Finite Group Databasebeta

← #6811812

Subgroups — #6811812

Order 512.

1961 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C212yesmaximal, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic
3#34929C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic, complemented
5#30779C214yesmaximal, complemented
6#30635C214yesmaximal, complemented
7#29691C212yesmaximal, complemented
8#28813C212yesmaximal, complemented
9#28511C211yesmaximal, characteristic, complemented
10#28284C211yesmaximal, characteristic, complemented
11#27769C212yesmaximal, complemented
12#26236C212yesmaximal
13#25016C212yesmaximal
14#24702C212yesmaximal, complemented
15#24302C212yesmaximal, complemented
16#22795C211yesmaximal, characteristic
17C4×#477C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C2×#533C2212yes—
20C22×C4⋊Q8C2212yes—
21#3230—216no—
22C2×#500C2214yes—
23C2×#487C2214yes—
24C2×#484C2214yes—
25C2×#484C2214yes—
26C2×#477C2212yes—
27C2×#477C2212yes—
28C2×#477C2212yes—
29C2×#477C2212yes—
30C2×#477C2211yescharacteristic
31C2×#431C2212yes—
32#2868C2211yescharacteristic
33#2857C2212yes—
34#2556—22no—
35#2551—21no—
36#2551C2212yes—
37#2551C2211yescharacteristic
38#2546—21no—
39#2546C2211yescharacteristic
40#2542C2211yescharacteristic
41#2538C2211yescharacteristic
42#2506—21no—
43#2492C2212yes—
44#2452—21no—
45#2440—22no—
46#2440C2214yes—
47#2434—22no—
48#2429C2212yes—
49#2396—22no—
50D4×C4⋊C4—21no—
51D4×C4⋊C4C2212yes—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C22⋊Q8C2212yes—
56#2293—21no—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2095C2214yes—
60#2076C2214yes—
61#2068C2214yes—
62#2057C2214yes—
63#2057C2214yes—
64#2043C2214yes—
65#2036C2214yes—
66#2036C2214yes—
67#2036C2214yes—
68#2024C2214yes—
69#1987C2211yescharacteristic
70#1987C2212yes—
71#1981C2212yes—
72#1956C2212yes—
73#1954C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1947C2212yes—
77#1947C2212yes—
78#1942C2214yes—
79#1933C2214yes—
80#1923C2214yes—
81#1921C2212yes—
82#1872C2214yes—
83#1863C2212yes—
84#1852C2214yes—
85#1845C2212yes—
86#1842C2211yescharacteristic
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C8×C4⋊C4C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8—22no—
97D4×Q8—48no—
98#533—28no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—44no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2314yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×#125—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C4×Q8—22no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—22no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4C2314yes—
134C22×C42C2312yes—
135#500—216no—
136#487—216no—
137#484—216no—
138#484—216no—
139#477—28no—
140#477—28no—
141#477—28no—
142#477—44no—
143#477D418yescomplemented
144#431—28no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#107C2314yes—
150C2×#107C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2314yes—
167#400—22no—
168#399—22no—
169#398—22no—
170#398—22no—
171#398—24no—
172#396—22no—
173#396—22no—
174#395—22no—
175#395—24no—
176#395—24no—
177#393—22no—
178#391—21no—
179#391—22no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#390—21no—
184#390—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—22no—
195#389—22no—
196#389C2312yes—
197#389C2312yes—
198#389C2314yes—
199#388—22no—
200#388—22no—
201#387—22no—
202#385—22no—
203#385—24no—
204#385—24no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2312yes—
211#384C2312yes—
212#380—22no—
213#380C2312yes—
214#380C2314yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—22no—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4C2312yes—
233C43—21no—
234C43C2311yescharacteristic
235#340C2312yes—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—48no—
249C22×Q8—48no—
250C22×Q8—84no—
251C22×Q8C2412yes—
252C22×Q8C2411yescharacteristic
253C22×D4C2×D411yescharacteristic
254C23×C4—24no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—28no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—48no—
272C4⋊Q8—42no—
