Finite Group Databasebeta

← #6816637

Subgroups — #6816637

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3238C212yesmaximal, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#36114C211yesmaximal, characteristic
4#33316C211yesmaximal, characteristic, complemented
5#31104C214yesmaximal, complemented
6#31095C214yesmaximal, complemented
7#29883C212yesmaximal, complemented
8#28726C211yesmaximal, characteristic, complemented
9#28693C211yesmaximal, characteristic, complemented
10#26046C212yesmaximal, complemented
11#26044C212yesmaximal, complemented
12#25816C212yesmaximal, complemented
13#25679C212yesmaximal, complemented
14#25014C212yesmaximal, complemented
15#25010C212yesmaximal
16C4×#489C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#529C2212yes—
21#3238—216no—
22C2×#501C2212yes—
23C2×#501C2212yes—
24C2×#501C2212yes—
25C2×#489C2212yes—
26C2×#489C2211yescharacteristic
27C2×#483C2214yes—
28C2×#480C2214yes—
29C2×#477C2214yes—
30C2×#476C2214yes—
31C2×#432C2212yes—
32#2868C2211yescharacteristic
33#2859C2211yescharacteristic
34#2857C2212yes—
35#2851C2212yes—
36#2551—21no—
37#2551C2212yes—
38#2546—21no—
39#2538C2211yescharacteristic
40#2537C2211yescharacteristic
41#2496—21no—
42#2470—22no—
43#2466—22no—
44#2440—22no—
45#2440C2214yes—
46#2434—22no—
47#2422—21no—
48#2421C2212yes—
49#2420C2212yes—
50#2361—21no—
51Q8×C4⋊C4C2211yescharacteristic
52D4×C4⋊C4—21no—
53D4×C4⋊C4C2211yescharacteristic
54D4×C4⋊C4C2212yes—
55#2344—21no—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#124C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2088C2214yes—
60#2087C2214yes—
61#2085C2214yes—
62#2077C2214yes—
63#2075C2214yes—
64#2072C2212yes—
65#2071C2212yes—
66#2058C2214yes—
67#2057C2214yes—
68#2035C2214yes—
69#2033C2214yes—
70#1993C2212yes—
71#1993C2211yescharacteristic
72#1992C2212yes—
73#1959C2211yescharacteristic
74#1954C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1939C2212yes—
78#1936C2212yes—
79#1935C2212yes—
80#1924C2214yes—
81#1864C2214yes—
82#1863C2214yes—
83#1853C2214yes—
84#1852C2214yes—
85#1847C2214yes—
86#1845C2212yes—
87#1844C2212yes—
88#1836C2211yescharacteristic
89#1803C2212yes—
90#1802C2212yes—
91#1767C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C4×#97C2211yescharacteristic
96C23×Q8—22no—
97D4×Q8—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108#529—28no—
109#529—44no—
110C2×#126—22no—
111C2×#125C2312yes—
112C2×#124C2311yescharacteristic
113C2×#123—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2314yes—
133C22×C42C2312yes—
134#501—28no—
135#501—28no—
136#501—44no—
137#489—28no—
138#489D418yescomplemented
139#483—216no—
140#480—216no—
141#477—216no—
142#476—216no—
143#432—28no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×#97C2311yescharacteristic
163C2×#97C2311yescharacteristic
164C2×C4×C8C2312yes—
165#400—22no—
166#398—24no—
167#397—22no—
168#397—22no—
169#396—22no—
170#395—24no—
171#395—24no—
172#394—22no—
173#394—22no—
174#391—21no—
175#391—22no—
176#391C2311yescharacteristic
177#390—21no—
178#390C2311yescharacteristic
179#389—21no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2314yes—
193#388—22no—
194#388—21no—
195#388—21no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—22no—
200#385—22no—
201#385—22no—
202#385—24no—
203#385—24no—
204#385—22no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2312yes—
210#384C2312yes—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#382C2311yescharacteristic
214#382C2312yes—
215#381—22no—
216#380—22no—
217#380—21no—
218#380—21no—
219#380C2312yes—
220#380C2314yes—
221#380C2312yes—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4C2312yes—
233C43C2311yescharacteristic
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#340C2312yes—
