Finite Group Databasebeta

← #6514001

Subgroups — #6514001

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34977C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#27678C212yesmaximal, complemented
4#27644C212yesmaximal, complemented
5#27627C212yesmaximal, complemented
6#27627C212yesmaximal, complemented
7#26318C212yesmaximal, complemented
8#26300C212yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26230C212yesmaximal, complemented
11#26214C214yesmaximal, complemented
12#26142C214yesmaximal, complemented
13#21893C211yesmaximal, characteristic, complemented
14#21859C211yesmaximal, characteristic, complemented
15#21849C211yesmaximal, characteristic, complemented
16#21362C211yesmaximal, characteristic, complemented
17#21171C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#480C2214yes—
22C2×#431C2211yescharacteristic
23C2×#428C2212yes—
24C2×#423C2211yescharacteristic
25#2610—22no—
26#2567C2211yescharacteristic
27#2563C2211yescharacteristic
28#2548C2211yescharacteristic
29#2541C2211yescharacteristic
30#2500—21no—
31#2431—21no—
32#2414C2212yes—
33#2414C2212yes—
34#2388—22no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2212yes—
41#2317—22no—
42#2317—21no—
43C4×#127—21no—
44C4×#124—21no—
45C4×#121C2212yes—
46C4×#121C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2212yes—
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2094C2214yes—
54#2083C2214yes—
55#2077C2214yes—
56#2077C2214yes—
57#2071C2214yes—
58#2033C2214yes—
59#2033C2214yes—
60#1990C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1982C2212yes—
64#1957C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1948C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1947C2211yescharacteristic
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1926C2214yes—
75#1926C2214yes—
76#1921C2214yes—
77#1921C2214yes—
78#1895C2214yes—
79#1874C2214yes—
80#1874C2214yes—
81#1870C2214yes—
82#1854C2212yes—
83#1854C2212yes—
84#1853C2214yes—
85#1823C2214yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×#106C2212yes—
94#1736C2212yes—
95C22×#41—21no—
96C22×#41D411yescharacteristic
97C22×M16C2312yes—
98#532—24no—
99#532—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127—21no—
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×#124—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#121C2312yes—
109#520—48no—
110C4×#41—24no—
111C4×#41—24no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—24no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—22no—
126C2×#117—21no—
127C2×#117D412yes—
128C2×#117D411yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#480—48no—
140#431—24no—
141#428—44no—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#106C2312yes—
146C2×#106C2314yes—
147C2×#106C2312yes—
148#423—24no—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161#395—22no—
162#394—22no—
163#394—22no—
164#394—22no—
165#393—22no—
166#392—22no—
167#392—22no—
168#392—22no—
169#391—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—22no—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—24no—
179#389—22no—
180#389—24no—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#386—22no—
188#386—22no—
189#385—24no—
190#385—24no—
191#385—24no—
192#384—21no—
193#384—22no—
194#384—21no—
195#384—22no—
196#384—21no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2312yes—
202#382—21no—
203#382—22no—
204#382—21no—
205#380—22no—
206#380—21no—
207#380C2312yes—
208#380C2312yes—
209#380C2314yes—
210#380C2312yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223#376—21no—
224C43—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—44no—
241C2×#41—82no—
242C2×#41—82no—
243C22×Q8—21no—
244C22×Q8C2×D411yescharacteristic
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—42no—
264C2×M16—42no—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#127—22no—
271#127—22no—
272#127—44no—
273#124—44no—
274#124—44no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—42no—
280#121—42no—
281#121—44no—
282C4×Q8—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—48no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—416no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—84no—
298C4×D4—84no—
299#117—24no—
300#117—28no—
