Finite Group Databasebeta

← #6502460

Subgroups — #6502460

Order 512.

2036 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic, complemented
2#34206C211yesmaximal, characteristic, complemented
3#28117C212yesmaximal, complemented
4#27614C211yesmaximal, characteristic, complemented
5#27585C211yesmaximal, characteristic, complemented
6#27427C214yesmaximal, complemented
7#26999C214yesmaximal, complemented
8#26888C214yesmaximal, complemented
9#26382C214yesmaximal, complemented
10#24305C212yesmaximal, complemented
11#24061C211yesmaximal, characteristic, complemented
12#23930C211yesmaximal, characteristic
13#22076C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#447C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#520—21no—
19C22×C4×D4C2211yescharacteristic
20C2×#481C2212yes—
21C2×#478C2214yescomplemented
22C2×#447C2212yes—
23C2×C4×D8C2214yes—
24#2454—22no—
25#2454—22no—
26#2454C2214yes—
27#2454C2214yes—
28C2×#399—21no—
29C2×#396—21no—
30#2367—22no—
31#2366C2212yescomplemented
32#2365C2212yescomplemented
33#2354C2212yescomplemented
34#2352—22no—
35D4×C4⋊C4C2212yescomplemented
36#2325—21no—
37#2325—21no—
38#2311—21no—
39C4×#121C2212yes—
40C4×#121C2212yes—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46#2285C2211yescharacteristic
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2212yescomplemented
50#2094C2212yes—
51#2083C2212yescomplemented
52#2071C2212yes—
53#2065C2214yes—
54#2059C2214yescomplemented
55#2058C2212yescomplemented
56#2054C2214yes—
57#2041C2212yescomplemented
58#2025C2214yescomplemented
59#2023C2212yes—
60#2011C2212yes—
61#1989C2214yescomplemented
62#1941C2214yescomplemented
63#1934C2214yescomplemented
64#1926C2214yes—
65#1926C2214yes—
66#1921C2214yescomplemented
67#1920C2214yescomplemented
68#1916C2214yescomplemented
69#1916C2214yescomplemented
70#1879C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1864C2214yescomplemented
76#1864C2214yescomplemented
77#1862C2214yes—
78#1853C2212yes—
79#1852C2212yes—
80#1824C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1786C2212yes—
85C4×#102C2214yes—
86C4×C22⋊C8C2212yes—
87C22×#41—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×D8C2314yes—
91#532—28no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#125—22no—
95C2×#125—44no—
96C2×#125C2314yes—
97C2×#124—21no—
98C2×#121—21no—
99C2×#121C2312yes—
100#520—48no—
101C2×C4×Q8—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—22no—
107C2×C4×D4—44no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2314yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#481—28no—
133#478—48no—
134#447—28no—
135C4×D8—48no—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2314yes—
149#399—24no—
150#399—24no—
151#396—24no—
152#396—24no—
153#391—22no—
154#390—22no—
155#390—21no—
156#390—21no—
157#390—22no—
158#390C2312yes—
159#390C2314yes—
160#390C2312yes—
161#390C2312yes—
162#390C2312yes—
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#385—24no—
182#385—24no—
183#385—22no—
184#385—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—21no—
188#380—21no—
189#380—22no—
190#380—22no—
191#380—24no—
192#380—24no—
193#380C2312yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C2×#41—28no—
222C2×#41—44no—
223C2×#41—44no—
224C22×Q8—21no—
225C22×D4—22no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4C2412yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—24no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—82no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×D8—28no—
252C2×D8—28no—
253C2×D8—44no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—24no—
257#127—24no—
258#125—28no—
259#125—44no—
260#125—48no—
261#124—28no—
262#121—28no—
263#121—24no—
264#121#4118yes—
265C4×Q8—48no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—28no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4—48no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—48no—
277#117—48no—
278#117—44no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349#107—28no—
350#106—28no—
351#106—44no—
352C4⋊C8—44no—
353#102—28no—
354#102—28no—
355#102—28no—
356#102—28no—
357#102—44no—
358#102—44no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C4×C8—44no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385#41—432no—
386C2×Q8—24no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—416no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4C2×#4114yes—
409C2×D42−1+414yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—84no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—816no—
456C22×C4#13514yes—
457C22×C4AGL(1,ℤ/8)14yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C2×#4114yes—
468C22×C4C2511yescharacteristic
469D8—416no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—84no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4C2×#4114yes—
502C4⋊C4H(5,𝔽2)14yes—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C22⋊C4—82no—
515C22⋊C4—84no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—44no—
529C42—44no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—24no—
535C23—24no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23#52512yes—
556C23C2×#13512yes—
557C23C2×AGL(1,ℤ/8)12yes—
558C23C23×D411yescharacteristic
559Q8—48no—
560D4—28no—
561D4—28no—
562D4—48no—
563D4—48no—
564D4—416no—
565D4—44no—
566D4—44no—
567D4—84no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—416no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—816no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—164no—
622C2×C4—164no—
623C2×C4#52512yes—
624C2×C4C2×#13512yes—
625C2×C4C2×#13512yes—
626C2×C4C2×AGL(1,ℤ/8)14yes—
627C2×C4C22×#4112yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+412yes—
630C8—44no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—816no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22#310514yes—
654C22C2×#52511yescharacteristic
655C22#348411yescharacteristic
656C22#348611yescharacteristic
657C22C22×#13511yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22#358511yescharacteristic
660C22#358511yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—24no—
665C4—44no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—41no—
671C4—41no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#352014yes—
677C2—24no—
678C2—84no—
679C2—84no—
680C2#3535711yescharacteristic
681C2C2×#310511yescharacteristic
682C2#6056811yescharacteristic
683C2#6156611yescharacteristic
684C2#6156911yescharacteristic
685C2#6181911yescharacteristic
686C2#6182511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23930C2
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
1. C2×#2454C2semidirect
2. #34206C2semidirect
3. #28117C2semidirect
4. #27614C2semidirect
5. #27585C2semidirect
6. #27427C2semidirect
7. #26999C2semidirect
8. #26888C2semidirect
9. #26382C2semidirect
10. #24305C2semidirect
11. #24061C2semidirect
13. #22076C2semidirect
14. #21912C2semidirect
15. C4×#447C2semidirect
21. C2×#478C22semidirect
31. #2366C22semidirect
32. #2365C22semidirect
33. #2354C22semidirect
35. D4×C4⋊C4C22semidirect
49. #2100C22semidirect
51. #2083C22semidirect
54. #2059C22semidirect
55. #2058C22semidirect
57. #2041C22semidirect
58. #2025C22semidirect
61. #1989C22semidirect
62. #1941C22semidirect
63. #1934C22semidirect
66. #1921C22semidirect
67. #1920C22semidirect
68. #1916C22semidirect
69. #1916C22semidirect
75. #1864C22semidirect
76. #1864C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
#342062
#281172
#276142
#275852
#274272
#269992
#268882
#263822
#243052
#240612
#239302
#220762
#219122
C4×#4472