Finite Group Databasebeta

← #6809594

Subgroups — #6809594

Order 512.

1955 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C212yesmaximal
2#36133C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic
4#33109C211yesmaximal, characteristic
5#30957C214yesmaximal, complemented
6#30816C214yesmaximal, complemented
7#29642C212yesmaximal
8#28813C212yesmaximal, complemented
9#27910C211yesmaximal, characteristic
10#27769C212yesmaximal, complemented
11#27648C211yesmaximal, characteristic
12#26466C212yesmaximal, complemented
13#26046C212yesmaximal, complemented
14#24602C212yesmaximal
15#24064C212yesmaximal
16#22599C211yesmaximal, characteristic
17C4×#484C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2212yes—
21#3230—216no—
22C2×#500C2214yes—
23C2×#487C2214yes—
24C2×#484C2212yes—
25C2×#484C2212yes—
26C2×#484C2212yes—
27C2×#484C2212yes—
28C2×#484C2211yescharacteristic
29C2×#477C2214yes—
30C2×#477C2214yes—
31C2×#431C2212yes—
32#2857C2211yescharacteristic
33#2855C2212yes—
34#2551—21no—
35#2551—22no—
36#2551—21no—
37#2551C2211yescharacteristic
38#2551C2214yes—
39#2551C2212yes—
40#2551C2212yes—
41#2546C2211yescharacteristic
42#2506—21no—
43#2462—22no—
44#2451C2212yes—
45#2446C2212yes—
46#2440—21no—
47#2438—22no—
48C2×#395—22no—
49#2365—21no—
50#2365C2211yescharacteristic
51Q8×C4⋊C4C2211yescharacteristic
52#2344—21no—
53C4×C22⋊Q8C2211yescharacteristic
54C2×#389—21no—
55C2×#389C2212yes—
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2095C2214yes—
60#2087C2214yes—
61#2083C2212yes—
62#2072C2212yes—
63#2068C2214yes—
64#2057C2214yes—
65#2057C2214yes—
66#2057C2214yes—
67#2043C2214yes—
68#2024C2214yes—
69#2024C2214yes—
70#1981C2212yes—
71#1956C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1942C2214yes—
75#1939C2212yes—
76#1933C2214yes—
77#1928C2212yes—
78#1928C2214yes—
79#1921C2212yes—
80#1916C2214yes—
81#1872C2211yescharacteristic
82#1872C2212yes—
83#1863C2214yes—
84#1863C2214yes—
85#1852C2211yescharacteristic
86#1845C2212yes—
87#1824C2212yes—
88#1824C2211yescharacteristic
89#1817C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#107C2212yes—
93C8×C4⋊C4C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8—22no—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—44no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2314yes—
113C2×#125—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C4×Q8—22no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×#117—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#500—216no—
135#487—216no—
136#484—28no—
137#484—28no—
138#484—28no—
139#484—44no—
140#484D418yes—
141#477—216no—
142#477—216no—
143#431—28no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2312yes—
148C2×#107C2314yes—
149C2×#107C2312yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2311yescharacteristic
162C2×C4×C8C2312yes—
163C2×C4×C8C2314yes—
164#400—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#395—22no—
169#395—24no—
170#395—24no—
171#395—24no—
172#395—24no—
173#395—22no—
174#391—22no—
175#391C2312yes—
176#391C2311yescharacteristic
177#391C2312yes—
178#391C2312yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—24no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#389C2314yes—
194#389C2314yes—
195#389C2311yescharacteristic
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—21no—
201#387—22no—
202#387—22no—
203#385—22no—
204#385—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2312yes—
209#384C2312yes—
210#384C2311yescharacteristic
211#381—22no—
212#380—21no—
213#380—21no—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—21no—
230C2×#95—21no—
231C2×#95—22no—
232C2×#95D412yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#336C2312yes—
241#336C2314yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—48no—
248C22×Q8—48no—
249C22×Q8—84no—
250C22×Q8C2412yes—
251C22×Q8C2411yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—28no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273C4⋊Q8—42no—
274C4⋊Q8—416no—
275C4⋊Q8—48no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278#125—48no—
279C22⋊Q8—22no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—48no—
283C22⋊Q8C2×D414yes—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—48no—
290#117—48no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4Q1618yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—44no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—28no—
363#107—28no—
364#107—28no—
365#107—24no—
366#107—44no—
367#107C2×D418yes—
368C4⋊C8—28no—
369C4⋊C8—24no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—24no—
374#103—28no—
375#103—44no—
376#103C2×D418yes—
377C22⋊C8—24no—
378C22⋊C8—22no—
379C22⋊C8C2×D414yes—
380C4×C8—28no—
381C4×C8—24no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24C22×D411yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—44no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—48no—
416C2×Q8—44no—
417C2×Q8—82no—
418C2×Q8—84no—
419C2×Q8—84no—
420C2×Q8—832no—
421C2×Q8C22×D412yes—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4C2×Q1618yes—
453C22×C4C2×Q1614yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4H(5,𝔽2)12yes—
465C22×C4C2511yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8C22×D414yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4C2×Q1618yes—
513C4⋊C4C2×Q1614yes—
514C4⋊C4C2×Q1614yes—
515C4⋊C4AGL(1,ℤ/8)14yes—
516C4⋊C4C22×D414yes—
517C4⋊C4C22×D412yes—
518C4⋊C42−1+414yes—
519C22⋊C4—24no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—44no—
536C42—42no—
537C422−1+414yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23D4212yes—
543C23C22×Q1612yes—
544C23#57812yes—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—42no—
548Q8—44no—
549Q8—44no—
550Q8—88no—
551Q8—816no—
552Q8—816no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—216no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—88no—
607C2×C4#53612yes—
608C2×C4#53612yes—
609C2×C4D4212yes—
610C2×C4C22×Q1614yes—
611C2×C4C22×Q1614yes—
612C2×C4C22×Q1612yes—
613C2×C4C22×Q1612yes—
614C2×C4C2×AGL(1,ℤ/8)14yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+412yes—
619C2×C4C2×2−1+412yes—
620C8—24no—
621C8—44no—
622C8—42no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22D4×Q1614yes—
630C22C2×#53611yescharacteristic
631C22C2×D4211yescharacteristic
632C22#348611yescharacteristic
633C22C23×Q1611yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C2×#57811yescharacteristic
636C22#358511yescharacteristic
637C4—24no—
638C4—24no—
639C4—24no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—44no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—41no—
649C4—84no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4#323014yescomplemented
655C4#349912yes—
656C4#354412yes—
657C2—24no—
658C2#3933711yescharacteristic
659C2C2×#323011yescharacteristic
660C2#6087411yescharacteristic
661C2C2×D4×Q1611yescharacteristic
662C2D4×AGL(1,ℤ/8)11yescharacteristic
663C2#6182111yescharacteristic
664C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #30957C2semidirect
6. #30816C2semidirect
8. #28813C2semidirect
10. #27769C2semidirect
12. #26466C2semidirect
13. #26046C2semidirect
654. C4#3230semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#361332
C2×#25512
#331092
#309572
#308162
#296422
#288132
#279102
#277692
#276482
#264662
#260462
#246022
#240642
#225992
C4×#4842