Finite Group Databasebeta

← #6500828

Subgroups — #6500828

Order 512.

2170 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2416C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#27613C211yesmaximal, characteristic, complemented
4#27610C211yesmaximal, characteristic, complemented
5#27602C212yesmaximal, complemented
6#27585C211yesmaximal, characteristic, complemented
7#27582C211yesmaximal, characteristic, complemented
8#26229C214yesmaximal, complemented
9#26225C214yesmaximal, complemented
10#26157C214yesmaximal, complemented
11#26153C214yesmaximal, complemented
12#24009C212yesmaximal, complemented
13#21767C212yesmaximal, complemented
14#21742C211yesmaximal, characteristic, complemented
15#20844C212yesmaximal, complemented
16C22×#121—21no—
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#482C2214yescomplemented
22C2×#428C2214yescomplemented
23C2×#407C2212yes—
24#2416—22no—
25#2416—22no—
26#2416C2214yes—
27#2416C2214yes—
28C2×#396—21no—
29C2×#394—21no—
30#2357—22no—
31#2357—22no—
32#2354C2212yescomplemented
33#2354C2212yescomplemented
34#2354C2212yescomplemented
35#2354C2212yescomplemented
36#2312—21no—
37#2312—21no—
38#2309—21no—
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#386—21no—
42#2285C2211yescharacteristic
43#2285C2211yescharacteristic
44#2280C2212yes—
45#2280C2212yes—
46C2×#380C2211yescharacteristic
47#2274—21no—
48#2100C2212yescomplemented
49#2099C2212yescomplemented
50#2094C2212yescomplemented
51#2093C2212yescomplemented
52#2084C2212yescomplemented
53#2083C2212yescomplemented
54#2077C2214yescomplemented
55#2076C2214yescomplemented
56#2071C2212yescomplemented
57#2071C2212yescomplemented
58#2034C2214yescomplemented
59#2033C2214yescomplemented
60#1926C2214yescomplemented
61#1925C2214yescomplemented
62#1919C2214yescomplemented
63#1918C2214yescomplemented
64#1895C2214yescomplemented
65#1895C2214yescomplemented
66#1875C2214yescomplemented
67#1875C2214yescomplemented
68#1875C2214yescomplemented
69#1870C2214yescomplemented
70#1870C2214yescomplemented
71#1854C2214yescomplemented
72#1853C2212yes—
73#1853C2212yes—
74#1853C2214yescomplemented
75#1852C2212yes—
76#1852C2212yes—
77#1852C2214yescomplemented
78#1825C2214yescomplemented
79#1823C2212yes—
80#1823C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1786C2212yes—
84#1786C2212yes—
85#1779C2212yes—
86#1736C2214yescomplemented
87C23×D4—21no—
88C23×D4—21no—
89C23×D4D411yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C22×M16C2314yes—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121—28no—
100C2×#121—44no—
101C2×#121—48no—
102C2×#121C2314yes—
103C2×#121C2314yes—
104C2×C4×D4—28no—
105C2×C4×D4D412yes—
106C2×C4×D4D412yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—21no—
110C2×#117—21no—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—44no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42C2311yescharacteristic
132#482—48no—
133#428—48no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142#407—28no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148#396—24no—
149#396—24no—
150#394—24no—
151#394—24no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2312yes—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#386—24no—
173#386—24no—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#382—21no—
179#382—21no—
180#381—22no—
181#381—22no—
182#380—21no—
183#380—21no—
184#380—22no—
185#380C2312yes—
186#380C2314yes—
187#380C2312yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192#379—22no—
193#379—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—21no—
207C2×#95—21no—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2312yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C25C2×D411yescharacteristic
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—24no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—82no—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—24no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—24no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—82no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4#4118yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263C2×M16—28no—
264C2×M16—44no—
265C2×M16—44no—
266C22×C8C2414yes—
267C22×C8C2414yes—
268#121—28no—
269#121—28no—
270#121—28no—
271#121—44no—