301#117—22no—
302#117—22no—
303#117—41no—
304#117—41no—
305#117—48no—
306#117—44no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4—84no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42#4114yes—
374C2×C42#4114yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#107—24no—
380#107—24no—
381#106—42no—
382#106—42no—
383#106—44no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—22no—
387C4⋊C8—22no—
388#102—44no—
389#102—44no—
390#102—44no—
391#102—44no—
392C22⋊C8—24no—
393C22⋊C8—44no—
394C22⋊C8—44no—
395C4×C8—24no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—22no—
406#95—24no—
407#95—22no—
408C24—22no—
409C24C22×D411yescharacteristic
410C24C22×D412yes—
411#41—232no—
412#41—816no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—42no—
416C2×Q8—81no—
417C2×Q8—81no—
418C2×Q8#13714yes—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4AGL(1,ℤ/8)14yes—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—82no—
483C22×C4#12314yes—
484C22×C4#12314yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2511yescharacteristic
496M16—48no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—82no—
536C4⋊C4—84no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4H(5,𝔽2)12yes—
540C4⋊C42−1+412yes—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—42no—
563C42—42no—
564C42—48no—
565C42—44no—
566C42—42no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42#12314yes—
571C42#12314yes—
572C42C2×#4112yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C23—22no—
577C23—21no—
578C23—22no—
579C23—21no—
580C23—22no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—21no—
586C23—22no—
587C23—21no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23C2×#12312yes—
596C23C2×AGL(1,ℤ/8)14yes—
597C23C23×D411yescharacteristic
598Q8—28no—
599Q8—84no—
600D4—216no—
601D4—28no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—84no—
607D4—88no—
608D4—88no—
609D4—88no—
610D4—84no—
611C2×C4—216no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—48no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—88no—
662C2×C4—168no—
663C2×C4—164no—
664C2×C4C2×#12312yes—
665C2×C4C2×#12312yes—
666C2×C4C2×#12312yes—
667C2×C4C2×#12312yes—
668C2×C4C2×#12312yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4C2×#13714yes—
671C2×C4C2×#13712yes—
672C2×C4#57612yes—
673C2×C4#57712yes—
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×2−1+411yescharacteristic
678C2×C4#58511yescharacteristic
679C2×C4#58511yescharacteristic
680C8—44no—
681C8—42no—
682C8—42no—
683C22—24no—
684C22—22no—
685C22—24no—
686C22—22no—
687C22—24no—
688C22—22no—
689C22—41no—
690C22—41no—
691C22—42no—
692C22—42no—
693C22—41no—
694C22—42no—
695C22—41no—
696C22—42no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22C22×#12311yescharacteristic
705C22#348811yescharacteristic
706C22#348911yescharacteristic
707C22C22×AGL(1,ℤ/8)11yescharacteristic
708C22C22×#13711yescharacteristic
709C22C2×#57611yescharacteristic
710C22C2×#57711yescharacteristic
711C4—22no—
712C4—24no—
713C4—22no—
714C4—22no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—82no—
720C4—84no—
721C4—81no—
722C4—82no—
723C4—82no—
724C4—81no—
725C4#309214yes—
726C4#309614yes—
727C4#350611yescharacteristic
728C4#350711yescharacteristic
729C4#351111yescharacteristic
730C4#351211yescharacteristic
731C2—44no—
732C2—42no—
733C2—162no—
734C2#3551711yescharacteristic
735C2C2×#309211yescharacteristic
736C2C2×#309611yescharacteristic
737C2#6197911yescharacteristic
738C2#6200211yescharacteristic
739C2#6212811yescharacteristic
740C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34977C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #27678C2semidirect
4. #27644C2semidirect
5. #27627C2semidirect
6. #27627C2semidirect
7. #26318C2semidirect
8. #26300C2semidirect
9. #26282C2semidirect
10. #26230C2semidirect
11. #26214C2semidirect
12. #26142C2semidirect
13. #21893C2semidirect
14. #21859C2semidirect
15. #21849C2semidirect
16. #21362C2semidirect
17. #21171C2semidirect

Maximal subgroups

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

subgroupindex
#349772
#341832
#276782
#276442
#276272
#263182
#263002
#262822
#262302
#262142
#261422
#218932
#218592
#218492
#213622
#211712