Finite Group Databasebeta

← #7533238

Subgroups — #7533238

Order 512.

1955 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59822C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic, complemented
3#31413C214yesmaximal, complemented
4#29219C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28607C212yesmaximal, complemented
7#28149C212yesmaximal, complemented
8#26301C212yesmaximal, complemented
9#26218C214yesmaximal, complemented
10#25567C212yesmaximal, complemented
11#24967C212yesmaximal, complemented
12#24117C212yesmaximal, complemented
13#23176C211yesmaximal, characteristic, complemented
14#22503C212yesmaximal, complemented
15#22500C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic, complemented
18C2×#555C2212yes—
19C2×#547C2212yes—
20C2×#532—21no—
21C2×#486C2212yes—
22C2×#476C2211yescharacteristic
23C2×#459C2211yescharacteristic
24C2×C4×D8C2212yes—
25C2×#436C2212yes—
26C2×#436C2212yes—
27#2868C2211yescharacteristic
28#2857—21no—
29#2733—22no—
30#2548—22no—
31#2547C2212yes—
32#2541—21no—
33#2541C2212yes—
34#2537C2212yes—
35#2537C2212yes—
36#2537C2212yes—
37#2432—22no—
38#2422—22no—
39#2422—22no—
40#2414—22no—
41#2414C2214yes—
42#2354—21no—
43#2354C2212yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C42×D4C2211yescharacteristic
50#2099C2214yes—
51#2077C2214yes—
52#2076C2214yes—
53#2071C2214yes—
54#2034C2214yes—
55#2034C2214yes—
56#1999C2214yes—
57#1999C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1987C2211yescharacteristic
63#1982C2211yescharacteristic
64#1954C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1925C2212yes—
69#1920C2212yes—
70#1895C2214yes—
71#1895C2214yes—
72#1895C2214yes—
73#1879C2212yes—
74#1875C2214yes—
75#1874C2214yes—
76#1874C2214yes—
77#1870C2214yes—
78#1864C2212yes—
79#1854C2214yes—
80#1834C2212yes—
81#1825C2214yes—
82#1824C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87C4×#106C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#102C2212yes—
91C4×#102C2212yes—
92#1736C2214yes—
93C22×#41—22no—
94C22×D8C2312yes—
95C22×M16C2314yes—
96#555—28no—
97#555—44no—
98#547—28no—
99#547—44no—
100#532—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127—22no—
106C2×#127D411yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#124—22no—
109C2×#124—24no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×#121C2314yes—
118C2×#121C2314yes—
119C2×C4×Q8—21no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×#117—22no—
132C2×#117C2314yes—
133C22×C4⋊C4C2314yes—
134C22×C42—21no—
135#486—44no—
136#476—24no—
137#459—24no—
138C4×D8—44no—
139#436—44no—
140#436—44no—
141C2×#107C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#397—24no—
162#394—24no—
163#394—24no—
164#394—24no—
165#394—24no—
166#392—24no—
167#392—24no—
168#391—22no—
169#391—21no—
170#390—22no—
171#390—22no—
172#390—24no—
173#390—21no—
174#390—21no—
175#390D412yes—
176#390D412yes—
177#390D412yes—
178#390C2312yes—
179#389—22no—
180#389—22no—
181#389—21no—
182#389C2312yes—
183#388—24no—
184#388—24no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384D411yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#380—22no—
195#380—22no—
196#380—22no—
197#380C2314yes—
198#380C2314yes—
199#380C2314yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C43C2311yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#336C2312yes—
223#336C2312yes—
224C2×#41—48no—
225C2×#41—48no—
226C2×#41—84no—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2×D412yes—
235C22×D4C2×D412yes—
236C22×D4C2×D412yes—
237C22×D4C2412yes—
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2414yes—
245C2×D8—24no—
246C2×D8—42no—
247C2×D8—42no—
248C2×M16—44no—
249C2×M16—44no—
250C2×M16—44no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yescomplemented
261#127—24no—
262#127—22no—
263#127—22no—
264#127—84no—
265#124—416no—
266#124—48no—
267#123—416no—
268C22⋊Q8—48no—
269#121—28no—
270#121—28no—
271#121—44no—
272#121—44no—
273#121—48no—
274#121—48no—
275#121—48no—
276#121—44no—
277#121—44no—
278#121—84no—
279C4×Q8—82no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—41no—
290C4×D4—41no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—44no—
296C4×D4—82no—
297C4×D4#4114yes—
298#117—28no—
299#117—44no—
300#117—48no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C42—24no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—42no—
355#106—24no—
356#106—42no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361#102—24no—
362#102—24no—
363#102—42no—
364#102—42no—
365#102—42no—
366#102—42no—
367#102—44no—
368#102—44no—
369C22⋊C8—44no—
370C22⋊C8—44no—
371C22⋊C8—44no—
372C4×C8—24no—
373C4×C8—24no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382C24—21no—
383C24—22no—
384C24C22×D414yes—
385#41—832no—
386C2×Q8—24no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4C2×#4112yes—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—48no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—84no—
456C22×C4C22×D414yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464D8—48no—
465M16—88no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C2×#4112yes—
506C4⋊C4C2×#4114yes—
507C22⋊C4—24no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—42no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—41no—
528C42—48no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42AGL(1,ℤ/8)14yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42H(5,𝔽2)12yes—
537C42H(5,𝔽2)12yes—
538C23—24no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C23×D411yescharacteristic
560Q8—84no—
561Q8—88no—
562D4—22no—
563D4—22no—
564D4—48no—
565D4—48no—
566D4—42no—
567D4—42no—
568D4—41no—
569D4—42no—
570D4—41no—
571D4—42no—
572D4—48no—
573D4—84no—
574D4—816no—
575D4—84no—
576D4—168no—
577D4—168no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—816no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4#54612yes—
623C2×C4#54612yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4#57612yes—
628C2×C4#57712yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4#58511yescharacteristic
634C2×C4#58511yescharacteristic
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C2×#54611yescharacteristic
661C22C2×#54611yescharacteristic
662C22#348511yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C2×#57611yescharacteristic
665C22C2×#57711yescharacteristic
666C22#358611yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—41no—
672C4—44no—
673C4—42no—
674C4—81no—
675C4—82no—
676C4—84no—
677C4—82no—
678C4—81no—
679C4#332514yes—
680C4#333214yes—
681C4#348011yescharacteristic
682C4#350611yescharacteristic
683C4#351711yescharacteristic
684C4#354611yescharacteristic
685C2—84no—
686C2—82no—
687C2—162no—
688C2#5903311yescharacteristic
689C2C2×#332511yescharacteristic
690C2C2×#333211yescharacteristic
691C2#6119911yescharacteristic
692C2#6120911yescharacteristic
693C2#6171711yescharacteristic
694C2#6178811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59822C2semidirect
2. #36127C2semidirect
3. #31413C2semidirect
4. #29219C2semidirect
5. #28716C2semidirect
6. #28607C2semidirect
7. #28149C2semidirect
8. #26301C2semidirect
9. #26218C2semidirect
10. #25567C2semidirect
11. #24967C2semidirect
12. #24117C2semidirect
13. #23176C2semidirect
14. #22503C2semidirect
15. #22500C2semidirect
16. C4×#476C2semidirect
17. C4×#459C2semidirect
260. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#598222
#361272
#314132
#292192
#287162
#286072
#281492
#263012
#262182
#255672
#249672
#241172
#231762
#225032
#225002
C4×#4762
C4×#4592