Finite Group Databasebeta

← #6570696

Subgroups — #6570696

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36259C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29937C211yesmaximal, characteristic, complemented
4#29914C212yesmaximal, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27862C212yesmaximal, complemented
7#27677C211yesmaximal, characteristic, complemented
8#27611C212yesmaximal, complemented
9#26281C212yesmaximal, complemented
10#26219C214yesmaximal, complemented
11#24991C212yesmaximal, complemented
12#24954C214yesmaximal, complemented
13#23755C211yesmaximal, characteristic, complemented
14#23747C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22500C212yesmaximal, complemented
17#22473C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#436C2212yes—
22#2857C2211yescharacteristic
23#2854C2212yes—
24#2546C2211yescharacteristic
25#2538C2211yescharacteristic
26#2486C2212yes—
27#2429—21no—
28#2425—22no—
29#2416C2212yes—
30#2415—21no—
31#2413—22no—
32#2355—21no—
33#2355C2212yes—
34#2354—21no—
35#2354C2212yes—
36#2352—22no—
37#2352C2214yes—
38D4×C4⋊C4C2211yescharacteristic
39#2339—21no—
40#2314—22no—
41#2311—21no—
42C4×#121C2211yescharacteristic
43C2×#389C2212yes—
44#2285C2211yescharacteristic
45#2285C2212yes—
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48#2274—21no—
49#2101C2214yes—
50#2101C2214yes—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2099C2212yes—
54#2099C2212yes—
55#2099C2212yes—
56#2094C2211yescharacteristic
57#2093C2212yes—
58#2085C2214yes—
59#2084C2211yescharacteristic
60#2084C2212yes—
61#2077C2214yes—
62#2077C2212yes—
63#2073C2214yes—
64#2071C2211yescharacteristic
65#2034C2212yes—
66#1990C2212yes—
67#1988C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1926C2214yes—
72#1921C2214yes—
73#1920C2214yes—
74#1920C2214yes—
75#1895C2214yes—
76#1875C2214yes—
77#1874C2212yes—
78#1870C2214yes—
79#1854C2214yes—
80#1853C2214yes—
81#1852C2212yes—
82#1834C2212yes—
83#1824C2214yes—
84#1823C2214yes—
85#1817C2212yes—
86#1809C2214yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2212yes—
90C4×#102C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41—21no—
96C22×#41D411yescharacteristic
97C22×M16C2314yes—
98#547—28no—
99#547—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#123—22no—
103C2×#123—22no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×#121C2311yescharacteristic
111C4×#41—28no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D412yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—22no—
123C2×#117D412yes—
124C2×#117D411yescharacteristic
125C2×#117C2314yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#436—28no—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147#398—22no—
148#398—22no—
149#397—22no—
150#396—22no—
151#396—22no—
152#394—22no—
153#391—22no—
154#391—21no—
155#391—21no—
156#390—22no—
157#390C2311yescharacteristic
158#390C2312yes—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#389C2311yescharacteristic
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#388—24no—
169#388—24no—
170#388—22no—
171#388—22no—
172#388—21no—
173#387—24no—
174#387—24no—
175#386—24no—
176#386—22no—
177#386—21no—
178#386—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2311yescharacteristic
187#382—21no—
188#382—22no—
189#382C2314yes—
190#381—22no—
191#380—22no—
192#380—22no—
193#380—21no—
194#380—21no—
195#380—22no—
196#380—21no—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200#380C2312yes—
201#380C2311yescharacteristic
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—24no—
212C4×C22⋊C4—21no—
213C2×#95—22no—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2314yes—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—42no—
235C2×#41—44no—
236C2×#41—44no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×D4—22no—
241C22×D4—21no—
242C22×D4C2×D412yes—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263#123—48no—
264#123—48no—
265#123—48no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—28no—
270#121—22no—
271#121—41no—
272#121—48no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×D4—24no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4#4118yes—
282#117—24no—
283#117—28no—
284#117—44no—
285#117—44no—
286#117—44no—
287#117—84no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—28no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#106—24no—
357#106—44no—
358#106—42no—
359C4⋊C8—24no—
360#102—42no—
361#102—44no—
362#102—42no—
363C22⋊C8—28no—
364C22⋊C8—42no—
365C22⋊C8—42no—
366C4×C8—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382C24—21no—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385#41—416no—
386#41—816no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—81no—
408C2×D4#12314yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4AGL(1,ℤ/8)14yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4118yes—
457C22×C4C2511yescharacteristic
458M16—88no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4#12314yes—
503C4⋊C42−1+412yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—82no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4#12318yes—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—22no—
525C42—24no—
526C42—42no—
527C42—44no—
528C42—84no—
529C42—84no—
530C42C2×#4114yes—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—22no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C2×#12314yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23C23×D411yescharacteristic
553Q8—44no—
554Q8—84no—
555D4—28no—
556D4—48no—
557D4—42no—
558D4—42no—
559D4—84no—
560D4—88no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4—164no—
614C2×C4C2×#12314yes—
615C2×C4C2×#12312yes—
616C2×C4C2×#12312yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4#57512yes—
620C2×C4#57712yes—
621C2×C4C22×#4112yes—
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58512yes—
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—44no—
627C8—42no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—22no—
633C22—24no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—41no—
641C22—41no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C22×#12311yescharacteristic
650C22#348911yescharacteristic
651C22#350011yescharacteristic
652C22C22×AGL(1,ℤ/8)11yescharacteristic
653C22C2×#57511yescharacteristic
654C22C2×#57711yescharacteristic
655C22#358611yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—41no—
664C4—81no—
665C4—84no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#309414yes—
672C4#351512yes—
673C4#352112yes—
674C2—42no—
675C2—44no—
676C2—161no—
677C2#3606711yescharacteristic
678C2C2×#309411yescharacteristic
679C2#6052211yescharacteristic
680C2#6210911yescharacteristic
681C2#6214511yescharacteristic
682C2#6254111yescharacteristic
683C2#6259111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36259C2
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. #34188C2semidirect
3. #29937C2semidirect
4. #29914C2semidirect
5. #27960C2semidirect
6. #27862C2semidirect
7. #27677C2semidirect
8. #27611C2semidirect
9. #26281C2semidirect
10. #26219C2semidirect
11. #24991C2semidirect
12. #24954C2semidirect
13. #23755C2semidirect
14. #23747C2semidirect
15. #22715C2semidirect
16. #22500C2semidirect
17. #22473C2semidirect

Maximal subgroups

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

subgroupindex
#362592
#341882
#299372
#299142
#279602
#278622
#276772
#276112
#262812
#262192
#249912
#249542
#237552
#237472
#227152
#225002
#224732