611yescharacteristic
721C2—44no—
722C2—82no—
723C2#3891011yescharacteristic
724C2C2×#320611yescharacteristic
725C2#6083711yescharacteristic
726C2C2×#329811yescharacteristic
727C2#6115711yescharacteristic
728C2#6194811yescharacteristic
729C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. D4×C4⋊Q8C2semidirect
3. #33162C2semidirect
4. #31288C2semidirect
5. #30986C2semidirect
6. #29922C2semidirect
7. #28748C2semidirect
8. #28400C2semidirect
9. #28283C2semidirect
10. #27749C2semidirect
11. #25884C2semidirect
12. #24651C2semidirect
14. #24209C2semidirect
15. #22795C2semidirect
16. #21804C2semidirect
17. C4×#461C2semidirect
312. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#349322
#331622
#312882
#309862
#299222
#287482
#284002
#282832
#277492
#258842
#246512
#242392
#242092
#227952
#218042
C4×#4612