272#121—48no—
273#121—416no—
274#121—416no—
275#121—48no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282#117—44no—
283#117—44no—
284#117—44no—
285#117—44no—
286#117—44no—
287#117—44no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—28no—
360#107—44no—
361#106—28no—
362#106—44no—
363#102—44no—
364#102—44no—
365#102—44no—
366#102—44no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369C22⋊C8—28no—
370C22⋊C8—44no—
371C22⋊C8—44no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24—41no—
396C24—41no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24#12314yes—
405C24C22×D412yes—
406C24C22×D412yes—
407C24C22×D411yescharacteristic
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—416no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—42no—
427C2×D4—88no—
428C2×D4—84no—
429C2×D4—816no—
430C2×D4—816no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—416no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—88no—
474C22×C4—84no—
475C22×C4—816no—
476C22×C4#12314yes—
477C22×C4#12318yes—
478C22×C4AGL(1,ℤ/8)14yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2511yescharacteristic
490M16—416no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4H(5,𝔽2)14yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—44no—
547C42—44no—
548C42—82no—
549C42—82no—
550C42—82no—
551C42—82no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—28no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—82no—
587C23—82no—
588C23—82no—
589C23—82no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23C2×#12314yes—
598C23C2×#12312yes—
599C23C2×AGL(1,ℤ/8)12yes—
600C23#57512yes—
601C23C23×D411yescharacteristic
602D4—48no—
603D4—48no—
604D4—44no—
605D4—44no—
606D4—84no—
607D4—816no—
608D4—816no—
609C2×C4—28no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—84no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—816no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4C2×#12312yes—
663C2×C4C2×#12314yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C22×#4112yes—
667C2×C4C2×H(5,𝔽2)12yes—
668C2×C4#58512yes—
669C8—44no—
670C8—44no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—416no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—88no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—168no—
701C22#309414yes—
702C22C22×#12311yescharacteristic
703C22#348411yescharacteristic
704C22#348511yescharacteristic
705C22C22×AGL(1,ℤ/8)11yescharacteristic
706C22C2×#57511yescharacteristic
707C22#358511yescharacteristic
708C22#358611yescharacteristic
709C4—22no—
710C4—24no—
711C4—22no—
712C4—22no—
713C4—42no—
714C4—41no—
715C4—41no—
716C4—44no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4#350514yes—
724C2—24no—
725C2—44no—
726C2—82no—
727C2—162no—
728C2#3535111yescharacteristic
729C2C2×#309411yescharacteristic
730C2#6052011yescharacteristic
731C2#6157811yescharacteristic
732C2#6158211yescharacteristic
733C2#6171311yescharacteristic
734C2#6171711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2416C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #27613C2semidirect
4. #27610C2semidirect
5. #27602C2semidirect
6. #27585C2semidirect
7. #27582C2semidirect
8. #26229C2semidirect
9. #26225C2semidirect
10. #26157C2semidirect
11. #26153C2semidirect
12. #24009C2semidirect
13. #21767C2semidirect
14. #21742C2semidirect
15. #20844C2semidirect
21. C2×#482C22semidirect
22. C2×#428C22semidirect
32. #2354C22semidirect
33. #2354C22semidirect
34. #2354C22semidirect
35. #2354C22semidirect
48. #2100C22semidirect
49. #2099C22semidirect
50. #2094C22semidirect
51. #2093C22semidirect
52. #2084C22semidirect
53. #2083C22semidirect
54. #2077C22semidirect
55. #2076C22semidirect
56. #2071C22semidirect
57. #2071C22semidirect
58. #2034C22semidirect
59. #2033C22semidirect
60. #1926C22semidirect
61. #1925C22semidirect
62. #1919C22semidirect
63. #1918C22semidirect
64. #1895C22semidirect
65. #1895C22semidirect
66. #1875C22semidirect
67. #1875C22semidirect
68. #1875C22semidirect
69. #1870C22semidirect
70. #1870C22semidirect
71. #1854C22semidirect
74. #1853C22semidirect
77. #1852C22semidirect
78. #1825C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24162
#341862
#276132
#276102
#276022
#275852
#275822
#262292
#262252
#261572
#261532
#240092
#217672
#217422
#208442