239#340C2314yes—
240#340C2312yes—
241#340C2314yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—48no—
248C22×Q8—48no—
249C22×Q8—84no—
250C22×Q8C2411yescharacteristic
251C22×D4—22no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2411yescharacteristic
254C23×C4—24no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272C4⋊Q8—82no—
273C4⋊Q8C2×D414yes—
274#126—48no—
275#125—24no—
276#125—24no—
277#124—22no—
278#124C2×D414yes—
279#123—48no—
280C22⋊Q8—24no—
281C22⋊Q8—416no—
282C22⋊Q8—42no—
283C22⋊Q8—48no—
284#121—24no—
285#121—42no—
286#121—44no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—48no—
298C4×D4—48no—
299#117—48no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—28no—
367#107—28no—
368#107—44no—
369#106—28no—
370#106—28no—
371#106—44no—
372C4⋊C8—28no—
373C4⋊C8—24no—
374#103—28no—
375#103—24no—
376#103C2×D418yes—
377#102—28no—
378#102—24no—
379#102C2×D418yes—
380C22⋊C8—28no—
381C22⋊C8—28no—
382#97—24no—
383#97—24no—
384#97—22no—
385#97C2×D414yes—
386C4×C8—24no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—22no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—21no—
405C24C22×D412yes—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—48no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—84no—
418C2×Q8—832no—
419C2×Q8C22×D412yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—82no—
428C2×D4AGL(1,ℤ/8)14yes—
429C2×D4C22×D412yes—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—84no—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8C22×D414yes—
475C2×C82−1+414yes—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4#13714yes—
513C4⋊C4C22×D414yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—82no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—41no—
536C42—41no—
537C42—44no—
538C42—42no—
539C42—44no—
540C42C22×D412yes—
541C42H(5,𝔽2)12yes—
542C422−1+414yes—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C2×AGL(1,ℤ/8)14yes—
557C23C23×D411yescharacteristic
558Q8—24no—
559Q8—48no—
560Q8—42no—
561Q8—88no—
562Q8—816no—
563D4—28no—
564D4—24no—
565D4—42no—
566D4—88no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—88no—
616C2×C4—816no—
617C2×C4—84no—
618C2×C4#53612yes—
619C2×C4#53612yes—
620C2×C4D4212yes—
621C2×C4D4212yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4C2×#13714yes—
624C2×C4C2×#13712yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+412yes—
628C2×C4C2×2−1+412yes—
629C8—24no—
630C8—24no—
631C8—44no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22C2×#53611yescharacteristic
647C22C2×D4211yescharacteristic
648C22#349911yescharacteristic
649C22C22×AGL(1,ℤ/8)11yescharacteristic
650C22C22×#13711yescharacteristic
651C22#358511yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—24no—
656C4—22no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—81no—
668C4—81no—
669C4#323814yescomplemented
670C4#349912yes—
671C4#355212yes—
672C2—44no—
673C2—82no—
674C2#3936011yescharacteristic
675C2C2×#323811yescharacteristic
676C2#6088711yescharacteristic
677C2D4×AGL(1,ℤ/8)11yescharacteristic
678C2D4×#13711yescharacteristic
679C2#6250311yescharacteristic
680C2#6251011yescharacteristic
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. C2×#3238C2semidirect
2. D4×C4⋊Q8C2semidirect
4. #33316C2semidirect
5. #31104C2semidirect
6. #31095C2semidirect
7. #29883C2semidirect
8. #28726C2semidirect
9. #28693C2semidirect
10. #26046C2semidirect
11. #26044C2semidirect
12. #25816C2semidirect
13. #25679C2semidirect
14. #25014C2semidirect
16. C4×#489C2semidirect
17. #21270C2semidirect
138. #489D4semidirect
669. C4#3238semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
D4×C4⋊Q82
#361142
#333162
#311042
#310952
#298832
#287262
#286932
#260462
#260442
#258162
#256792
#250142
#250102
C4×#4892
#212702