273C4⋊Q8—48no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—48no—
277C4⋊Q8—82no—
278C4⋊Q8C2×D414yes—
279#125—48no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—416no—
284C22⋊Q8—42no—
285C22⋊Q8—42no—
286C22⋊Q8—42no—
287C22⋊Q8—48no—
288C22⋊Q8—48no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—48no—
296C4×Q8—41no—
297C4×Q8—44no—
298C4×Q8C2×D414yes—
299C4×D4—24no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—48no—
303C4×D4Q1618yes—
304#117—48no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—28no—
374#107—28no—
375#107—28no—
376#107—24no—
377#107—44no—
378#107C2×D418yes—
379C4⋊C8—24no—
380C4⋊C8—22no—
381C4⋊C8—24no—
382C4⋊C8—42no—
383C4⋊C8C2×D414yes—
384#103—28no—
385#103—28no—
386#103—28no—
387#103—42no—
388#103—44no—
389#103—44no—
390#103C2×D418yes—
391C22⋊C8—28no—
392C22⋊C8—44no—
393C4×C8—28no—
394C4×C8—24no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95C2×D412yes—
409#95C2×D412yes—
410#95C2×D412yes—
411C24C22×D412yes—
412C2×Q8—24no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—44no—
425C2×Q8—48no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—44no—
433C2×Q8—82no—
434C2×Q8—84no—
435C2×Q8—832no—
436C2×Q8C22×D412yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4C2×Q1614yes—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—84no—
463C22×C4C2×Q1618yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2511yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8C22×D414yes—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4C2×Q1614yes—
523C4⋊C4AGL(1,ℤ/8)14yes—
524C4⋊C4C22×D414yes—
525C4⋊C4C22×D412yes—
526C4⋊C42−1+412yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—82no—
534C22⋊C4C2×Q1618yes—
535C42—24no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—41no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—81no—
552C42C2×Q1614yes—
553C42C22×D412yes—
554C42H(5,𝔽2)12yes—
555C422−1+412yes—
556C422−1+412yes—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23C22×Q1614yes—
564C23C23×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—48no—
568Q8—41no—
569Q8—41no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—88no—
575Q8—816no—
576Q8—88no—
577D4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—216no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—816no—
628C2×C4#53612yes—
629C2×C4#53612yes—
630C2×C4D4212yes—
631C2×C4D4212yes—
632C2×C4C22×Q1612yes—
633C2×C4C22×Q1614yes—
634C2×C4C22×Q1612yes—
635C2×C4C22×Q1612yes—
636C2×C4C2×AGL(1,ℤ/8)14yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4#57812yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4#58511yescharacteristic
645C8—24no—
646C8—44no—
647C8—42no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22C2×#53611yescharacteristic
656C22C2×D4211yescharacteristic
657C22#348911yescharacteristic
658C22C23×Q1611yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C2×#57811yescharacteristic
661C22#358511yescharacteristic
662C4—24no—
663C4—22no—
664C4—24no—
665C4—22no—
666C4—22no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—84no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4#323014yescomplemented
679C4D4×Q1614yes—
680C4#349711yescharacteristic
681C4#349911yescharacteristic
682C4#354711yescharacteristic
683C4#355011yescharacteristic
684C2—44no—
685C2#3934311yescharacteristic
686C2C2×#323011yescharacteristic
687C2#6089311yescharacteristic
688C2C2×D4×Q1611yescharacteristic
689C2D4×AGL(1,ℤ/8)11yescharacteristic
690C2#6210011yescharacteristic
691C2#6214811yescharacteristic
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×#3230C2semidirect
4. #33198C2semidirect
5. #30779C2semidirect
6. #30635C2semidirect
7. #29691C2semidirect
8. #28813C2semidirect
9. #28511C2semidirect
10. #28284C2semidirect
11. #27769C2semidirect
14. #24702C2semidirect
15. #24302C2semidirect
17. C4×#477C2semidirect
143. #477D4semidirect
678. C4#3230semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
D4×C4⋊Q82
#349292
#331982
#307792
#306352
#296912
#288132
#285112
#282842
#277692
#262362
#250162
#247022
#243022
#227952
C4×#4772